AM  Vol.9 No.11 , November 2018
Two Very Accurate and Efficient Methods for Solving Time-Dependent Problems
Abstract: In this paper, collocation method based on Bernoulli and Galerkin method based on wavelet are proposed for solving nonhomogeneous heat and wave equations. The two methods have the linear systems solved by suitable solvers. Several examples are given to examine the performance of these methods and a comparison is made.
Cite this paper: El-Gamel, M. , Adel, W. and El-Azab, M. (2018) Two Very Accurate and Efficient Methods for Solving Time-Dependent Problems. Applied Mathematics, 9, 1270-1280. doi: 10.4236/am.2018.911083.

[1]   Wang, Y., Chenb, X. and He, Z. (2012) A Second-Generation Wavelet-Based Finite Element Method for the Solution of Partial Di_erential Equations. Applied Mathematics Letters, 25, 1608- 1613.

[2]   Xu, J. and Shann, W. (1992) Galerkin-Wavelet Methods for Two- Point Boundary Value Problems. Numerische Mathematik, 63, 123-144.

[3]   Avudainayagam, A. (2000)Wavelet-Galerkin Method for Integro- Di_erential Equations. Applied Numerical Mathematics, 32, 247- 254.

[4]   Beylkin, G., Coifman, R. and Rokhlin, V. (1991) Fast Wavelet Transform and Numerical Algorithm I. Communications on Pure and Applied Mathematics, 44, 141-183.

[5]   Fang, W.,Wang, Y. and Xu, Y. (2004) An Implementation of Fast Wavelet-Galerkin Methods for Integral Equations of the Second Kind. Journal of Scienti_c Computing, 20, 277-302.

[6]   Alpert, B. (1993) A Class of Bases in l2 for the Sparse Representation of Integral Operators. SIAM Journal on Mathematical Analysis, 24, 246-262.

[7]   El-Gamel, M. (2004) Wavelet Algorithm for the Numerical Solution of Nonhomogeneous Time-Dependent Problems. Journal of Di_erence Equations and Applications, 9, 169-185.

[8]   El-Gamel, M. (2006) A Wavelet-Galerkin Method for a Singularly Perturbed Convection-Dominated Di_usion Equation. Applied Mathematics and Computation, 181, 1635-1644.

[9]   El-Azab, M. and El-Gamel, M. (2007) A Numerical Algorithm for the Solution of Telegraph Equations. Applied Mathematics and Computation, 190, 757-764.

[10]   Panigrahi, B. and Nelakanti, G. (2012) Wavelet-Galerkin Method for Eigenvalue Problem of a Compact Integral Operator. Applied Mathematics and Computation, 218, 1222-1232.

[11]   Rathish, B. and Priyadarshi, G. (2018) Wavelet Galerkin Method for Fourth-Order Multi-Dimensional Elliptic Partial Di_erential Equations. International Journal of Wavelets, Multiresolution and Information Processing, 16, 185-205.

[12]   Priyadarshi, G. and Rathish, B. (2018) Wavelet-Galerkin Method for Fourth Order Linear and Nonlinear Di_erential Equations. Applied Mathematics and Computation, 327, 8-21.

[13]   Mohmmadi, F. (2016) Wavelet-Galerkin Method for Solving Stochastic Fractional Di_erential Equations. Journal of Fractional Calculus and Applications, 7, 73-86.

[14]   Wang, J., Liu, X. and Zhou, Y. (2018) A High-Order Accurate Wavelet Method for Solving Schrdinger Equations with General Nonlinearity. Applied Mathematics and Mechanics, 39, 275-290.

[15]   Yang, Z. and Liao, S. (2018) On the Generalized Wavelet- Galerkin Method. Journal of Computational and Applied Mathe- matics, 331, 178-195.

[16]   El-Gamel, M. and Zayed, A. (2002) A Comparison between the Wavelet-Galerkin and the Sinc-Galerkin Methods in Solving Nonhomogeneous Heat Equations. In: Nashed, Z. and Scherzer, O., Eds., Contemporary Mathematics of the American Mathematical Society, Series, Inverse Problem, Image Analysis, and Medical Imaging, Vol. 313, AMS, Providence.

