(PDF) A novel wavelet based hybrid metho

Ain Shams Engineering Journal xxx (2018) xxx–xxx Contents lists available at ScienceDirect Ain Shams Engineering Journal journal homepage: www.sciencedirect.com A novel wavelet based hybrid method for finding the solutions of higher order boundary value problems Geeta Arora a, Ratesh Kumar a,⇑, Harpreet Kaur b a Department of Mathematics, School of Physical Sciences, Lovely Professional University, Phagwara 144411, Punjab, India b Department of Mathematics, I. K. Gujral Punjab Technical University, Kapurthala, Punjab, India a r t i c l e i n f o a b s t r a c t Article history: In this paper, a new wavelet based hybrid method is developed for obtaining the solution of higher order Received 18 September 2017 linear and nonlinear boundary value problems. The proposed method is based on approximation of solu- Revised 9 December 2017 tion by non-dyadic wavelets family with dilation factor 3. Discretization of domain is done by collocation Accepted 31 December 2017 method. The nonlinearities in boundary value problems are tackled by Quasi-linearization technique. Available online xxxx Eleven numerical experiments are performed on linear and nonlinear boundary value problems with order ranging from eighth to twelfth to prove the successful application of the proposed method. Also, Keywords: the obtained solutions are compared with exact and numerical solutions available in the literature to Non-dyadic wavelets Quasi-linearization prove the efficiency of the method over other methods. Collocation method and boundary value Ó 2018 Ain Shams University. Production and hosting by Elsevier B.V. This is an open access article under problems (65L10) the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 1. Introduction (HPM) [13], Quintic B-Spline Collocation Method (QBSCM) [14], Haar Wavelet Colocation Method (HWCM) [15] with dilation fac- Many physical phenomena like hydro dynamic and hydromag- tor 2, Modified Adomian Decomposition Method (MADM) [16] netic stability [1], induction motor with two rotor circuits [2], vis- etc. coelastic flows in fluid dynamics etc. are governed by the higher Wavelet based numerical techniques are one of the latest order boundary value problems. Higher order boundary value techniques in mathematical theory of approximation which are problems (HOBVPs) have been a major concern for the research- in considerable qualitative progress in comparison with other ers, especially when these are nonlinear or higher order linear methods. Majority of the work has been done by using dyadic ODE with variable coefficients. Existence and uniqueness of solu- wavelets. Till date no literature is available for the use of tion for HOBVPs has already been established by Agarwal in his non-dyadic wavelets in finding the solution of higher order book [3]. But general closed form solution for these kinds of prob- boundary value problems. The existence of non-dyadic wavelets lems has yet not been established. Therefore, researchers are have been proved by Chui and Lian [17] in 1995 in the study of using numerical techniques to find the solutions of HOBVPs. construction of wavelets. This motivates and inspires us to use Many numerical mechanisms have been developed by the non-dyadic wavelet with collocation method for the solution of researchers to solve these problems such as Variational Iteration HOBVPs. In the present study, a new wavelet based hybrid Decomposition Method (VIDM) [4], Optimal Homotopy Asymp- method is developed by using non-dyadic wavelet with colloca- totic Method (OHAM) [5], Galerkin Method with Quintic B- tion method. splines (GMQBS) [6], Legendre Galerkin Method (LGM) [7], Repro- The main objective of our work is to establish a non-dyadic ducing Kernel Space Method (RKSM) [8], Variational Iteration Haar wavelet based collocation technique for numerical solution Method (VIM) [9], Modified Variational Iteration Method (MVID) of linear and nonlinear HOBVPs emerging in many physical phe- [10], Sextic B-splines Collocation Method (SBSCM) [11], Petrov- nomena. To test the efficiency and accuracy of the method, we con- Galerkin Method (PGM) [12], Homotopy Perturbation Method sider the general HOBVPs of the type 0 00 xn ðtÞ ¼ f t; x; x ; x xn1 atb ð1:1Þ Peer review under responsibility of Ain Shams University. ⇑ Corresponding author. with the following types of constraints on the solution at the E-mail address:

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(R. Kumar). boundary points https://doi.org/10.1016/j.asej.2017.12.006 2090-4479/Ó 2018 Ain Shams University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: Arora G et al. A novel wavelet based hybrid method for finding the solutions of higher order boundary value problems. Ain Shams Eng J (2018), https://doi.org/10.1016/j.asej.2017.12.006 2 G. Arora et al. / Ain Shams Engineering Journal xxx (2018) xxx–xxx 9 xðaÞ ¼ m1 ; xðbÞ ¼ r1 ; x ðaÞ ¼ m2 ; x ðbÞ ¼ r2 ; ; xð21Þ ðaÞ ¼ mn2 ; xð21Þ ðbÞ ¼ r2n ; if n is even integer > 0 0 n n > > = 0 0 xðaÞ ¼ c1 ; xðbÞ ¼ d1 ; x ðaÞ ¼ c2 ; x ðbÞ ¼ d2 ; ; x 2 ð n11 Þ ðn1 Þ ðaÞ ¼ cn1 ; x 2 ðbÞ ¼ dnþ1 ; if n is odd integer ð1:2Þ 2 > > > 2 m1 ; m mn ; r1 ; r2 ; ; rn ; c ; c ; ; cnþ1 ; d1 ; d2 ; ; dn1 2 2 2 1 2 are the real constants: ; 2 2 The rest of the manuscript is prearranged into the sections as aÞ /ðt Þ 2 V 0 ) / 3 j t 2 V j; follows. In Section 2, non-dyadic Haar wavelet and its integrals are briefly described. In this section, wavelet family has also been bÞ /ðtÞ 2 V 0 ) u 3 j t k 2 V j generated with help of multiresolution analysis. Approximation of cÞ wi ðt Þ 2 W i0 ; i ¼ 1; 2 ) wi 3 j t 2 W ij solution by non-dyadic Haar wavelets is briefly described in the Section 3. Convergence of the method is given in the Section 4. dÞ wi ðt Þ 2 W i0 ; i ¼ 1; 2 ) wi 3 j t k 2 W ij To validate the proposed method, eleven non-linear and linear eÞ W J ¼ W 1j W 2j ¼ W ij ; i ¼ 1; 2 higher order boundary value problems are considered in Section 5. In this section performance of the proposed method is compared f Þ V0 V1 V2 V3 V4 with other methods to demonstrate the efficiency and accuracy gÞ ? W 0 ? W 1 ? W 2 ? W 3 ? W 4 ? P Pj1 2 of the method. Error analysis and convergence of the proposed hÞ V j ¼ V 0 þ j1 1 i¼0 W j þ i¼0 W j method have also been discussed for each of the examples at dif- iÞ /ðt Þ 2 V 0 ) /ðt kÞ 2 V 0 ; k 2 Z is a Riesz Basis in V 0 ferent level of resolution in Section 5. In the last section, conclu- sions are drawn from the results of numerical experiments and Now by applying MRA, generalized form of non-dyadic Haar idea for future research is given. wavelet family is obtained as follows: 1 0t<1 hi ðt Þ ¼ uðt Þ ¼ fori ¼ 1 2. Haar wavelet and its integrals 0 elsewhere The explicit mathematical expressions of Haar scaling function hi ðt Þ ¼ w1 3 j t k and mother wavelets for non-dyadic Haar wavelet family with 8 > 1 a1 ðiÞ t < a2 ðiÞ dilation factor three [17,18] are given below > > 1 < 2 a2 ðiÞ t < a3 ðiÞ ð2:6Þ 1 0t<1 ¼ pﬃﬃﬃ ; for i ¼ 2; 4; 3p 1 Haar scaling function uðtÞ ¼ ð2:3Þ 2> > > 1 a3 ðiÞ t < a4 ðiÞ 0 elsewhere : 0 elsewhere 8 > 1 0 t < 13 hi ðt Þ ¼ w2 3 j t k > > > < 8 1 2 13 t < 23 1 a1 ðiÞ t < a2 ðiÞ Haar symmetric wavelet function w1 ðt Þ ¼ pﬃﬃﬃ rﬃﬃﬃ>> > 2> > 1 23 t < 1 > 3< 0 a2 ðiÞ t < a3 ðiÞ ð2:7Þ > : ¼ ; for i ¼ 3; 6; 3p 0 elsewhere 2> > 1 a3 ðiÞ t < a4 ðiÞ > : ð2:4Þ 0 elsewhere 8 where a1 ðiÞ ¼ pk, a2 ðiÞ ¼ 3kþ1, a3 ðiÞ ¼ ð3kþ2Þ ,a4 ðiÞ ¼ kþ1, p ¼ 3 j, > 1 0t<3 1 3p 3p p > rﬃﬃﬃ> > > j ¼ 0; 1; 2; ,k ¼ 0; 1; 2; ; p 1. 