Research Article
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A Numerical Method for Solving Singularly Perturbed Quasilinear Boundary Value Problems on Shishkin Mesh

Year 2022, Volume: 14 Issue: 1, 145 - 154, 30.06.2022
https://doi.org/10.47000/tjmcs.1010528

Abstract

In this paper, singularly perturbed quasilinear boundary value problems are taken into account. With this purpose, a finite difference scheme is proposed on Shishkin-type mesh (S-mesh). Quasilinearization technique and interpolating quadrature rules are used to establish the numerical scheme. Then, an error estimate is derived. A numerical experiment is demonstratedto verify the theory.

References

  • Aga Bullo, T., Degla, G.A., Duressa, G.F., Uniformly convergent higher-order finite difference scheme for singularly perturbed parabolic problems with non-smooth data, Journal of Applied Mathematics and Computational Mechanics, 20(1)(2021), 5-16.
  • Alquran, M.T., DoĞan, N., Variationaliteration method for solving two-parameter singularly perturbed two point boundary value problem, Applications and Applied Mathematics: An International Journal (AAM), 5(1)(2010), 81-95.
  • Amiraliyev, G.M., Mamedov, Y.D., Difference schemes on the uniform mesh for singularly perturbed pseudo-parabolic equations, Turkish Journal of Mathematics, 19(1995), 207-222.
  • Cakir, M., Gunes, B., Duru, H., A novel computational method for solving nonlinear Volterra integro-differential equation, Kuwait Journal of Science, 48(1)(2021), 1-9.
  • Cakir, M., Gunes, B., Exponentially fitteddifference scheme for singularly perturbed mixed integro-differential equations, Georgian Mathematical Journal, (2022).
  • Cassani, D., Wang, Y., Zhang, J., A unifed approach to singularly perturbed quasilinear Schrödinger equations, Milan Journal of Mathematics, 88(2020), 507-534.
  • Chen, S-B., Soradi-Zeid, S., Dutta, H., Mesrizadeh, M., Jahanshahi, H. et al., Reproducing kernel Hilbert space method for nonlinear second order singularly perturbed boundary value problems with time-delay, Chaos, Solitons and Fractals, 144(2021), 110674.
  • Çakır, M., Güneş, B., A new difference method for the singularly perturbed Volterra-Fredholm integro-differential equations on a Shishkin mesh, Hacettepe Journal of Mathematics and Statistics, 51(3)(2022), 787-799.
  • Doğan, N., Ertürk, V.S., Momani, S., Akın, Ö., Yıldırım, A., Differential transform method for solving singularly perturbed Volterra integral equations, Journal of King Saud University-Science, 23(2011), 223-228.
  • Doğan, N., Ertürk, V.S., Momani, S., He's variational iteration method for solving the singularly perturbed Volterra integral equations, World Applied Sciences Journal, 22(11)(2013), 1657-1661.
  • Duru, H., Gunes, B., Numerical solutions for singularly perturbed nonlinear reaction diffusion problems on the piecewise equidistant mesh, Erzincan University Journal of Science and Technology, 12(1)(2019), 425-436.
  • Duru, H., Gunes, B., The finite difference method on adaptive mesh for singularly perturbed nonlinear 1D reaction-diffusion boundary value problems, Journal of Applied Mathematics and Computational Mechanics, 19(4)(2020), 45-56.
  • Erdogan, F., Sakar, M.G., A quasilinearization technique for the solution of singularly perturbed delay differential equation, Mathematics in Natural Science, 2(2018), 1-7.
  • Erdogan, F., Sakar, M.G., Saldır, O., A finite difference method on layer-adapted mesh for singularly perturbed delay differential equations, Applied Mathematics and Nonlinear Sciences, 5(1)(2020), 425-436.
  • Gunes, B., Chianeh, A.B., Demirbas, M., Comparison of multiple scales method and finite difference method for solving singularly perturbed convection diffusion problem}, Gumushane University Journal of Science and Technology Institute, 10(4)(2020), 1169-1181.
  • Konyaev, Y.A., Workneh, A.Z., Estimating the norm of solution of singularly perturbed quasilinear problems for ODE systems with nonlinear normal matrices on the semiaxis, Discrete and Continuous Models and Applied Computational Science, 4(2013), 5-10.
  • Kumar, S., Layer-adapted methods for quasilinear singularly perturbed delay differential problems, Applied Mathematics and Computation, 233(1)(2014), 214-221.
  • Lin, R., Ye, X., Zhang, S., Zhu, P., A weak Galerkin finite element method for singularly perturbed convection-diffusion-reaction problems, SIAM J. Numer. Anal., 56(3)(2018), 1482-1497.
  • Nefedov, N.N., Davydova, M.A., Constrast structures in singularly perturbed quasilinear reactions-diffusion-advection equations, Differential Equations, 49(2013), 688-706.
  • Ni, M., Pang, Y., Levashova, N.T., Nikolaeva, O.A., Internal layers for a singularly perturbed second-order quasilinear differential equation with discontinuous right-hand side, Differential Equations, 53(2017), 1567-1577.
  • Samarski, A.A., The Theory of Difference Schemes, Moscow M.V. Lomonosov State University, Russia, 2001.
  • Sekar, E., Tamilselvan, A., Third order singularly perturbed delay differential equation of reaction diffusion type with integral boundary condition, Journal of Applied Mathematics and Computational Mechanics, 18(2)(2019), 99-110.
  • Selvakumar, K., Lazarus, G.P., A fitted operator and fitted mesh method for singularly perturbed convection diffusion problem, International Journal of Recent Research Aspects, (2018), 544550.
  • Shishkin, G.I., Higher-order accurate method for a quasilinear singularly perturbed elliptic convection-diffusion equation, Sib. Zh. Vychisl. Mat., 9(1)(2006), 81-108.
  • Voevodin, A.F., Factorization method for linear and quasilinear singularly perturbed boundary value problems for ordinary differential equations, Numerical Analysis and Applications, 2(2009), 1-12.
  • Xie, S., Zhu, P., Wang, X., Error analysis of weak Galerkin finite element methods for time-dependent convection-diffusion equations, Applied Numerical Mathematics, 137(2019), 19-33.
  • Zheng, Q., Ye, F., Numerical solution of quasilinear singularly perturbed problems by the principle of equidistribution, Journal of Applied Mathematics and Physics, 8(10)(2020), 103603.
Year 2022, Volume: 14 Issue: 1, 145 - 154, 30.06.2022
https://doi.org/10.47000/tjmcs.1010528

