Numerical Approximations of a Class of Nonlinear Second-Order Boundary Value Problems using Galerkin-Compact Finite Difference Method



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Submitted Apr 15, 2023
Published Aug 31, 2023
10.24018/ejmath.2023.4.4.235

Abstract viewed = 144 times

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In this study, we examine numerical approximations for 2nd-order linear-nonlinear differential equations with diverse boundary conditions, followed by the residual corrections of the first approximations. We first obtain numerical results using the Galerkin weighted residual approach with Bernstein polynomials. The generation of residuals is brought on by the fact that our first approximation is computed using numerical methods. To minimize these residuals, we use the compact finite difference scheme of 4th-order convergence to solve the error differential equations in accordance with the error boundary conditions. We also introduce the formulation of the compact finite difference method of fourth-order convergence for the nonlinear BVPs. The improved approximations are produced by adding the error values derived from the approximations of the error differential equation to the weighted residual values. Numerical results are compared to the exact solutions and to the solutions available in the published literature to validate the proposed scheme, and high accuracy is achieved in all cases.

Keywords: Compact Finite Difference Method Galerkin Method Nonlinear BVPs Residual Correction

References

  1. Bender CM, Orszag S, Orszag SA. Advanced mathematical methods for scientists and engineers I: Asymptotic methods and perturbation theory. Springer Science & Business Media; 1999.
     Google Scholar
  2. Collatz L. The numerical treatment of differential equations. Springer Science & Business Media; 2012.
     Google Scholar
  3. Kanth AR, Reddy YN. Cubic spline for a class of singular two-point boundary value problems. Applied mathematics and computation. 2005;170(2):733-40.
     Google Scholar
  4. Kanth AR, Bhattacharya V. Cubic spline for a class of non-linear singular boundary value problems arising in physiology. Applied Mathematics and Computation. 2006;174(1):768-74.
     Google Scholar
  5. Motsa SS, Sibanda P, Shateyi S. A new spectral-homotopy analysis method for solving a nonlinear second order BVP. Communications in Nonlinear Science and Numerical Simulation. 2010;15(9):2293-302.
     Google Scholar
  6. Islam MD, Shirin A. Numerical solutions of a class of second order boundary value problems on using Bernoulli polynomials. arXiv preprint arXiv:1309.6064. 2013.
     Google Scholar
  7. Burden RL, Faires JD, Burden AM. Numerical analysis. Cengage learning; 2015.
     Google Scholar
  8. Pandey PK. Solution of two-point boundary value problems, a numerical approach: parametric difference method. Applied Mathematics and Nonlinear Sciences. 2018;3(2):649-58.
     Google Scholar
  9. Ramos H, Rufai MA. A third-derivative two-step block Falkner-type method for solving general second-order boundary-value systems. Mathematics and Computers in Simulation. 2019;165:139-55.
     Google Scholar
  10. Baccouch M. A finite difference method for stochastic nonlinear second-order boundary-value problems driven by additive noises. International Journal of Numerical Analysis & Modeling. 2020;17(3).
     Google Scholar
  11. Abbasbandy S, Hajiketabi M. A simple, efficient and accurate new Lie--group shooting method for solving nonlinear boundary value problems. International Journal of Nonlinear Analysis and Applications. 2021;12(1):761-81.
     Google Scholar
  12. Mustafa G, Ejaz ST, Kouser S, Ali S, Aslam M. Subdivision collocation method for one-dimensional Bratu’s problem. Journal of Mathematics. 2021;2021:1-8.
     Google Scholar
  13. Boyd JP. Chebyshev polynomial expansions for simultaneous approximation of two branches of a function with application to the one-dimensional Bratu equation. Applied Mathematics and Computation. 2003;143(2-3):189-200.
     Google Scholar