3< 0 3 t < 3 1 2 Here i > 1 represents wavelet number calculated from the rela- Haar antisymmetric wavelet function w2 ðt Þ ¼ 2> > 1 23 t < 1 tions i 1 ¼ p þ 2k (for even i) and i 2 ¼ p þ 2k (for odd i), j rep- > > > : resents the level of dilation/resolution of the wavelet (as we 0 elsewhere increase the value of j support of wavelet decreases) and k repre- ð2:5Þ sents the translation parameters of the wavelet. The function The main difference between the dyadic and non-dyadic Haar h1 ðtÞ is called father wavelet, h2 ðt Þ and h3 ðtÞ are mother wavelets wavelet family is that, in the construction of dyadic wavelet family, and all other functions h4 ðtÞ; h5 ðtÞ; h6 ðt Þ; are generated from we get only one mother wavelet to generate whole wavelet family, translation and dilation of the mother wavelets are called daughter but in the case of non-dyadic wavelet family, we get more than one wavelets. mother wavelets to generate whole wavelet family, which Using the explicit mathematical expression of non-dyadic Haar increases the rate of convergence of the solution. In case of dilation wavelet family [Eqs. (2.3)–(2.7)], we can integrate over the interval factor 3, two wavelets represented by Eqs. (2.4) and (2.5) are [0,1) as many time as required by using formula given below Z t Z t Z t Z t obtained to generate the whole wavelet family. The construction m of non-dyadic Haar wavelet family is done by using the properties qm;i ðtÞ ¼ m times hi ðxÞðdxÞ 0 0 0 0 of Multi-resolution analysis which are described below. Z t 1 ¼ ðt xÞm1 hi ðxÞdx ð2:8Þ ðm 1Þ! 0 2.1. Multi-resolution analysis(MRA) m ¼ 1; 2; 3 ; i ¼ 1; 2; 3; 3p A multi-resolution analysis(MRA) of L2 ðRÞis defined as a sequence of closed subspace W j ; V j L2 ðRÞ; j 2 Z with the follow- After evaluating the above integrals, we get ing properties (where L2 ðRÞis vector space of square integral tb qb;i ðt Þ ¼ for i¼1 ð2:9Þ functions) b! Please cite this article in press as: Arora G et al. A novel wavelet based hybrid method for finding the solutions of higher order boundary value problems. Ain Shams Eng J (2018), https://doi.org/10.1016/j.asej.2017.12.006 G. Arora et al. / Ain Shams Engineering Journal xxx (2018) xxx–xxx 3 qb;i ðt Þ0 s for i ¼ 2; 4; 6; 8; ; 3p 1 are given below qb;i ðt Þ ¼ First Integral of first nine members of haar wavelet family P1 to P9 8 9 1 > > 0for 0 t a1 ðiÞ > > > > > > 0.5 > > > > > > 1 ðt a1 ðiÞÞb for a1 ðiÞ t a2 ðiÞ > > > > b! > > 0 > > 1 < 1 ½ðt a1 ðiÞÞb þ 3ðt a2 ðiÞÞb for a2 ðiÞ t a3 ðiÞ = 0.5 0 0.1 0.2 0.3 0.4 0.5 0.5 0.6 0.7 0.8 0.9 1 pﬃﬃﬃ b! P1 2> > > > 0 > > > > -0.5 0 > > 1 ½ðt a1 ðiÞÞb þ 3ðt a2 ðiÞÞb 3ðt a3 ðiÞÞb for a3 ðiÞ t a4 ðiÞ > > > > b! > > 0 0.5 1 0 0.5 1 > > > > > :1 b b b b > ; P2 P3 ½ðt a1 ðiÞÞ þ 3ðt a2 ðiÞÞ 3ðt a3 ðiÞÞ þ ðt a4 ðiÞÞ for a4 ðiÞ t 1 b! 0.1 0.2 ð2:10Þ 0 0 -0.1 -0.2 0 0.5 1 0 0.5 1 qb;i ðt Þ0 s for i ¼ 3; 5; 7; 9; ; 3p are given below P4 P5 8 9 0.1 0.2 > 0for 0 t a1 ðiÞ > > > > > 0 0 > > > > > > 1 ðt b a1 ðiÞÞ for a1 ðiÞ t a2 ðiÞ > > -0.1 -0.2 > > 0 0.5 1 0 0.5 1 rﬃﬃﬃ> > > > b! > > 3< 1 ½ðt a1 ðiÞÞ ðt a2 ðiÞÞb for a2 ðiÞ t a3 ðiÞ b = P6 P7 qb;i ðt Þ ¼ 2> > b! > > 0.1 0.2 > > > > 0 0 > > 1 ½ðt a1 ðiÞÞb ðt a2 ðiÞÞb ðt a3 ðiÞÞb for a3 ðiÞ t a4 ðiÞ > > > > b! > > -0.1 -0.2 > > > > 0 0.5 1 0 0.5 1 > :1 b b b b > ; b! ½ðt a1 ðiÞÞ ðt a2 ðiÞÞ ðt a3 ðiÞÞ þ ðt a4 ðiÞÞ for a4 ðiÞ t 1 P8 P9 ð2:11Þ Fig. 2. First integral of wavelets. First nine members of the Haar wavelet family and their first integrals are shown in Figs. 1 and 2 respectively. Take N ¼ 1 ) n m ¼ 1 ) m ¼ n 1, by putting m ¼ n 1 we 3. Approximation of function by non-dyadic Haar wavelets get Theorem 3.1. Let xðt Þ be any square integrable function over the X 3p xn1 ðt Þ ¼ ai q1;i ðt Þ þ xn1 ðAÞ interval ½A; BÞ whose highest order derivative is expressible as a linear i¼1 P combination of Haar wavelet family as xn ðtÞ ¼ 3p a h ðtÞ. Then all i¼0 i i derivatives of x(t) of order less than n are given by which is same as calculated above. Hence the result is true for N = 1 Now assume that the result is true for N ¼ n m ¼ k. X X mþm Put N ¼ k ) n m ¼ k ) m ¼ n k in ð t AÞ m 3p nm1 xm ð t Þ ¼ ai qnm;i ðt Þ þ x ðAÞ for i¼1 m¼0 m! X 3p X k1 ðt AÞm ðnkÞþm xnk ðt Þ ¼ ai qk;i ðt Þ þ x ðAÞ m ¼ 0; 1; 2; 3 n 2; n 1 i¼1 m¼0 m! To prove the result for N ¼ k þ 1, integrating the above equation P3p Proof. We have xn ðtÞ ¼ a h ðtÞ. i¼0 i i within the limits A to t, we get Integrating xn ðt Þone-time w.r.t within the limits A to t we get X X ðnkÞþm ðt AÞmþ1 3p k1 xnk1 ðtÞ ¼ ai qkþ1;i ðt Þ þ x ðAÞ þ xnk1 ðAÞ X 3p ðm þ 1Þ! m¼0 x n1 ðt Þ ¼ ai q1;i ðtÞ þ x n1 ð AÞ i¼1 i¼1 X 3p The theorem is proved by using the principal of mathematical xnk1 ðtÞ ¼ ai qkþ1;i ðtÞ þ xnk1 ðAÞ induction onNð¼ n mÞ i¼1 " ðnkÞ ðnkÞþ1 ðt AÞ1 ð t AÞ 2 þ x ðAÞ þ x ðAÞ 1! 2! First nine members of haar wavelet family h1 to h9 3 1.5 ðnkÞþðk1Þ ðt AÞk 1 þ þ x ðAÞ5 0.5 k! 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 2 0 h1 0 X 3p ðt AÞ0 nðkþ1Þ -2 -2 0 0.5 1 0 0.5 1 xnk1 ðtÞ ¼ ai qkþ1;i ðtÞ þ x ðAÞ 0! h2 h3 i¼1 " ðnðkþ1ÞÞþ1 ðnðkþ1Þþ2 2 2 0 0 ðt AÞ1 ðt AÞ2 -2 -2 þ x ðAÞ þ x ð AÞ 0 0.5 1 0 0.5 1 1! 2! h4 h5 3 ðnðkþ1Þþk 2 2 ðt AÞk 0 0 þ þ x ðAÞ5 -2 -2 k! 0 0.5 1 0 0.5 1 h6 h7 2 2 X X ðkþ1Þ1 ðnðkþ1ÞÞþm ð t AÞ m 3p 0 0 -2 -2 xnðkþ1Þ ðt Þ ¼ ai qkþ1;i ðtÞ þ x ð AÞ 0 0.5 1 0 0.5 1 i¼1 m¼0 m! h8 h9 Hence the result is true for N ¼ n m ¼ K þ 1. Therefore, result Fig. 1. Haar wavelets. is true for all the derivatives of x(t) which completes the proof. Please cite this article in press as: Arora G et al. A novel wavelet based hybrid method for finding the solutions of higher order boundary value problems. Ain Shams Eng J (2018), https://doi.org/10.1016/j.asej.2017.12.006 4 G. Arora et al. / Ain Shams Engineering Journal xxx (2018) xxx–xxx 0 00 0 0 Since the members of family of non-dyadic Haar wavelet are xð0Þ ¼ 1; x ð0Þ ¼ 0; x ð0Þ ¼ 1; x 00 ð0Þ ¼ 2; xð1Þ ¼ 0; x ð1Þ orthogonal to each other, thus by using theorem 3.1 and the prop- 00 0 ¼ e; x ð1Þ ¼ 2e; x 00 ð1Þ ¼ 3e erties of wavelets, any square integrable function xðtÞ over the interval [0,1) can be expressed as Analytic solution of the problem is xðtÞ ¼ ð1 tÞet . X1 The present method is applied on the above problem to test the xð t Þ ¼ a h ðtÞ i¼0 i i X efficiency of the method and following solution is obtained: X ¼ a1 hi ðt Þ þ ai w1 3 j t k þ ai w2 3 j t k ð3:11Þ ev eni oddi X 3p 0 xðt Þ ¼ ai ½q8;i ðt Þ f 1 ðt Þq8;i ð1Þ f 2 ðt Þq7;i ð1Þ Here ai s are Haar wavelet coefficients whose values can be cal- R1 i¼1 culated as ai ¼ 0 xðtÞhi ðtÞdt,i ¼ 1; 2; 3; 3p. In practice only finite 1 number of terms are considered, hence considering the first 3p f 3 ðt Þq6;i ð1Þf 4 ðtÞq5;i ð1Þ f 1 ðtÞ ðe 2Þf 2 ðtÞ 6 terms, where p ¼ 3 j ,j ¼ 0; 1; 2; to approximate the function t3 t2 xðtÞ we get ð2e 3Þf 3 ðt Þð3e 2Þf 4 ðtÞ þ 1 2 2 X3p xðtÞ x3p ¼ a h ðtÞ i¼0 i i ð3:12Þ where 4. Convergence analysis f 1 ðtÞ ¼ 20t7 þ 70t 6 84t5 þ 35t4 , f 2 ðtÞ ¼ 10t7 34t6 þ 39t 5 15t 4 , It has been proved by the Mittal and Pandit [18] that if xðtÞ is f 3 ðtÞ ¼ 2t 7 þ 6:5t 6 7t5 þ 2:5t4 , 0 any differentiable function such that x ðtÞ M 8t 2 ð0; 1Þ for some f 4 ðtÞ ¼ 16 t7 12 t6 þ 12 t 5 16 t 4 , positive real constant M and x(t) is approximated by Haar wavelet family as given below: ai 0 s are the wavelets coefficients and qj;i 0 s are the wavelets X3p integrals. x3p ðtÞ ¼ a h ðtÞ i¼0 i i Above solution is also presented in form of tables and figures for Then the error bound calculated for Haar wavelet approxima- different collocation points at different level of resolution for better tion of function xðtÞ by L2 -norm is given by visibility of results. It is shown in Fig. 3 and Table 1 that the results obtained from present method are in good agreement with the M 1 1 exact solution for different value of j (level of resolution). From jjxðt Þ x3p ðtÞjj pﬃﬃﬃﬃﬃﬃ j ¼ o ð4:13Þ 24 3 p Fig. 4 and Table 1a, it can be concluded that the errors are reducing by increasing the level of resolution which ensures the conver- which means if we know the exact value of M then we can get the gence of the method. Moreover,L2 and L1 errors at j = 1 are exact error bound for the approximation. Also with the increase in 5.60E10 and 7.90E10 respectively which is less than error the level of resolution (the value of j or p=3 j ) error decreased which obtained in case of dyadic Haar wavelet shown in Table 1b. proves the convergence for approximate solutions to the exact solu- Table 1b shows the comparison of accuracy of results obtained tion. This concept of convergence is also presented by the numerical by present method, Variational Iteration Decomposition Method experiments performed below. (VIDM) [4] and optimal Homotopy asymptotic method (OHAM) [5]. It can be concluded from the obtained results, that the present 5. Numerical experiments and error analysis method is providing better results as compared to other methods, which verifies the efficiency and reliability of the method. To describe the applicability and effectiveness of the proposed mechanism, eleven linear and nonlinear higher order boundary value problems have been solved. To check the efficiency of the Numerical Experiment No.1 1 proposed method L2 , L1 and absolute errors are calculated which are defined as 0.9 Absolute error ¼ juexact ðt l Þ unum ðtl Þj ð5:14Þ 0.8 L1 ¼ max juexact ðt l Þ unum ðtl Þj ð5:15Þ 0.7 l sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 0.6 P3p juexact ðt l Þ unum ðt l Þj2 x (t) 0.5 l¼1 L2 ¼ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð5:16Þ 0.4 P3p 2 juexact ðt l Þj l¼1 0.3 where t l represents the collocation points of the domain. 