Research Article
BibTex RIS Cite

The Approximate Solution of Singularly Perturbed Burger-Huxley Equation with RDTM

Year 2023, Volume: 36 Issue: 4, 1647 - 1656, 01.12.2023

Derya Arslan

https://doi.org/10.35378/gujs.935885

Abstract

In this paper, the numerical integral method on a uniform mesh is used to solve the singularly perturbed problem with integral boundary value. This method also includes the trapezoid method, the finite difference method, and the Thomas algorithm. The problem is converted to finite difference problem by using finite difference approximations and trapezoid method. Finally, the convergence of the presented method is analyzed through sample application. Thus, the accuracy and efficiency of the method are shown.

Keywords

the finite difference method SingularLy Perturbed Problem Trapezoidal Method boundary layer Numerical integral method Singularly perturbed problem Finite difference method Trapezoidal method Uniform mesh

References

  • Reference 1 Cakir, M., “A numerical study on the difference solution of singularly perturbed semilinear problem with integral boundary condition”, Mathematical Modelling and Analysis, 2 (5):644-658, (2016).
  • Reference 2 Arslan, D., “A numerical solution for singularly perturbed multi-point boundary value problems with the numerical integration method”, BEU Journal of Science, 9(1) :157-167, (2020).
  • Reference 3 Arslan, D., “Stability and convergence analysis on shishkin mesh for a nonlinear singularly perturbed problem with three-point boundary condition”, Quaestiones Mathematicae 43(11):1-14, (2020).
  • Reference 4 Cimen, E., Cakir, M., “Numerical treatment of nonlocal boundary value problem with layer behaviour”, Bull. Belg. Math. Soc. Simon Stevin, 24:339-352, (2017).
  • Reference 5 Arslan, D., “On the generation for numerical solution of singularly perturbed problem with right boundary layer”, Turkish Journal of Mathematics and Computer Science, 12 (1), 31-38, (2020).
  • Reference 6 Arslan, D., “An effective approximation for singularly perturbed problem with multi-point boundary value”, New Trends in Mathematical Sciences, 9 (2):15-25, (2021).
  • Reference 7 Kudu, M., Amiraliyev, G. M., “Finite difference method for a singularly perturbed differential equations with integral boundary condition”, Int. J. Math. Comput., 26 (3):71-79, (2015).
  • Reference 8 Arslan, D., “An effective numerical method for singularly perturbed nonlocal boundary value problem on bakhvalov mesh”, Journal of Informatics and Mathematical Sciences, 11(3-4):253–264, (2019).
  • Reference 9 Chegis,R., “On the dierence schemes for problems with nonlocal boundary conditions”, Informatica, 2(2):155170, (1991).
  • Reference 10 Bitsadze, A. V., Samarskii, A. A., “On some simpler generalization of linear elliptic boundary value problems”, Doklady Akademii Nauk SSSR, 18:739-40, (1969).
  • Reference 11 Belarbi, A., Benchohra, M., Existence results for nonlinear boundary value problems with integral boundary conditions, Electronıc J. Diff. Equat., 2005(6), 1-10, (2005).
  • Reference 12 ankowski, T., “Existence of solutions of differential equations with nonlinear multipoint boundary conditions”, Comput. Math. Appl., 4:1095-1103, (2004).
  • Reference 13 Farell, P. A., Miller, J. J. H., O'Riordan, E., Shishkin, G. I., “A uniformly convergent finite difference scheme for a singularly perturbed semilinear equation”, SIAM Journal on Numerical Analysis, 3:1135-1149, (1996).
  • Reference 14 Gupt, C. P., Trofmchuk, S. I., “A sharper condition for the solvability of a three-point second order boundary value problem”, Journal of Mathematical Analysis and Applications, 205:586-597, (1997).
  • Reference 15 Herceg, D., Surla, K., “Solving a nonlocal singularly perturbed nonlocal problem by splines in tension”, Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Math., 21(2):119-132, (1991).
  • Reference 16 Miller, J. J. H., O'Riordan, E., Shishkin, G. I., “Fitted Numerical Methods for Singular Perturbation Problems”. World Scientific, Singapore, (1996).
  • Reference 17 Nayfeh, A. H., “Introduction to Perturbation Techniques”. Wiley, New York, (1993).
  • Reference 18 Reddy, Y. N., “A Numerical Integration Method for Solving Singular Perturbation Problems”, Applied Mathematics and Computation, 37: 83-95, (1990).
  • Reference 19 Andargie, A., Reddy, Y. N., “Numerical integration method for singular perturbation problems with mixed boundary conditions”, J. Appl. Math. & Informatics, 26 (5-6): 1273-1287, (2008).
  • Reference 20 Soujanya, G., Phnaeendra, K., “ Numerical intergration method for singular-singularly perturbed two-point boundary value problems”, Procedia Engineering, 127:545-552, (2015).
  • Reference 21 Ranjan, R., Prasad, H.S., “An efficient method of numerical integration for a class of singularly perturbed two point boundary value problems”, Wseas Transactions on Mathematics, 17:265-273, (2018).
  • Reference 22 Amirali, G., Amirali, I. “Nümerik Analiz Teori ve Uygulamalarla”. Seçkin Yayıncılık, Ankara, Türkiye, (2018).