[17]   El-Gamel, M. (2007) Comparison of the Solutions Obtained by Adomian Decomposition and Wavelet-Galerkin Methods of Boundary-Value Problems. Applied Mathematics and Computa- tion, 186, 652-664.

[18]   Singh, S., Kumar Patel, V., Kumar Singh, V. and Tohidi, E. (2018) Application of Bernoulli Matrix Method for Solving Two-Dimensional Hyperbolic Telegraph Equations with Dirichlet Boundary Conditions. Computers & Mathematics with Applica- tions, 75, 2280-2294.

[19]   Toutounian, F. and Tohidi, E. (2013) A New Bernoulli Matrix Method for Solving Second Order Linear Partial Di_erential Equations with the Convergence Analysis. Applied Mathematics and Computation, 223, 298-310.

[20]   Tohidi, E., Bhrawy, A. and Erfani, K. (2007) A Collocation Method Based on Bernoulli Operational Matrix for Numerical Solution of Generalized Pantograph Equation. Applied Mathemat- ical Modelling, 37, 4283-4294.

[21]   Sahu, P. and SahaRay, S. (2017) A New Bernoulli Wavelet Method for Accurate Solutions of Nonlinear Fuzzy Hammerstein- Volterra Delay Integral Equations. Fuzzy Sets and Systems, 309, 131-144.

[22]   Rahimkhan, P., Ordokhani, Y. and Babolian, E. (2017) Fractional-Order Bernoulli Functions and Their Applications in Solving Fractional Fredholem-Volterra Integro-Di_erential Equations. Applied Numerical Mathematics, 122, 66-81.

[23]   Calvert, V. and Razzaghi, M. (2017) Solutions of the Blasius and MHD Falkner-Skan Boundary-Layer Equations by Modi_ed Rational Bernoulli Functions. International Journal of Numerical Methods for Heat & Fluid Flow, 27, 1687-1705.

[24]   Zoghe, B., Tohidi, E. and Shatey, S. (2017) Bernoulli-Collocation Method for Solving Linear Multidimensional Di_usion and Wave Equations with Dirichlet Boundary Conditions. Advances in Mathematical Physics, 2017, Article ID: 5691452.

[25]   Bokhari, A., Amir, A., Bahri, S. and Belgacem, F. (2017) A Generalized Bernoulli Wavelet Operational Matrix of Derivative Applications to Optimal Control Problems. Nonlinear Studies, 24, 775-790.

[26]   Mohamed, A. and Mohamed, R. (2018) Numerical Solution of Fuzzy Integral Equations via a New Bernoulli-Wavelet Method. International Journal of Modern Mathematical Sciences, 16, 37- 50.

[27]   El-Gamel, M. and Adel, W. (2018) Numerical Investigation of the Solution of Higher-Order Boundary Value Problems via Euler Matrix Method. SeMA Journal, 75, 349-364.

[28]   Amaratunga, K., Williams, J., Qian, S. and Weiss, J. (1994) Wavelet-Galerkin Solutions for One-Dimensional Partial Di_erential Equations. International Journal for Numerical Methods in Engineering, 37, 2703-2716.

[29]   Mallat, S. (1989) Multiresoltion Approximations andWavelet Orthonormal Bases of L2(R) . Transactions of the American Math- ematical Society, 315, 69-87.

[30]   Daubechies, I. (1992) Ten Lectures on Wavelets. Captial City Press, Vermont.

[31]   Qian, S. and Weiss, J. (1993) Wavelets and the Numerical Solution of Boundary Value Problems. Applied Mathematics Letters, 6, 47-52.

[32]   Chen, M., Wang, C. and Shin, Y. (1996) The Computation of Wavelet-Galerkin Approximation on a Bounded Interval. International Journal for Numerical Methods in Engineering, 39, 2921-2944. AID-NME983i3.0.CO;2-D