0.2 Numerical Experiment No. 1: Consider the eighth order linear 0.1 Exact Solution differential equation Numerical Solution 8 0 d xðtÞ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 xðtÞ ¼ 8et ; 0t1 ð5:17Þ t dt8 with the following types of constraints on the solution at the Fig. 3. Exact and numerical solution of numerical experiment no.1 at level of resolution j = 2. boundary points Please cite this article in press as: Arora G et al. A novel wavelet based hybrid method for finding the solutions of higher order boundary value problems. Ain Shams Eng J (2018), https://doi.org/10.1016/j.asej.2017.12.006 G. Arora et al. / Ain Shams Engineering Journal xxx (2018) xxx–xxx 5 Table 1 Numerical Experiment No. 2: Consider the eighth order linear Exact and approximated solution by NDHWCM for j = 1 for numerical experiment differential equation given as: no 1. 8 7 6 5 4 3 2 xðtÞ Exact solution Solution by NDHWCM d xðtÞ d xðtÞ d xðtÞ d xðtÞ d xðtÞ d xðtÞ d xðt Þ þ þ þ þ þ þ 0.0556 0.998398425606890 0.998398425605433 dt 8 dt7 dt6 dt 5 dt 4 dt 3 dt2 0.1667 0.984467010721372 0.984467010648896 dxðtÞ 0.2778 0.953472569424642 0.953472569105286 þ þ xðt Þ dt 0.3889 0.901597042799825 0.901597042162876 0.5000 0.824360635350064 0.824360634559944 ¼ 14 cos t 16 sin t 4tsint; 0 t 1 ð5:18Þ 0.6111 0.716519011851887 0.716519011198242 with the following types of constraints on the solution at the 0.7222 0.571945470614687 0.571945470278510 0.8333 0.383495981815471 0.383495981737315 boundary points 0.9444 0.142854690821557 0.142854690819942 0 00 0 0 xð0Þ ¼ 0; x ð0Þ ¼ 1; x ð0Þ ¼ 0; x 00 ð0Þ ¼ 7; xð1Þ ¼ 0; x ð1Þ 00 0 ¼ 2sin1; x ð1Þ ¼ 4 cos 1 þ 2 sin 1; x 00 ð1Þ ¼ 6 cos 1 6 sin 1 -9 Numerical Experiment No.1 x 10 9 Analytic solution of the problem is xðtÞ ¼ t2 1 sin t. j=2 We proposed the following approximate solution of this prob- 8 j=3 j=4 lem based on the non -dyadic Haar wavelet mechanism which sat- (t)| 7 isfies the given boundary conditions numerical X 3p 6 xðt Þ ¼ ai ½q8;i ðt Þ f 1 ðt Þq8;i ð1Þ f 2 ðt Þq7;i ð1Þ (t) - x i¼1 5 1 5 exact f 3 ðtÞq6;i ð1Þf 4 ðt Þq5;i ð1Þ f 1 ðtÞ þ 2 sin 1 f ðt Þ 6 2 2 Absolute error =|x 4 þ ð4 cos 1 þ 2 sin 1 7Þf 3 ðt Þþð6 cos 1 6 sin 1 7Þf 4 ðt Þ 3 7t 3 þ t 2 6 where 1 0 f 1 ðt Þ ¼ 20t7 þ 70t6 84t5 þ 35t4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 f 2 ðt Þ ¼ 10t7 34t 6 þ 39t 5 15t 4 t f 3 ðt Þ ¼ 2t 7 þ 6:5t6 7t5 þ 2:5t4 Fig. 4. Absolute error at different level of resolution for numerical experiment no. 1. f 4 ðt Þ ¼ 16 t7 12 t6 þ 12 t5 16 t 4 Table 1a L2 and L1 errors at different level of resolution for numerical experiment no. 1. Level of resolution (j) L2 -error L2 -error L1 -error L1 -error Dyadic Haar wavelet [15] Non-Dyadic Haar wavelet Dyadic Haar wavelet [15] Non-Dyadic Haar wavelet 0 1.33E08 5.40E09 1.36E08 7.16E09 1 2.79E09 5.60E10 3.06E09 7.90E10 2 7.08E10 6.23E11 9.45E10 8.79E11 3 1.77E10 6.92E12 2.47E10 9.77E12 4 4.43E11 7.69E13 6.25E11 1.09E12 5 1.11E11 8.54E14 1.56E11 1.21E13 6 2.77E12 9.52E15 3.91E12 1.38E14 Table 1b Comparison of numerical results at random collocation points available in literature for Exp. No. 1. x(t) Exact solution Approximated solution NDHWCM(E*) VIDM(E*) [4] OHAM(E*) [5] 0.1 0.994653826268083 0.994653826215959 5.21E11 6.71E08 2.55E09 0.2 0.977122206528136 0.977122206231891 2.96E10 1.27E07 2.84E09 0.3 0.944901165303202 0.944901165118273 1.85E10 1.75E07 3.12E09 0.4 0.895094818584762 0.895094819504096 9.19E10 2.06E07 3.40E09 0.5 0.824360635350064 0.824360637948572 2.60E09 2.18E07 3.67E09 0.6 0.728847520156204 0.728847523706210 3.55E09 2.08E07 3.94E09 0.7 0.604125812241143 0.604125815143196 2.90E09 1.78E07 4.20E09 0.8 0.445108185698494 0.445108186965076 1.27E09 1.29E07 4.45E09 0.9 0.245960311115695 0.245960311272646 1.57E10 6.66E08 4.70E09 E* (Absolute Error) = Exact Solution-Approximate Solution. Please cite this article in press as: Arora G et al. A novel wavelet based hybrid method for finding the solutions of higher order boundary value problems. Ain Shams Eng J (2018), https://doi.org/10.1016/j.asej.2017.12.006 6 G. Arora et al. / Ain Shams Engineering Journal xxx (2018) xxx–xxx ai 0 s are the wavelets coefficients and qj;i 0 s are the wavelets x 10 -10 Numerical Experiment No.2 7 integrals. j=2 Obtained solution are compared with exact solution at different j=3 level of resolutions in Table 2 and Fig. 5. Note that obtained solu- 6 j=4 (t)| tion and exact solution are roughly coincided, which explains the numerical high accuracy obtained by proposed method for small number of 5 grid points. L2 and L1 errors at j = 1 are 1.16E08, 5.53E09 (t) - x respectively which are less than the error obtained in case of dya- 4 dic Haar wavelet shown in Table 2a. It can be concluded from exact Table 2a and Fig. 6 that with the increase in the level of resolution Absolute error =|x j, the errors between exact solution and obtained solution 3 decreases which ensures the convergence of proposed solution to exact solution. In Table 2b performance of proposed method is 2 compared with the Galerkin Method with Septic B-splines [6] and Legendre Galerkin method [7]. We infer that our method is 1 working better than the methods [6,7] given in Table 2b. 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Table 2 t Exact and approximated solution by NDHWCM for j = 1 for numerical experiment no 2. Fig. 6. Absolute error at different level of resolution for numerical experiment no. 2. xðtÞ Exact solution Approximated solution by NDHWCM Numerical Experiment No. 3: Consider the eighth order non- 0.0556 0.055355602430645 0.055355602419651 linear differential equation 0.1667 0.161287906785265 0.161287906248417 0.2778 0.253060393438294 0.253060391112554 8 d xðtÞ 0.3889 0.321818328942119 0.321818324390107 ¼ ðxðtÞÞ2 et ; 0 t 1 ð5:19Þ 0.5000 0.359569153953152 0.359569148418721 dt 8 0.6111 0.359496601052921 0.359496596569607 with the following types of constraints on the solution at the 0.7222 0.316244836086907 0.316244833830374 0.8333 0.226165149587678 0.226165149074205 boundary points 0.9444 0.087518515001869 0.087518514991516 0 00 0 0 00 xð0Þ ¼ 1; x ð0Þ ¼ 1; x ð0Þ ¼ 1; x 00 ð0Þ ¼ 1; xð1Þ ¼ e; x ð1Þ ¼ e; x ð1Þ 0 ¼ e; x 00 ð1Þ ¼ e: Numerical Experiment No.2 Analytic solution of the problem is xðtÞ ¼ et . 0 In this problem, non-linearity in the differential equation is tackled by Quasilinearization technique. By using Quasilineariza- -0.05 tion technique, above non-linear differential equation is trans- formed into a sequence of linear differential equations as -0.1 8 d xrþ1 ðtÞ ¼ 2xr xrþ1 et ; 0 t 1 r ¼ 0; 1; 2 ð5:20Þ -0.15 dt8 subjected to the boundary conditions x (t) -0.2 0 00 000 xrþ1 ð0Þ ¼ 1; xrþ1 ð0Þ ¼ 1; xrþ1 ð0Þ ¼ 1; xrþ1 ð0Þ ¼ 1; xrþ1 ð1Þ ¼ e; 0 00 000 -0.25 xrþ1 ð1Þ ¼ e; xrþ1 ð1Þ ¼ e; xrþ1 ð1Þ ¼ e where xrþ1 ðtÞ is the ðr þ 1Þth approximation for xðtÞ. -0.3 Then by applying the present method on the sequence of linear differential equations, we proposed the following solution: -0.35 Exact Solution X 3p Numerical Solution xðt Þ ¼ ai ½q8;i ðt Þ f 1 ðt Þq8;i ð1Þ f 2 ðt Þq7;i ð1Þ -0.4 i¼1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t 8 5 f 3 ðt Þq6;i ð1Þf 4 ðtÞq5;i ð1Þ þ e f 1 ðt Þ þ e f ðt Þ 3 2 2 Fig. 5. Exact and numerical solution of numerical experiment no. 2 at level of resolution j = 2. t3 t2 þ ðe 2Þf 3 ðtÞþðe 1Þf 4 ðt Þ þ þ þtþ1 3! 