  14. Caglar H, Caglar N, Özer M, Valarıstos A, Anagnostopoulos AN. B-spline method for solving Bratu's problem. International Journal of Computer Mathematics. 2010;87(8):1885-91.
     Google Scholar
  15. Khuri SA. A new approach to Bratu’s problem. Applied mathematics and computation. 2004;147(1):131-6.
     Google Scholar
  16. Ben-Romdhane M, Temimi H, Baccouch M. An iterative finite difference method for approximating the two-branched solution of Bratu's problem. Applied Numerical Mathematics. 2019;139:62-76.
     Google Scholar
  17. Qin Y. Simulation based on Galerkin method for solidification of water through energy storage enclosure. Journal of Energy Storage. 2022;50:104672.
     Google Scholar
  18. Li X, Li S. A fast element-free Galerkin method for the fractional diffusion-wave equation. Applied Mathematics Letters. 2021;122:107529.
     Google Scholar
  19. Abdelhakem M, Alaa-Eldeen T, Baleanu D, Alshehri MG, El-Kady M. Approximating real-life BVPs via Chebyshev polynomials’ first derivative pseudo-Galerkin method. Fractal and Fractional. 2021;5(4):165.
     Google Scholar
  20. Ruman U, Islam MS. Galerkin Weighted Residual Method for Solving Fourth Order Fractional Differential and Integral Boundary Value Problems. Journal of Applied Mathematics and Computation. 2022;6(2):410-422.
     Google Scholar
  21. Ali H, Kamrujjaman M, Islam MS. Numerical computation of Fitzhugh-Nagumo equation: a novel Galerkin finite element approach. International Journal of Mathematical Research. 2020;9(1):20-7.
     Google Scholar
  22. Lele SK. Compact finite difference schemes with spectral-like resolution. Journal of Computational Physics. 1992;103(1):16-42.
     Google Scholar
  23. Mehra M, Patel KS. Algorithm 986: a suite of compact finite difference schemes. ACM Transactions on Mathematical Software (TOMS). 2017;44(2):1-31.
     Google Scholar
  24. Malele J, Dlamini P, Simelane S. Highly Accurate Compact Finite Difference Schemes for Two-Point Boundary Value Problems with Robin Boundary Conditions. Symmetry. 2022;14(8):1720.
     Google Scholar
  25. Lewis PE, Ward JP. The finite element method: principles and applications. Wokingham: Addison-Wesley; 1991.
     Google Scholar
  26. Pettigrew MF, Rasmussen H. A compact method for second-order boundary value problems on nonuniform grids. Computers & Mathematics with Applications. 1996;31(9):1-6.
     Google Scholar
  27. Oliveira FA. Collocation and residual correction. Numerische Mathematik. 1980;36:27-31.
     Google Scholar
  28. Çelik İ. Collocation method and residual correction using Chebyshev series. Applied Mathematics and Computation. 2006;174(2):910-20.
     Google Scholar
  29. Tyler JG. Analysis and implementation of high-order compact finite difference schemes. Brigham Young University; 2007.
     Google Scholar
  30. MÜLLENHEIM G. Solving two-point boundary value problems with spline functions. IMA Journal of Numerical Analysis. 1992;12(4):503-18.
     Google Scholar
  31. Sohel MN, Islam MS, Islam MS, Kamrujjaman M. Galerkin Method and Its Residual Correction with Modified Legendre Polynomials. Contemporary Mathematics. 2022:188-202.
     Google Scholar
  32. Abbasbandy S, Hashemi MS, Liu CS. The Lie-group shooting method for solving the Bratu equation. Communications in Nonlinear Science and Numerical Simulation. 2011;16(11):4238-49.
     Google Scholar
  33. Deeba E, Khuri SA, Xie S. An Algorithm for Solving Boundary Value Problems. Journal of Computational Physics. 2001;1(170):448.
     Google Scholar
  34. Ala’yed O, Batiha B, Abdelrahim R, Jawarneh AA. On the numerical solution of the nonlinear Bratu type equation via quintic B-spline method. Journal of Interdisciplinary Mathematics. 2019;22(4):405-13.
     Google Scholar
  35. Roul P, Thula K. A fourth-order B-spline collocation method and its error analysis for Bratu-type and Lane–Emden problems. International Journal of Computer Mathematics. 2019;96(1):85-104.
     Google Scholar
  36. Farzana H, Islam MS. Computation of some second order Sturm-Liouville BVPs using Chebyshev-Legendre collocation method. GANIT: Journal of Bangladesh Mathematical Society. 2015;35:95-112.
     Google Scholar