Abstract

References

  • Aga Bullo, T., Degla, G.A., Duressa, G.F., Uniformly convergent higher-order finite difference scheme for singularly perturbed parabolic problems with non-smooth data, Journal of Applied Mathematics and Computational Mechanics, 20(1)(2021), 5-16.
  • Alquran, M.T., DoĞan, N., Variationaliteration method for solving two-parameter singularly perturbed two point boundary value problem, Applications and Applied Mathematics: An International Journal (AAM), 5(1)(2010), 81-95.
  • Amiraliyev, G.M., Mamedov, Y.D., Difference schemes on the uniform mesh for singularly perturbed pseudo-parabolic equations, Turkish Journal of Mathematics, 19(1995), 207-222.
  • Cakir, M., Gunes, B., Duru, H., A novel computational method for solving nonlinear Volterra integro-differential equation, Kuwait Journal of Science, 48(1)(2021), 1-9.
  • Cakir, M., Gunes, B., Exponentially fitteddifference scheme for singularly perturbed mixed integro-differential equations, Georgian Mathematical Journal, (2022).
  • Cassani, D., Wang, Y., Zhang, J., A unifed approach to singularly perturbed quasilinear Schrödinger equations, Milan Journal of Mathematics, 88(2020), 507-534.
  • Chen, S-B., Soradi-Zeid, S., Dutta, H., Mesrizadeh, M., Jahanshahi, H. et al., Reproducing kernel Hilbert space method for nonlinear second order singularly perturbed boundary value problems with time-delay, Chaos, Solitons and Fractals, 144(2021), 110674.
  • Çakır, M., Güneş, B., A new difference method for the singularly perturbed Volterra-Fredholm integro-differential equations on a Shishkin mesh, Hacettepe Journal of Mathematics and Statistics, 51(3)(2022), 787-799.
  • Doğan, N., Ertürk, V.S., Momani, S., Akın, Ö., Yıldırım, A., Differential transform method for solving singularly perturbed Volterra integral equations, Journal of King Saud University-Science, 23(2011), 223-228.
  • Doğan, N., Ertürk, V.S., Momani, S., He's variational iteration method for solving the singularly perturbed Volterra integral equations, World Applied Sciences Journal, 22(11)(2013), 1657-1661.
  • Duru, H., Gunes, B., Numerical solutions for singularly perturbed nonlinear reaction diffusion problems on the piecewise equidistant mesh, Erzincan University Journal of Science and Technology, 12(1)(2019), 425-436.
  • Duru, H., Gunes, B., The finite difference method on adaptive mesh for singularly perturbed nonlinear 1D reaction-diffusion boundary value problems, Journal of Applied Mathematics and Computational Mechanics, 19(4)(2020), 45-56.
  • Erdogan, F., Sakar, M.G., A quasilinearization technique for the solution of singularly perturbed delay differential equation, Mathematics in Natural Science, 2(2018), 1-7.
  • Erdogan, F., Sakar, M.G., Saldır, O., A finite difference method on layer-adapted mesh for singularly perturbed delay differential equations, Applied Mathematics and Nonlinear Sciences, 5(1)(2020), 425-436.
  • Gunes, B., Chianeh, A.B., Demirbas, M., Comparison of multiple scales method and finite difference method for solving singularly perturbed convection diffusion problem}, Gumushane University Journal of Science and Technology Institute, 10(4)(2020), 1169-1181.
  • Konyaev, Y.A., Workneh, A.Z., Estimating the norm of solution of singularly perturbed quasilinear problems for ODE systems with nonlinear normal matrices on the semiaxis, Discrete and Continuous Models and Applied Computational Science, 4(2013), 5-10.
  • Kumar, S., Layer-adapted methods for quasilinear singularly perturbed delay differential problems, Applied Mathematics and Computation, 233(1)(2014), 214-221.
  • Lin, R., Ye, X., Zhang, S., Zhu, P., A weak Galerkin finite element method for singularly perturbed convection-diffusion-reaction problems, SIAM J. Numer. Anal., 56(3)(2018), 1482-1497.
  • Nefedov, N.N., Davydova, M.A., Constrast structures in singularly perturbed quasilinear reactions-diffusion-advection equations, Differential Equations, 49(2013), 688-706.
  • Ni, M., Pang, Y., Levashova, N.T., Nikolaeva, O.A., Internal layers for a singularly perturbed second-order quasilinear differential equation with discontinuous right-hand side, Differential Equations, 53(2017), 1567-1577.
  • Samarski, A.A., The Theory of Difference Schemes, Moscow M.V. Lomonosov State University, Russia, 2001.
  • Sekar, E., Tamilselvan, A., Third order singularly perturbed delay differential equation of reaction diffusion type with integral boundary condition, Journal of Applied Mathematics and Computational Mechanics, 18(2)(2019), 99-110.
  • Selvakumar, K., Lazarus, G.P., A fitted operator and fitted mesh method for singularly perturbed convection diffusion problem, International Journal of Recent Research Aspects, (2018), 544550.
  • Shishkin, G.I., Higher-order accurate method for a quasilinear singularly perturbed elliptic convection-diffusion equation, Sib. Zh. Vychisl. Mat., 9(1)(2006), 81-108.
  • Voevodin, A.F., Factorization method for linear and quasilinear singularly perturbed boundary value problems for ordinary differential equations, Numerical Analysis and Applications, 2(2009), 1-12.
  • Xie, S., Zhu, P., Wang, X., Error analysis of weak Galerkin finite element methods for time-dependent convection-diffusion equations, Applied Numerical Mathematics, 137(2019), 19-33.
  • Zheng, Q., Ye, F., Numerical solution of quasilinear singularly perturbed problems by the principle of equidistribution, Journal of Applied Mathematics and Physics, 8(10)(2020), 103603.
There are 27 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Hakkı Duru 0000-0002-3179-3758