2! Table 2a L2 and L1 errors at different level of resolution for numerical experiment no. 2. Level of resolution L2 -error Dyadic Haar wavelet L2 -error Non-Dyadic Haar L1 -error Dyadic Haar wavelet L1 -error Non-Dyadic Haar (j) [15] wavelet [15] wavelet 0 2.00E07 1.10E07 6.89E08 4.93E08 1 5.79E08 1.16E08 2.14E08 5.53E09 2 1.47E08 1.29E09 6.60E09 6.16E10 3 3.67E09 1.43E10 1.73E09 6.85E11 4 9.19E10 1.59E11 4.37E10 7.61E12 5 2.30E10 1.77E12 1.10E10 8.45E13 6 5.74E11 1.97E13 2.74E11 9.41E14 Please cite this article in press as: Arora G et al. A novel wavelet based hybrid method for finding the solutions of higher order boundary value problems. Ain Shams Eng J (2018), https://doi.org/10.1016/j.asej.2017.12.006 G. Arora et al. / Ain Shams Engineering Journal xxx (2018) xxx–xxx 7 Table 2b Comparison of numerical results at random collocation points available in Literature for Exp. No. 2. x(t) Exact Approximated NDHWCM(E*) GMQBS(E*) [6] LGM(E*) [7] 0.1 0.098835082480360 0.098835083008578 5.28E10 3.80E07 5.04E08 0.2 0.190722557563259 0.190722561614420 4.05E09 2.15E06 5.14E07 0.3 0.268923388061819 0.268923396590150 8.53E09 5.63E06 1.56E10 0.4 0.327111407539266 0.327111416381471 8.84E09 9.75E06 2.71E06 0.5 0.359569153953152 0.359569157955952 4.00E09 1.14E05 3.26E06 0.6 0.361371182972823 0.361371181204998 1.77E09 1.01E05 2.82E06 0.7 0.328551020491222 0.328551016592605 3.90E09 7.27E06 1.68E06 0.8 0.258248192723828 0.258248190503417 2.22E09 3.87E06 5.78E07 0.9 0.148832112829222 0.148832112522090 3.07E10 1.43E06 5.88E08 E* (Absolute Error) = Exact Solution-Approximate Solution. Table 3 where Exact and approximated solution by NDHWCM for j = 1 for numerical experiment no 3. f 1 ðt Þ ¼ 20t7 þ 70t6 84t5 þ 35t4 xðtÞ Exact solution Approximated solution by NDHWCM f 2 ðt Þ ¼ 10t7 34t 6 þ 39t 5 15t 4 0.0556 1.057127744760230 1.057127744760390 f 3 ðt Þ ¼ 2t 7 þ 6:5t6 7t5 þ 2:5t4 0.1667 1.181360412865640 1.181360412873280 f 4 ðt Þ ¼ 16 t7 12 t6 þ 12 t5 16 t 4 0.2778 1.320192788434120 1.320192788467740 0.3889 1.475340615490620 1.475340615557600 ai 0 s are the wavelets coefficients and qj;i 0 s are the wavelets 0.5000 1.648721270700120 1.648721270783100 0.6111 1.842477459047700 1.842477459116250 integrals. 0.7222 2.059003694212870 2.059003694248070 Solution obtained by the present method is explained with help 0.8333 2.300975890892820 2.300975890900990 of tables and figures. Table 3 and Fig. 7 shows the comparison 0.9444 2.571384434788030 2.571384434788190 between the exact and obtained solution for the Eq. (5.19) at j = 1 which explains the high accuracy obtained by proposed method for small number of grid points (in this case only 9 grid pts).L2 and L1 errors at j = 1 are 2.54E11, 8.30E11 respectively Numerical Experiment No.3 which is less than error obtained in case of dyadic Haar wavelet. 2.8 Table 3a and Fig. 8 ensures the convergence of present solution to exact solution. The performance of present method is compared 2.6 with the other methods in Table 3b and it can be concluded that 2.4 present method is working better than the other methods [8,9]. Numerical Experiment No. 4: Consider the ninth order linear 2.2 differential equation 9 2 d xð t Þ ¼ xðtÞ 9et ; 0t1 ð5:21Þ x (t) dt 9 1.8 with the following types of constraints on the solution at the 1.6 boundary points xð0Þ ¼ 1; x ð0Þ ¼ 0; x ð0Þ ¼ 1; x 00 ð0Þ ¼ 2; xiv ð0Þ 0 00 0 1.4 0 ¼ 3; xð1Þ ¼ 0; x ð1Þ ¼ e 1.2 Exact Solution Numerical Solution 00 0 1 x ð1Þ ¼ 2e; x 00 ð1Þ ¼ 3e 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t Analytic solution of the problem is xðtÞ ¼ ð1 tÞet . The present method is applied on the linear differential equa- Fig. 7. Exact and numerical solution of numerical experiment no. 3 at level of tion Eq. (5.21) and we proposed the following solution as resolution j = 2. Table 3a L2 and L1 errors at different level of resolution for numerical experiment no. 3. Level of resolution L2 -error Dyadic Haar wavelet L2 -error Non-Dyadic Haar L1 -error Dyadic Haar wavelet L1 -error Non-Dyadic Haar (j) [15] wavelet [15] wavelet 0 6.31E10 2.47E10 1.48E09 7.50E10 1 1.27E10 2.54E11 3.20E10 8.30E11 2 3.22E11 2.83E12 9.92E11 9.23E12 3 8.05E12 3.14E13 2.60E11 1.03E12 4 2.01E12 3.50E14 6.56E12 1.14E13 5 5.03E13 3.96E15 1.64E12 1.33E14 6 1.26E13 5.54E16 4.11E13 3.11E15 Please cite this article in press as: Arora G et al. A novel wavelet based hybrid method for finding the solutions of higher order boundary value problems. Ain Shams Eng J (2018), https://doi.org/10.1016/j.asej.2017.12.006 8 G. Arora et al. / Ain Shams Engineering Journal xxx (2018) xxx–xxx x 10 -11 Numerical Experiment No.3 Numerical Experiment No.4 1 1 j=2 0.9 j=3 0.9 j=4 (t)| 0.8 0.8 numerical 0.7 0.7 (t) - x 0.6 0.6 exact x (t) 0.5 0.5 Absolute error =|x 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 Exact Solution Numerical Solution 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t t Fig. 8. Absolute error at different level of resolution for numerical experiment no. 3. Fig. 9. Exact and numerical solution of numerical experiment no. 4 at level of resolution j = 2. X 3p xð t Þ ¼ ai ½q9;i ðtÞ f 1 ðt Þq9;i ð1Þ f 2 ðt Þq8;i ð1Þ Table 4 i¼1 Exact and approximated solution by NDHWCM for j = 1 for numerical experiment no 1 5 4. f 3 ðtÞq7;i ð1Þf 4 ðt Þq6;i ð1Þ f 1 ðt Þ þ e f ðt Þ 24 2 2 x(t) Exact solution Approximated solution by NDHWCM 4 3 9 t t t2 0.0556 0.998398425606890 0.998398425606881 þ 2e f 3 ðtÞþð5 3eÞf 4 ðt Þ þ 1 2 8 3 2 0.1667 0.984467010721372 0.984467010719958 0.2778 0.953472569424642 0.953472569414279 0.3889 0.901597042799825 0.901597042770934 where 0.5000 0.824360635350064 0.824360635304049 0.6111 0.716519011851887 0.716519011805412 0.7222 0.571945470614687 0.571945470586460 f 1 ðt Þ ¼ 35t8 þ 120t 6 140t 5 þ 56t 4 0.8333 0.383495981815471 0.383495981807899 f 2 ðt Þ ¼ 15t8 50t 7 þ 56t 6 21t5 0.9444 0.142854690821557 0.142854690821380 f 3 ðt Þ ¼ 2:5t8 þ 8t7 8:5t6 þ 3t 5 f 4 ðt Þ ¼ 16 t 8 12 t7 þ 12 t6 16 t5 . Numerical Experiment No. 5: Consider the ninth order non- ai 0 s are the wavelets coefficients and qj;i 0 s are the wavelets linear differential equation integrals. 9 d xðtÞ dxðtÞ Below Fig. 9 and Table 4 shows the numerical solution obtained ðxðtÞÞ2 ¼ cos3 t; 0t1 ð5:22Þ from the present method at j = 1 are in good agrement with the dt 9 dt exact solution which explains the high accuracy obtained by pre- with the following types of constraints on the solution at the sent method for small number of grid points (in this case only 9 boundary points Grid pts). L2 and L1 errors at j = 1 are 3.36E11, 4.65E11 respec- xð0Þ ¼ 0; x ð0Þ ¼ 1; x ð0Þ ¼ 0; x ð0Þ ¼ 1; x v ð0Þ 0 00 000 0 tively which are less than error obtained in case of dyadic Haar 0 00 wavelet. It can be concluded that rate of convergence of non- ¼ 0; xð1Þ ¼ sin 1; x ð1Þ ¼ cos 1; x ð1Þ dyadic wavelet is faster than dyadic wavelets. Table 4a and 000 ¼ sin 1; x ð1Þ ¼ cos 1 Fig. 10 ensure the convergence of proposed solution to exact solu- tion. The performance of present method is compared with the Analytic solution of the problem is ðtÞ ¼ sin t. other methods and it can be concluded that present method is In this problem, non-linearity in the differential equation is working better than the other methods [10,11] given in Table 4b tackled by Quasilinearization technique. By using Quasilineariza- Table 3b Comparison of numerical results at random collocation points available in literature for Exp. No. 3. x(t) Exact Approximated NDHWCM (E*) Reproducing Kernel(E*) [8] VIT(E*) [9] 0.1 1.105170918075640 