A Robust Numerical Approach for Singularly Perturbed Problem with Integral Boundary Condition

Year 2023, Volume: 36 Issue: 4, 1647 - 1656, 01.12.2023

Derya Arslan

https://doi.org/10.35378/gujs.935885

Abstract

In this research, the numerical integral method procedure on uniform mesh is used to solve the singularly perturbed problem which has integral boundary value. This method also includes the trapezoid method, the finite difference method, and the Thomas algorithm. The problem is converted to finite difference problem by using finite difference approximations and trapezoid method. Finally, the convergence of the presented method is analyzed through sample application. Thus, the correctness and sufficiency of the method are shown.

Keywords

Numerical integral method Singularly perturbed problem Finite difference method Trapezoidal method Uniform mesh

References

  • Reference 1 Cakir, M., “A numerical study on the difference solution of singularly perturbed semilinear problem with integral boundary condition”, Mathematical Modelling and Analysis, 2 (5):644-658, (2016).
  • Reference 2 Arslan, D., “A numerical solution for singularly perturbed multi-point boundary value problems with the numerical integration method”, BEU Journal of Science, 9(1) :157-167, (2020).
  • Reference 3 Arslan, D., “Stability and convergence analysis on shishkin mesh for a nonlinear singularly perturbed problem with three-point boundary condition”, Quaestiones Mathematicae 43(11):1-14, (2020).
  • Reference 4 Cimen, E., Cakir, M., “Numerical treatment of nonlocal boundary value problem with layer behaviour”, Bull. Belg. Math. Soc. Simon Stevin, 24:339-352, (2017).
  • Reference 5 Arslan, D., “On the generation for numerical solution of singularly perturbed problem with right boundary layer”, Turkish Journal of Mathematics and Computer Science, 12 (1), 31-38, (2020).
  • Reference 6 Arslan, D., “An effective approximation for singularly perturbed problem with multi-point boundary value”, New Trends in Mathematical Sciences, 9 (2):15-25, (2021).
  • Reference 7 Kudu, M., Amiraliyev, G. M., “Finite difference method for a singularly perturbed differential equations with integral boundary condition”, Int. J. Math. Comput., 26 (3):71-79, (2015).
  • Reference 8 Arslan, D., “An effective numerical method for singularly perturbed nonlocal boundary value problem on bakhvalov mesh”, Journal of Informatics and Mathematical Sciences, 11(3-4):253–264, (2019).
  • Reference 9 Chegis,R., “On the dierence schemes for problems with nonlocal boundary conditions”, Informatica, 2(2):155170, (1991).
  • Reference 10 Bitsadze, A. V., Samarskii, A. A., “On some simpler generalization of linear elliptic boundary value problems”, Doklady Akademii Nauk SSSR, 18:739-40, (1969).
  • Reference 11 Belarbi, A., Benchohra, M., Existence results for nonlinear boundary value problems with integral boundary conditions, Electronıc J. Diff. Equat., 2005(6), 1-10, (2005).
  • Reference 12 ankowski, T., “Existence of solutions of differential equations with nonlinear multipoint boundary conditions”, Comput. Math. Appl., 4:1095-1103, (2004).
  • Reference 13 Farell, P. A., Miller, J. J. H., O'Riordan, E., Shishkin, G. I., “A uniformly convergent finite difference scheme for a singularly perturbed semilinear equation”, SIAM Journal on Numerical Analysis, 3:1135-1149, (1996).
  • Reference 14 Gupt, C. P., Trofmchuk, S. I., “A sharper condition for the solvability of a three-point second order boundary value problem”, Journal of Mathematical Analysis and Applications, 205:586-597, (1997).
  • Reference 15 Herceg, D., Surla, K., “Solving a nonlocal singularly perturbed nonlocal problem by splines in tension”, Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Math., 21(2):119-132, (1991).
  • Reference 16 Miller, J. J. H., O'Riordan, E., Shishkin, G. I., “Fitted Numerical Methods for Singular Perturbation Problems”. World Scientific, Singapore, (1996).
  • Reference 17 Nayfeh, A. H., “Introduction to Perturbation Techniques”. Wiley, New York, (1993).
  • Reference 18 Reddy, Y. N., “A Numerical Integration Method for Solving Singular Perturbation Problems”, Applied Mathematics and Computation, 37: 83-95, (1990).
  • Reference 19 Andargie, A., Reddy, Y. N., “Numerical integration method for singular perturbation problems with mixed boundary conditions”, J. Appl. Math. & Informatics, 26 (5-6): 1273-1287, (2008).
  • Reference 20 Soujanya, G., Phnaeendra, K., “ Numerical intergration method for singular-singularly perturbed two-point boundary value problems”, Procedia Engineering, 127:545-552, (2015).
  • Reference 21 Ranjan, R., Prasad, H.S., “An efficient method of numerical integration for a class of singularly perturbed two point boundary value problems”, Wseas Transactions on Mathematics, 17:265-273, (2018).
  • Reference 22 Amirali, G., Amirali, I. “Nümerik Analiz Teori ve Uygulamalarla”. Seçkin Yayıncılık, Ankara, Türkiye, (2018).
There are 22 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Mathematics
Authors