1.105170918082360 6.72E12 1.61E08 1.91E07 0.2 1.221402758160170 1.221402758200630 4.05E11 3.07E08 1.25E07 0.3 1.349858807576000 1.349858807613920 3.79E11 4.23E08 7.25E08 0.4 1.491824697641270 1.491824697560440 8.08E11 4.97E08 4.85E08 0.5 1.648721270700120 1.648721270427080 2.73E10 5.23E08 2.91E07 0.6 1.822118800390500 1.822118800001710 3.89E10 4.98E08 7.80E08 0.7 2.013752707470470 2.013752707147060 3.23E10 4.24E08 1.11E07 0.8 2.225540928492460 2.225540928350050 1.42E10 3.08E08 1.71E07 0.9 2.459603111156940 2.459603111139210 1.77E11 1.62E08 7.93E08 E* (Absolute Error) = Exact Solution-Approximate Solution. Please cite this article in press as: Arora G et al. A novel wavelet based hybrid method for finding the solutions of higher order boundary value problems. Ain Shams Eng J (2018), https://doi.org/10.1016/j.asej.2017.12.006 G. Arora et al. / Ain Shams Engineering Journal xxx (2018) xxx–xxx 9 Table 4a L2 and L1 errors at different level of resolution for numerical experiment no. 4. Level of resolution L2 -error Dyadic Haar wavelet L2 -error Non-Dyadic Haar L1 -error Dyadic Haar wavelet L1 -error Non-Dyadic Haar (j) [15] wavelet [15] wavelet 0 5.32E10 3.17E10 5.82E10 4.20E10 1 1.73E10 3.36E11 2.29E10 4.65E11 2 4.26E11 3.73E12 6.20E11 5.41E12 3 1.06E11 4.14E13 1.53E11 6.03E13 4 2.65E12 4.60E14 3.86E12 6.71E14 5 6.64E13 5.09E15 9.66E13 7.55E15 6 1.66E13 7.26E17 2.41E13 8.11E17 x 10 -12 Numerical Experiment No.4 X 3p 6 xðt Þ ¼ ai ½q9;i ðtÞ f 1 ðtÞq9;i ð1Þ f 2 ðt Þq8;i ð1Þ j=2 i¼1 j=3 5 5 j=4 f 3 ðt Þq7;i ð1Þf 4 ðtÞq6;i ð1Þ þ sin1 f 1 ðt Þ 6 (t)| numerical 1 t3 þ cos1 f ðtÞ þ ð1 sin1Þf 3 ðt Þþð1 cos 1Þf 4 ðtÞ þ t 4 2 2 6 (t) - x where exact 3 Absolute error =|x f 1 ðt Þ ¼ 35t 8 þ 120t 6 140t5 þ 56t 4 2 f 2 ðt Þ ¼ 15t 8 50t 7 þ 56t 6 21t 5 f 3 ðt Þ ¼ 2:5t 8 þ 8t7 8:5t6 þ 3t5 f 4 ðt Þ ¼ 16 t8 12 t7 þ 12 t6 16 t 5 1 ai 0 s are the wavelets coefficients and qj;i 0 s are the wavelets 0 integrals. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t Table 5 and Fig. 11 shows the comparison between the exact and approximated numerical solution at j = 1 for the given problem Fig. 10. Absolute error at different level of resolution for numerical experiment which explains the high accuracy obtained by present method for no. 4. small number of grid points (in this case only 9 Grid pts). L2 and L1 errors at j = 1 are 2.54E11, 8.43E11 respectively which are less than error obtained in case of dyadic Haar wavelet. Table 5a and tion technique, given non-linear differential equation is trans- Fig. 12 ensure the convergence of proposed solution to exact solu- formed into a sequence of linear differential equations as tion. The performance of present method is compared with the 9 d xrþ1 ðtÞ 0 0 0 2xr xr xrþ1 x2r xrþ1 þ 2x2r xr ¼ cos3 x; 0t Table 5 dt 9 Exact and approximated solution by NDHWCM for j = 1 for numerical experiment 1 r ¼ 0; 1; 2 ð5:23Þ no 5. subjected to the boundary conditions xðtÞ Exact solution Approximated solution by NDHWCM 0 00 000 0v xrþ1 ð0Þ ¼ 0; xrþ1 ð0Þ ¼ 1; xrþ1 ð0Þ ¼ 0; xrþ1 ð0Þ ¼ 1; xrþ1 ð0Þ ¼ 0; 0.0556 0.055526982004734 0.055526982003702 0 00 000 0.1667 0.165896132693415 0.165896132535308 xrþ1 ð1Þ ¼ sin 1; xrþ1 ð1Þ ¼ cos 1; xrþ1 ð1Þ ¼ sin 1; xrþ1 ð1Þ ¼ cos 1, 0.2778 0.274219289210727 0.274219288018338 where xrþ1 ðtÞ is the ðr þ 1Þth approximation for xðtÞ. 0.3889 0.379160503917260 0.379160500502594 Then by applying the present method on the sequence of linear 0.5000 0.479425538604203 0.479425533025153 differential equations, we proposed the following solution for 0.6111 0.573777826311066 0.573777820538742 given Eq. (5.22) as 0.7222 0.661053721884888 0.661053718298351 0.8333 0.740176853196037 0.740176852213119 0.9444 0.810171396017299 0.810171395993874 Table 4b Comparison of numerical results at random collocation points available in literature for Exp. No. 4. x(t) Exact Approximated NDHWCM(E*) MVIM(E*) [10] SBSCM(E*) [11] 0.1 0.994653826268083 0.994653826266204 1.88E12 2.00E10 1.08E06 0.2 0.977122206528136 0.977122206495194 3.29E11 2.00E10 5.19E06 0.3 0.944901165303202 0.944901165177828 1.25E10 2.00E10 6.13E06 0.4 0.895094818584762 0.895094818349846 2.35E10 2.00E10 1.23E05 0.5 0.824360635350064 0.824360635081231 2.69E10 2.00E10 1.07E05 0.6 0.728847520156204 0.728847519963689 1.93E10 6.00E10 4.91E06 0.7 0.604125812241143 0.604125812167427 7.37E11 1.00E09 9.95E06 0.8 0.445108185698494 0.445108185693307 5.19E12 2.00E09 1.65E06 0.9 0.245960311115695 0.245960311117830 2.13E12 3.04E09 2.00E06 E* (Absolute Error) = Exact Solution-Approximate Solution. Please cite this article in press as: Arora G et al. A novel wavelet based hybrid method for finding the solutions of higher order boundary value problems. Ain Shams Eng J (2018), https://doi.org/10.1016/j.asej.2017.12.006 10 G. Arora et al. / Ain Shams Engineering Journal xxx (2018) xxx–xxx Numerical Experiment No.5 x 10 -9 Numerical Experiment No.5 0.9 6 j=2 0.8 j=3 5 j=4 (t)| 0.7 numerical 0.6 4 (t) - x 0.5 exact x (t) 3 0.4 Absolute error =|x 0.3 2 0.2 1 0.1 Exact Solution Numerical Solution 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t t Fig. 11. Exact and numerical solution of numerical experiment no. 5 at level of Fig. 12. Absolute error at different level of resolution for numerical experiment resolution j = 2. no. 5. where other methods [11,12] and it has been found that present method is working better than the other methods [11,12] given in Table 5b. f 1 ðtÞ ¼ 70t9 315t8 þ 540t7 420t 6 þ 126t5 Numerical Experiment No. 6: Consider the tenth order linear f 2 ðtÞ ¼ 35x9 þ 155t8 260t 7 þ 196t 6 56t5 differential equation f 3 ðtÞ ¼ 7:5x9 32:5t8 þ 53t7 38:5t6 þ 10:5t5 10 d xðtÞ 00 f 4 ðtÞ ¼ 56 x9 þ 72 t8 11 2 t 7 þ 23 6 t6 t5 ¼ x ðt Þ 8et ; 0t1 ð5:24Þ f 5 ðtÞ ¼ 24 x 16 t 8 þ 14 t 7 16 t6 þ 24 1 9 1 5 t . dt10 with the following types of constraints on the solution at the ai 0 s are the wavelets coefficients and qj;i 0 s are the wavelets boundary points integrals. xð0Þ ¼ 1; x ð0Þ ¼ 0; x ð0Þ ¼ 1; x ð0Þ ¼ 2; xiv ð0Þ 0 00 000 Fig. 13 and Table 6 shows the comparison between the exact 0 and obtained solution for j = 1 which explains the high accuracy ¼ 3; xð1Þ ¼ 0; x ð1Þ ¼ e; obtained by present method for small number of grid points (in this case only 9 Grid pts). L2 and L1 errors at j = 1 are 1.79E12, x ð1Þ ¼ 2e; x ð1Þ ¼ 3e; xiv ð1Þ ¼ 4e 00 000 2.65E12 respectively which are less than error obtained in case Analytic solution of the problem is xðtÞ ¼ ð1 tÞet . of dyadic Haar wavelet. Table 6a and Fig. 14 ensure the conver- By applying the non-dyadic Haar wavelet mechanism for the gence of obtained solution (NDHWCM solution) to exact solution. solution of linear differential equation, we proposed the following The performance of present is compared with the other methods solution for the Eq. (5.24) [13,14] and it has been found that present method is working bet- ter than the other methods given in Table 6b. X 3p Numerical Experiment No. 7: Consider the tenth order non- xð t Þ ¼ ai ½q10;i ðt Þ f 1 ðt Þq10;i ð1Þ f 2 ðtÞq9;i ð1Þ linear differential equation i¼1 10 1 d xðtÞ f 3 ðtÞq8;i ð1Þf 4 ðt Þq7;i ð1Þf 5 ðt Þq6;i ð1Þ f ðt Þ ¼ ðxðtÞÞ2 et ; 0t1 ð5:25Þ 24 1 dt 10 5 9 with the following types of constraints on the solution at the þ e f ðt Þ þ 2e f 3 ðtÞþð5 3eÞf 4 ðt Þþð3 4eÞf 5 ðt Þ 2 2 2 boundary points xð0Þ ¼ 1; x ð0Þ ¼ 1;x ð0Þ ¼ 1;x ð0Þ ¼ 1;xiv ð0Þ ¼ 1;xv ð0Þ ¼ 1;xð1Þ ¼ e; 0 00 000 t4 t3 t2 þ1 8 3 2 x ð1Þ ¼ e; x ð1Þ ¼ e; x ð1Þ ¼ e; xiv ð1Þ ¼ e; xv ð1Þ ¼ e 0 00 000 Table 5a L2 and L1 errors at different level of resolution for numerical experiment no. 5. Level of resolution L2 -error Dyadic Haar wavelet L2 -error Non-Dyadic Haar L1 -error Dyadic Haar wavelet L1 -error Non-Dyadic Haar (j) [15] wavelet [15] wavelet 0 6.41E10 2.17E10 1.68E09 7.55E10 1 1.47E10 2.54E11 3.40E10 8.43E11 2 3.37E11 2.93E12 9.99E11 9.33E12 3 8.25E12 3.34E13 2.85E11 1.25E12 4 2.01E12 3.85E14 6.56E12 1.44E13 5 5.03E13 4.75E15 1.64E12 1.83E14 6 1.26E13 5.93E16 4.11E13 3.41E15 Please cite this article in press as: Arora G et al. A novel wavelet based hybrid method for finding the solutions of higher order boundary value problems. Ain Shams Eng J (2018), https://doi.org/10.1016/j.asej.2017.12.006 G. Arora et al. / Ain Shams Engineering Journal xxx (2018) xxx–xxx 11 Table 5b Comparison of numerical results at random collocation points available in literature for Exp. No. 5. x(t) Exact Approximated NDHWCM (E*) PGM(E*) [12] SBSCM(E*) [11] 0.1 0.099833416646828 0.099833416630007 1.68E11 1.86E07 2.85E06 0.2 0.198669330795061 0.198669330448275 3.47E10 7.30E07 1.35E06 0.3 0.295520206661340 0.295520205069110 1.59E09 9.83E07 4.09E06 0.4 0.389418342308651 0.389418338575155 3.73E09 1.22E06 1.05E06 0.5 0.479425538604203 0.479425532943724 5.66E09 8.34E07 3.45E05 0.6 0.564642473395035 0.564642467456141 5.94E09 3.87E06 3.46E05 0.7 0.644217687237691 0.644217683060221 4.18E09 5.66E06 2.09E05 0.8 0.717356090899523 0.717356089246222 1.65E09 4.89E06 2.01E05 0.9 0.783326909627483 0.783326909436309 1.91E10 2.86E06 3.84E06 E* (Absolute Error) = Exact Solution-Approximate Solution. Numerical Experiment No.6 -13 Numerical Experiment No.6 x 10 1 3.5 j=2 0.9 j=3 3 j=4 0.8 (t)| numerical 0.7 2.5 (t) - x 0.6 2 exact x (t) 0.5 0.4 Absolute error =|x 1.5 0.3 1 0.2 0.5 0.1 Exact Solution Numerical Solution 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t t Fig. 13. Exact and numerical solution of numerical experiment no. 6 at level of Fig. 14. Absolute error at different level of resolution for numerical experiment resolution j = 2. no. 6. Analytic solution of the problem is xðtÞ ¼ et . Table 6 Non-linearity in the differential equation is tackled by Quasilin- Exact and approximated solution by NDHWCM for j = 1 for numerical experiment no earization technique. By using Quasilinearization technique given 6. non-linear differential equation is transformed into a sequence of x(t) Exact solution Approximated solution by NDHWCM linear differential equations as 0.0556 0.998398425606890 0.998398425606891 0.1667 0.984467010721372 0.984467010721508 10 d xrþ1 ðtÞ 0.2778 0.953472569424642 0.953472569425507 ¼ 2xr xrþ1 et ; 0 t 1 r ¼ 0; 1; 2 ð5:26Þ 0.3889 0.901597042799825 0.901597042801862 dt 10 0.5000 0.824360635350064 0.824360635352717 0.6111 0.716519011851887 0.716519011853969 0.7222 0.571945470614687 0.571945470615589 subjected to the boundary conditions 0 00 000 v ð0Þ ¼ 1; xrþ1 ð0Þ ¼ 1; xrþ1 ð0Þ ¼ 1; xrþ1 ð0Þ ¼ 1; xrþ1 ð0Þ ¼ 1; xirþ1 0.8333 0.383495981815471 0.383495981815614 0.9444 0.142854690821557 0.142854690821560 0 00 v ð0Þ ¼ e xrþ1 ð1Þ ¼ e; xrþ1 ð1Þ ¼ e; xrþ1 ð1Þ ¼ e; xrþ1 ð1Þ ¼ e; xirþ1 000 where xrþ1 ðtÞ is the ðr þ 1Þth approximation for xðtÞ. Table 6a L2 and L1 errors at different level of resolution for numerical experiment no. 6. Level of resolution L2 -error Dyadic Haar wavelet L2 -error Non-Dyadic Haar L1 -error Dyadic Haar wavelet L1 -error Non-Dyadic Haar (j) [15] wavelet [15] wavelet 0 3.66E11 1.51E11 3.76E11 2.02E11 1 8.51E12 1.79E12 9.39E12 2.65E12 2 2.26E12 1.98E13 3.12E12 2.95E13 3 5.65E13 2.21E14 8.26E13 3.28E14 4 1.41E13 3.20E15 2.09E13 8.60E15 5 3.54E14 1.87E15 5.25E14 9.21E15 6 9.01E15 1.91E15 1.32E14 1.39E14 Please cite this article in press as: Arora G et al. A novel wavelet based hybrid method for finding the solutions of higher order boundary value problems. Ain Shams Eng J (2018), https://doi.org/10.1016/j.asej.2017.12.006 12 G. Arora et al. / Ain Shams Engineering Journal xxx (2018) xxx–xxx Table 6b Comparison of numerical results at random collocation points available in literature for Exp. No. 6. x(t) Exact Approximated NDHWCM (E*) HPM(E*) [13] QBSCM(E*) [14] 0.1 0.994653826268083 0.994653826268138 5.46E14 1.14E06 8.82E06 0.2 0.977122206528136 0.977122206528696 5.60E13 2.69E06 8.64E06 0.3 0.944901165303202 0.944901165303723 5.21E13 3.70E06 2.92E06 0.4 0.895094818584762 0.895094818582427 2.33E12 4.35E06 5.96E07 0.5 0.824360635350064 0.824360635342947 7.12E12 4.58E06 6.74E06 0.6 0.728847520156204 0.728847520146812 9.39E12 4.36E06 1.43E05 0.7 0.604125812241143 0.604125812234420 6.72E12 3.71E06 1.27E05 0.8 0.445108185698494 0.445108185696260 2.23E12 2.69E06 8.14E06 0.9 0.245960311115695 0.245960311115542 1.53E13 1.42E06 3.49E06 E* (Absolute Error) = Exact Solution-Approximate Solution. Table 7 X 3p Exact and approximated solution by NDHWCM for j = 1 for numerical experiment xðt Þ ¼ ai ½q10;i ðt Þ f 1 ðtÞq10;i ð1Þ f 2 ðt Þq9;i ð1Þ no 7. i¼1 65 x(t) Exact solution Approximated solution by NDHWCM f 3 ðt Þq8;i ð1Þf 4 ðtÞq7;i ð1Þf 5 ðt Þq6;i ð1Þ þ e f ðt Þ 24 1 0.0556 1.057127744760230 1.057127744760230 0.1667 1.181360412865640 1.181360412865630 8 5 þ e f ðt Þ þ e f ðtÞþðe 2Þf 4 ðt Þþðe 1Þf 5 ðt Þ 0.2778 1.320192788434120 1.320192788434040 3 2 2 3 0.3889 1.475340615490620 1.475340615490440 0.5000 1.648721270700120 1.648721270699890 t4 t3 t2 0.6111 1.842477459047700 1.842477459047520 þ þ þ þtþ1 4! 3! 2! 0.7222 2.059003694212870 2.059003694212790 0.8333 2.300975890892820 2.300975890892810 where 0.9444 2.57138443478803 2.57138443478803 f 1 ðtÞ ¼ 70t9 315t8 þ 540t7 420t 6 þ 126t5 f 2 ðtÞ ¼ 35x9 þ 155t8 260t 7 þ 196t 6 56t5 By applying the present method on the given linear differential f 3 ðtÞ ¼ 7:5x9 32:5t8 þ 53t7 38:5t6 þ 10:5t5 equation, we proposed the following solution f 4 ðtÞ ¼ 56 x9 þ 72 t8 11 t 7 þ 23 t6 t5 2 6 f 5 ðtÞ ¼ 24 x 16 t 8 þ 14 t 7 16 t6 þ 24 1 9 1 5 t . Numerical Experiment No.7 2.8 ai 0 s are the wavelets coefficients and qj;i 0 s are the wavelet integrals. 2.6 Proposed solution is compared with exact solution at different level of resolutions in Table 7 and Fig. 15. Note that obtained solu- 2.4 tion and exact solution are roughly coincided which explains the 2.2 high accuracy obtained by proposed method for small number of grid points. L2 and L1 errors at j = 1 are 3.22E13, 8.19E13 2 respectively which are less than error obtained in case of dyadic x (t) Haar wavelet shown in Table 7a. It can be concluded from 1.8 Table 7a and Fig. 16 that with the increase in the level of resolution j, the errors between exact solution and obtained solution 1.6 decreases which ensures the convergence of proposed solution to 1.4 exact solution. In Table 7b performance of proposed method is compared with the Homotopy Perturbation Method [13] and Quin- 1.2 Exact Solution tic B-Spline Collocation Method [14]. We infer that our method is Numerical Solution working better than the methods [13,14] given in Table 7b. 1 Numerical Experiment No. 8: Consider the eleventh order linear 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t differential equation ð11Þ Fig. 15. Exact and numerical solution of numerical experiment no. 7 at level of d xðt Þ ¼ 11et þ xðtÞ; 0t1 ð5:27Þ resolution j = 2. dt Table 7a L2 and L1 errors at different level of resolution for numerical experiment no. 7. Level of resolution L2 -error Dyadic Haar wavelet L2 -error Non-Dyadic Haar L1 -error Dyadic Haar wavelet L1 -error Non-Dyadic Haar (j) [15] wavelet [15] wavelet 0 5.73E13 1.44E12 1.75E12 3.36E12 1 6.75E14 3.22E13 2.32E13 8.19E13 2 7.91E15 8.53E14 2.73E14 2.72E13 3 1.31E15 2.17E14 4.66E15 7.33E14 4 6.01E16 5.75E15 2.22E15 1.98E14 5 5.27E16 1.78E15 2.22E15 6.22E15 6 5.23E17 8.07E16 3.55E16 2.89E15 Please cite this article in press as: Arora G et al. A novel wavelet based hybrid method for finding the solutions of higher order boundary value problems. Ain Shams Eng J (2018), https://doi.org/10.1016/j.asej.2017.12.006 G. Arora et al. / Ain Shams Engineering Journal xxx (2018) xxx–xxx 13 x 10 -14 Numerical Experiment No.7 Numerical Experiment No. 9: Consider the eleventh order non- linear differential equation j=2 ð11Þ j=3 d xðt Þ 2.5 j=4 þ ðxðtÞÞ2 ¼ 11ðcos t sin t Þ tðcos t þ sin tÞ dt (t)| numerical þ t2 ð1 sin 2tÞ; 0 t 1 ð5:28Þ 2 xð0Þ ¼ 0; x ð0Þ ¼ 1; x ð0Þ ¼ 2; x ð0Þ ¼ 3; xiv ð0Þ ¼ 4; xv ð0Þ (t) - x 0 00 000 exact 0 00 1.5 ¼ 5; xð1Þ ¼ sin 1 cos 1; x ð1Þ ¼ 2 sin 1; x ð1Þ Absolute error =|x ¼ 3 cos 1 þ sin 1; x ð1Þ ¼ 2 cos 1 4 sin 1; xiv ð1Þ 000 1 ¼ 5 cos 1 3 sin 1 0.5 Analytic solution of the problem is xðtÞ ¼ tðsin