Derya Arslan 0000-0001-6138-0607

Publication Date December 1, 2023
Published in Issue Year 2023 Volume: 36 Issue: 4

Cite

APA Arslan, D. (2023). A Robust Numerical Approach for Singularly Perturbed Problem with Integral Boundary Condition. Gazi University Journal of Science, 36(4), 1647-1656. https://doi.org/10.35378/gujs.935885
AMA Arslan D. A Robust Numerical Approach for Singularly Perturbed Problem with Integral Boundary Condition. Gazi University Journal of Science. December 2023;36(4):1647-1656. doi:10.35378/gujs.935885
Chicago Arslan, Derya. “A Robust Numerical Approach for Singularly Perturbed Problem With Integral Boundary Condition”. Gazi University Journal of Science 36, no. 4 (December 2023): 1647-56. https://doi.org/10.35378/gujs.935885.
EndNote Arslan D (December 1, 2023) A Robust Numerical Approach for Singularly Perturbed Problem with Integral Boundary Condition. Gazi University Journal of Science 36 4 1647–1656.
IEEE D. Arslan, “A Robust Numerical Approach for Singularly Perturbed Problem with Integral Boundary Condition”, Gazi University Journal of Science, vol. 36, no. 4, pp. 1647–1656, 2023, doi: 10.35378/gujs.935885.
ISNAD Arslan, Derya. “A Robust Numerical Approach for Singularly Perturbed Problem With Integral Boundary Condition”. Gazi University Journal of Science 36/4 (December 2023), 1647-1656. https://doi.org/10.35378/gujs.935885.
JAMA Arslan D. A Robust Numerical Approach for Singularly Perturbed Problem with Integral Boundary Condition. Gazi University Journal of Science. 2023;36:1647–1656.
MLA Arslan, Derya. “A Robust Numerical Approach for Singularly Perturbed Problem With Integral Boundary Condition”. Gazi University Journal of Science, vol. 36, no. 4, 2023, pp. 1647-56, doi:10.35378/gujs.935885.
Vancouver Arslan D. A Robust Numerical Approach for Singularly Perturbed Problem with Integral Boundary Condition. Gazi University Journal of Science. 2023;36(4):1647-56.