t cos tÞ. The non-linearity in the differential equation is tackled by Quasilinearization technique. By using Quasilinearization tech- 0 nique given non-linear differential equation is transformed into a 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t sequence of linear differential equations as 11 Fig. 16. Absolute error at different level of resolution for numerical experiment d xrþ1 ðtÞ þ 2xr xrþ1 x2r ¼ 11ðcos t sin t Þ tðcos t þ sin tÞ no. 7. dt 11 þ t2 ð1 sin 2tÞ; xð0Þ ¼ 1; x ð0Þ ¼ 0; x ð0Þ ¼ 1; x ð0Þ ¼ 2; xiv ð0Þ ¼ 3; xv ð0Þ 0 00 000 0 t 1 r ¼ 0; 1; 2 subjected to the boundary conditions ¼ 4; xð1Þ ¼ 0; ð5:29Þ 0 00 v ð0Þ ¼ 4; 000 xrþ1 ð0Þ ¼ 0; xrþ1 ð0Þ ¼ 1; xrþ1 ð0Þ ¼ 2; xrþ1 ð0Þ ¼ 3; xrþ1 0 x ð1Þ ¼ e; x ð1Þ ¼ 2e; x ð1Þ ¼ 3e; xiv ð1Þ ¼ 4e 0 00 000 xvrþ1 ð0Þ ¼ 5; xrþ1 ð1Þ ¼ sin 1 cos 1; xrþ1 ð1Þ ¼ 2 sin 1; xrþ1 ð1Þ ¼ 0 00 Analytic solution of the problem is xðt Þ ¼ ð1 tÞe . t 000 000 3 cos 1 þ sin 1; xrþ1 ð1Þ ¼ 2 cos 1 4 sin 1; xrþ1 ð1Þ ¼ 5 cos 1 3 sin 1 We introduced the following approximate solution for the solu- where xrþ1 ðtÞ is the ðr þ 1Þth approximation for xðtÞ. tion of given differential equation by applying the non-dyadic Haar Then by applying the non-dyadic Haar wavelet mechanism for wavelet mechanism the solution of linear differential equation on the sequence of lin- ear differential equations, we proposed the following solution for X 3p xð t Þ ¼ ai ½q11;i ðt Þ f 1 ðt Þq11;i ð1Þ f 2 ðtÞq10;i ð1Þ given equation. i¼1 X 3p 1 xðt Þ ¼ ai ½q11;i ðtÞ f 1 ðt Þq11;i ð1Þ f 2 ðt Þq10;i ð1Þ f 3 ðt Þq9;i ð1Þ f 3 ðtÞq9;i ð1Þf 4 ðt Þq8;i ð1Þf 5 ðtÞq7;i ð1Þ f ðt Þ 120 1 i¼1 7 8 f 4 ðtÞq8;i ð1Þf 5 ðt Þq7;i ð1Þ þ ððsin 1 cos 1Þ Þf 1 ðtÞ þ e f ðt Þ 24 3 2 31 t5 t4 13 13 þ 2e f 3 ðtÞþð7 3eÞf 4 ðt Þþð7 4eÞf 5 ðt Þ þ ð2 sin 1 Þf ðt Þ þ ð3 cos 1 þ sin 1Þ f ðt Þ 6 30 8 8 2 6 3 t3 t2 7 þ 1; þðð2 cos 1 4 sin 1Þ þ Þf ðtÞþð9 ð5 cos 1 þ 3 sin 1ÞÞf 5 ðtÞ 3 2 2 4 t2 t3 t4 t5 where f 1 ðtÞ ¼ 126t 10 560t9 þ945t8 720t 7 þ210t6 , f 2 ðt Þ ¼ 56t 10 þ tþ þ ; 2 3 8 30 245x9 405t 8 þ 300t 7 84t 6 , f 3 ðt Þ ¼ 10:5t 10 45x9 þ 72:5t 8 where 52t 14t , 7 6 f 4 ðt Þ ¼ t þ 6 x 13 10 25 9 2 t 8 þ 92 t 7 76 t 6 t 5 , f 5 ðt Þ ¼ 0 1 9 x 1 8 t þ þ 1 7 t 1 6 t 1 5 t . ai s are the wavelets coefficients and 24 6 4 6 24 f 1 ðt Þ ¼ 126t 10 560t9 þ 945t8 720t7 þ 210t 6 qj;i 0 s are the wavelets integrals. Table 7b Comparison of numerical results at random collocation points available in literature for Exp. No. 7. x(t) Exact Approximated NDHWCM HPM [13] QBSCM [14] 0.1 1.105170918075640 1.105170918075640 5.77E15 1.41E06 1.25E05 0.2 1.221402758160170 1.221402758160110 6.02E14 2.69E06 8.70E06 0.3 1.349858807576000 1.349858807575920 7.51E14 3.70E06 2.15E06 0.4 1.491824697641270 1.491824697641440 1.73E13 4.35E06 1.13E05 0.5 1.648721270700120 1.648721270700740 6.16E13 4.58E06 3.97E05 0.6 1.822118800390500 1.822118800391350 8.43E13 4.36E06 5.40E05 0.7 2.013752707470470 2.013752707471080 6.12E13 3.71E06 6.79E05 0.8 2.225540928492460 2.225540928492670 2.05E13 2.69E06 4.89E05 0.9 2.459603111156940 2.459603111156960 1.38E14 1.42E06 2.00E05 E* (Absolute Error) = Exact Solution-Approximate Solution. Please cite this article in press as: Arora G et al. A novel wavelet based hybrid method for finding the solutions of higher order boundary value problems. Ain Shams Eng J (2018), https://doi.org/10.1016/j.asej.2017.12.006 14 G. Arora et al. / Ain Shams Engineering Journal xxx (2018) xxx–xxx f 2 ðt Þ ¼ 56t10 þ 245x9 405t 8 þ 300t 7 84t6 Numerical Experiment No.9 0.3 f 3 ðt Þ ¼ 10:5t 10 45x9 þ 72:5t 8 52t 7 14t6 f 4 ðt Þ ¼ t10 þ 256 x9 13 2 t 8 þ 92 t7 76 t6 t5 0.2 f 5 ðt Þ ¼ 24 x 16 t8 þ 14 t 7 16 t 6 þ 24 1 9 1 5 t 0.1 ai 0 s are the wavelets coefficients and qj;i 0 s are the wavelets integrals. It can be concluded from Figs. 17, 18, Tables 8 and 9 that the 0 x (t) results obtained from present method are in good agreement with the exact solution. To have more detail of results, absolute error -0.1 obtained in the present solution is compared with the error obtained in other methods [16,19,20] in Tables 8a and 9a and we -0.2 infer our method provides better accuracy. Numerical Experiment No. 10: Consider the twelfth order linear -0.3 differential equation Exact Solution Numerical Solution ð12Þ -0.4 d xð t Þ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 þ txðt Þ ¼ ð120 þ 23t þ t3 Þet ; 0 t 1 ð5:30Þ dt t Fig. 18. Exact and numerical solution of numerical experiment no. 9 at level of xð0Þ ¼ 0; x ð0Þ ¼ 1; x ð0Þ ¼ 0; x ð0Þ ¼ 3; xiv ð0Þ ¼ 8; xv ð0Þ ¼ 15; 0 00 000 resolution j = 2. xð1Þ ¼ 0; x ð1Þ ¼ e; x ð1Þ ¼ 4e; x ð1Þ ¼ 9e; xiv ð1Þ 0 00 000 Table 8 ¼ 16e; xv ð1Þ ¼ 25e: Exact and approximated solution by NDHWCM for j = 1 for numerical experiment no 8. Analytic solution of the problem is xðt Þ ¼ tð1 tÞet . We pro- x(t) Exact solution Approximated solution by NDHWCM posed the following solution for the Eq. (5.30) by applying the pre- 0.0556 0.998398425606890 0.998398425606890 sent method for the solution of linear differential equation with 0.1667 0.984467010721372 0.984467010721374 variable coefficients 0.2778 0.953472569424642 0.953472569424665 0.3889 0.901597042799825 0.901597042799900 X 3p 0.5000 0.824360635350064 0.824360635350189 xð t Þ ¼ ai ½q12;i ðtÞ f 1 ðtÞq12;i ð1Þ f 2 ðt Þq11;i ð1Þ f 3 ðtÞq10;i ð1Þ 0.6111 0.716519011851887 0.716519011852007 0.7222 0.571945470614687 0.571945470614748 i¼1 0.8333 0.383495981815471 0.383495981815481 1 f 4 ðt Þq9;i ð1Þf 5 ðt Þq8;i ð1Þf 6 ðtÞq7;i ð1Þ f ðt Þ 0.9444 0.142854690821557 0.142854690821556 24 1 59 19 37 þ e f ðt Þ þ 4e f 3 ðt Þþ 9e f ðt Þ 24 2 2 2 4 Table 8a t5 t4 t3 Comparison of numerical results at random collocation points available in literature þð23 16eÞf 5 ðt Þþð15 25eÞf 6 ðt Þ þ t; for Exp. No. 8. 8 3 2 x Exact Approximated NDHWCM MADM(E*) where (t) (E*) [16] 0.1 0.994653826268083 0.994653826268084 1.44E15 3.80E09 0.2 0.977122206528136 0.977122206528185 4.93E14 4.60E07 0.3 0.944901165303202 0.944901165303448 2.46E13 5.30E06 Numerical Experiment No.8 0.4 0.895094818584762 0.895094818585289 5.27E13 3.10E05 1 0.5 0.824360635350064 0.824360635350693 6.29E13 1.20E04 0.6 0.728847520156204 0.728847520156638 4.35E13 3.60E04 0.9 0.7 0.604125812241143 0.604125812241291 1.48E13 9.30E04 0.8 0.445108185698494 0.445108185698504 9.88E15 2.10E03 0.8 0.9 0.245960311115695 0.245960311115694 1.55E15 4.30E03 0.7 E*(Absolute Error) = Exact Solution-Approximate Solution. 0.6 x (t) 0.5 Table 9 0.4 Exact and approximated solution by NDHWCM for j = 1 for numerical experiment no 9. 0.3 x(t) Exact solution Approximated solution by NDHWCM 0.2 0.0556 0.052385011388556 0.052385011388556 0.1667 0.136707849811585 0.136707849811584 0.1 Exact Solution 0.2778 0.190957749800737 0.190957749800731 Numerical Solution 0.3889 0.212399606414346 0.212399606414325 0 0.5000 0.199078511643085 0.199078511643052 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.6111 0.149864712744852 0.149864712744822 t 0.7222 0.064483504858708 0.064483504858693 0.8333 0.056470507594150 0.056470507594154 Fig. 17. Exact and numerical solution of numerical experiment no. 8 at level of 0.9444 0.211535090710309 0.211535090710311 resolution j = 2. Please cite this article in press as: Arora G et al. A novel wavelet based hybrid method for finding the solutions of higher order boundary value problems. Ain Shams Eng J (2018), https://doi.org/10.1016/j.asej.2017.12.006 G. Arora et al. / Ain Shams Engineering Journal xxx (2018) xxx–xxx 15 Table 9a Comparison of numerical results at random collocation points available in literature for Exp. No. 9. x(t) Exact Approximated NDHWCM (E*) DTM(E*) [20] VIT (E*) [19] 0.1 0.089517074863120 0.089517074863118 1.67E15 1.51E13 1.03E15 0.2 0.156279449409236 0.156279449409184 5.25E14 5.23E12 5.34E14 0.3 0.197944884739280 0.197944884739013 2.67E13 3.07E11 5.59E13 0.4 0.212657060677694 0.212657060677103 5.91E13 9.74E11 3.32E12 0.5 0.199078511643085 0.199078511642340 7.45E13 3.69E10 1.48E11 0.6 0.156415884908786 0.156415884908215 5.71E13 2.00E09 5.38E11 0.7 0.084437150032758 0.084437150032512 2.46E13 1.03E08 1.66E10 0.8 0.016519505241886 0.016519505241929 4.36E14 4.39E08 4.42E10 0.9 0.145545247221137 0.145545247221138 9.99E16 1.58E07 1.04E09 E* (Absolute Error) = Exact Solution-Approximate Solution. 12 ð3Þ f 1 ðt Þ ¼ 252t11 þ 1386t10 3080t9 þ 3465t8 1980t7 þ 462t6 d xrþ1 ðtÞ d xrþ1 ðt Þ 4et xr xrþ1 ¼ 2x2r et ; 0 t 1r f 2 ðt Þ ¼ 126t11 686t10 þ 1505x9 1665t 8 þ 930t7 210t6 dt 12 dt f 3 ðt Þ ¼ 28t11 þ 301 2 t 10 325x9 þ 705 2 t 8 192t7 42t6 ¼ 0; 1; 2 ð5:32Þ f 4 ðt Þ ¼ 72 t 11 37 t 10 þ 235 x9 83 t 8 þ 22t7 28 t6 2 6 2 6 subjected to the boundary conditions 10 f 5 ðt Þ ¼ 1 4 x11 þ 31 24 x 83 x9 þ 11 4 t 8 34 24 t7 þ 24 7 6 t 0 00 000 v ð0Þ ¼ 1; xrþ1 ð0Þ ¼ 1; xrþ1 ð0Þ ¼ 1; xrþ1 ð0Þ ¼ 1; xrþ1 ð0Þ ¼ 1; xirþ1 xvrþ1 ð0Þ ¼ 1; xrþ1 ð1Þ ¼ e1 ; xrþ1 ð1Þ ¼ e1 ; xrþ1 ð1Þ ¼ e1 ; xrþ1 ð1Þ ¼ 0 00 000 ai 0 s are the wavelets coefficients and qj;i 0 s are the wavelets v ð1Þ ¼ e1 ; xv ð1Þ ¼ e1 where x ðtÞ is the ðr þ 1Þth e1 ; xirþ1 rþ1 rþ1 integrals. approximation for xðtÞ. Numerical Experiment No. 11: Consider the twelfth order non- Then by applying the present method for the solution of linear linear differential equation differential equation on the sequence of linear differential equa- ð12Þ ð3 Þ d xð t Þ d xð t Þ tions as obtained above, we proposed the following solution ¼ 2et ðxðtÞÞ2 þ ; 0t1 ð5:31Þ dt dt xð0Þ ¼ 1; x ð0Þ ¼ 1; x ð0Þ ¼ 1; x ð0Þ ¼ 1; xiv ð0Þ ¼ 1; xv ð0Þ ¼ 1 0 00 000 Numerical Experiment No.11 1 1 0 1 00 1 000 1 iv 1 xð1Þ ¼ e ; x ð1Þ ¼ e ; x ð1Þ ¼ e ; x ð1Þ ¼ e ; x ð1Þ ¼ e ; xv ð1Þ ¼ e1 0.9 Analytic solution of the problem is xðt Þ ¼ et . 0.8 In this problem, non-linearity in the differential equation is tackled by Quasilinearization technique. By using Quasilineariza- tion technique given non-linear differential equation is trans- 0.7 x (t) formed into a sequence of linear differential equations as 0.6 0.5 Numerical Experiment No.10 0.4 Exact Solution 0.45 Numerical Solution 0.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t 0.35 Fig. 20. Exact and numerical solution of numerical experiment no. 11 at level of 0.3 resolution j = 2. 0.25 x (t) Table 10 0.2 Exact and approximated solution by NDHWCM for j = 1 for numerical experiment no 10. 0.15 x(t) Exact solution Approximated solution by NDHWCM 0.1 0.0556 0.055466579200383 0.055466579200383 Exact Solution 0.1667 0.164077835120229 0.164077835120226 0.05 0.2778 0.264853491506845 0.264853491506824 Numerical Solution 0.3889 0.350621072199932 0.350621072199874 0 0.5000 0.412180317675032 0.412180317674953 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.6111 0.437872729465042 0.437872729464982 t 0.7222 0.413071728777274 0.413071728777253 0.8333 0.319579984846226 0.319579984846204 Fig. 19. Exact and numerical solution of numerical experiment no. 10 at level of 0.9444 0.134918319109249 0.134918319109259 resolution j = 2. Please cite this article in press as: Arora G et al. A novel wavelet based hybrid method for finding the solutions of higher order boundary value problems. Ain Shams Eng J (2018), https://doi.org/10.1016/j.asej.2017.12.006 16 G. Arora et al. / Ain Shams Engineering Journal xxx (2018) xxx–xxx Table 10a Comparison of numerical results at random collocation points available in literature for Exp. No.10. x(t) Exact Approximated NDHWCM(E*) VIT(E*) [21] DTM(E*) [22] 0.1 0.099465382626808 0.099465382626808 4.58E16 9.52E13 1.64E15 0.2 0.195424441305627 0.195424441305619 8.13E15 1.25E13 2.08E13 0.3 0.283470349590961 0.283470349590953 7.77E15 3.35E13 3.44E12 0.4 0.358037927433905 0.358037927433964 5.92E14 5.38E13 2.46E11 0.5 0.412180317675032 0.412180317675209 1.77E13 8.04E13 1.10E10 0.6 0.437308512093722 0.437308512093943 2.21E13 1.14E12 3.67E10 0.7 0.422888068568800 0.422888068568938 1.38E13 3.93E13 9.89E10 0.8 0.356086548558795 0.356086548558830 3.50E14 1.23E13 2.28E09 0.9 0.221364280004126 0.221364280004120 5.47E15 8.25E13 4.68E09 E*(Absolute Error) = Exact Solution-Approximate Solution. X 3p It can be concluded from Figs. 19, 20, Tables 10 and 11 that the xð t Þ ¼ ai ½q12;i ðt Þ f 1 ðt Þq12;i ð1Þ f 2 ðtÞq11;i ð1Þ results obtained from present method are roughly coinciding with i¼1 the exact solution. To have more detail of results, absolute error f 3 ðtÞq10;i ð1Þf 4 ðtÞq9;i ð1Þf 5 ðt Þq8;i ð1Þf 6 ðtÞq7;i ð1Þ obtained in the present solution is compared with the error 44 9 obtained by the other methods [21,22] given in Tables 10a and þ e1 f 1 ðt Þ þ e1 f ðt Þ 11a and we conclude that our method provides better accuracy. 120 24 2 1 1 1 1 þ e f ðt Þþ e f ðt Þþe1 f 5 ðt Þþ 1 e1 f 6 ðt Þ 6. Conclusion 3 3 2 4 t5 t4 t3 t2 We have applied non-dyadic Haar wavelet collocation method þ þ tþ1 5! 4! 3! 2! (NDHWCM) to find the numerical solution of linear and nonlinear HOBVPs. Eleven numerical experiments are performed on linear where and nonlinear HOBVPs. L2 , L1 and absolute errors are calculated for each numerical experiment and following observations has f 1 ðt Þ ¼ 252t11 þ 1386t10 3080t 9 þ 3465t8 1980t 7 þ 462t6 been made for non-dyadic Haar wavelet collocation method f 2 ðt Þ ¼ 126t11 686t 10 þ 1505x9 1665t 8 þ 930t7 210t6 (NDHWCM) f 3 ðt Þ ¼ 28t11 þ 301 2 t 10 325x9 þ 705 2 t 8 192t 7 42t6 f 4 ðt Þ ¼ 72 t 11 37 t 10 þ 235 x9 83 t 8 þ 22t 7 28 t6 i. With the increase in the level of resolution, errors decrease 2 6 2 6 10 which proves the convergence of NDHWCM solution to the f 5 ðt Þ ¼ 1 4 x11 þ 31 24 x 83 x9 þ 11 4 t 8 34 24 t7 þ 24 7 6 t exact solution. ii. Higher level of accuracy is obtained by the proposed method ai 0 s are the wavelets coefficients and qj;i 0 s are the wavelets for a small number of grid points proves the reliability of this integrals. mechanism. (In Experiment No. 11, for nine grid points, level of accuracy obtained is of1016 ) Table 11 iii. The accurateness of solution is up to the level of 1016 and Exact and approximated solution by NDHWCM for j = 1 for numerical experiment can be increased by increasing the level of resolution. no 11. iv. Computational work is fully supportive and compatible with x(t) Exact solution Approximated solution by NDHWCM the proposed algorithm. Common subprograms can be used 0.0556 0.945959468906765 0.945959468906766 to calculate the wavelets matrices and the integrals of wave- 0.1667 0.846481724890614 0.846481724890614 lets for all the problems. 0.2778 0.757465128396966 0.757465128396966 v. Rate of convergence of non-dyadic Haar wavelet collocation 0.3889 0.677809578005450 0.677809578005451 method is faster than the rate of convergence of dyadic Haar 0.5000 0.606530659712633 0.606530659712634 0.6111 0.542747481164222 0.542747481164222 wavelet collocation method. 0.7222 0.485671785247712 0.485671785247712 vi. Non-dyadic Haar wavelet Matrices and matrices of their 0.8333 0.434598208507078 0.434598208507079 integrals are very sparse which makes the process faster 0.9444 0.388895563989223 0.388895563989223 and creates very less computation cost. Table 11a Comparison of numerical results at random collocation points available in literature for Exp. No.11. x(t) Exact Approximated NDHWCM(E*) DTM(E*) [22] VIT(E*) [21] 0.1 0.904837418035960 0.904837418035960 1.11E16 4.11E15 1.61E07 0.2 0.818730753077982 0.818730753077982 0.00E+00 1.30E13 3.07E07 0.3 0.740818220681718 0.740818220681718 2.22E16 6.75E13 4.22E07 0.4 0.670320046035639 0.670320046035639 7.77E16 1.53E12 4.97E07 0.5 0.606530659712633 0.606530659712633 4.44E16 1.98E12 5.22E07 0.6 0.548811636094027 0.548811636094026 1.11E16 1.57E12 4.97E07 0.7 0.496585303791410 0.496585303791410 1.11E16 7.17E13 4.22E07 0.8 0.449328964117222 0.449328964117222 1.11E16 1.42E13 3.07E07 0.9 0.406569659740599 0.406569659740600 3.33E16 4.16E15 1.61E07 E*(Absolute Error) = Exact Solution-Approximate Solution. Please cite this article in press as: Arora G et al. A novel wavelet based hybrid method for finding the solutions of higher order boundary value problems. Ain Shams Eng J (2018), https://doi.org/10.1016/j.asej.2017.12.006 G. 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