Mutlu Demirbaş 0000-0001-8187-3919

Publication Date June 30, 2022
Published in Issue Year 2022 Volume: 14 Issue: 1

Cite

APA Duru, H., & Demirbaş, M. (2022). A Numerical Method for Solving Singularly Perturbed Quasilinear Boundary Value Problems on Shishkin Mesh. Turkish Journal of Mathematics and Computer Science, 14(1), 145-154. https://doi.org/10.47000/tjmcs.1010528
AMA Duru H, Demirbaş M. A Numerical Method for Solving Singularly Perturbed Quasilinear Boundary Value Problems on Shishkin Mesh. TJMCS. June 2022;14(1):145-154. doi:10.47000/tjmcs.1010528
Chicago Duru, Hakkı, and Mutlu Demirbaş. “A Numerical Method for Solving Singularly Perturbed Quasilinear Boundary Value Problems on Shishkin Mesh”. Turkish Journal of Mathematics and Computer Science 14, no. 1 (June 2022): 145-54. https://doi.org/10.47000/tjmcs.1010528.
EndNote Duru H, Demirbaş M (June 1, 2022) A Numerical Method for Solving Singularly Perturbed Quasilinear Boundary Value Problems on Shishkin Mesh. Turkish Journal of Mathematics and Computer Science 14 1 145–154.
IEEE H. Duru and M. Demirbaş, “A Numerical Method for Solving Singularly Perturbed Quasilinear Boundary Value Problems on Shishkin Mesh”, TJMCS, vol. 14, no. 1, pp. 145–154, 2022, doi: 10.47000/tjmcs.1010528.
ISNAD Duru, Hakkı - Demirbaş, Mutlu. “A Numerical Method for Solving Singularly Perturbed Quasilinear Boundary Value Problems on Shishkin Mesh”. Turkish Journal of Mathematics and Computer Science 14/1 (June 2022), 145-154. https://doi.org/10.47000/tjmcs.1010528.
JAMA Duru H, Demirbaş M. A Numerical Method for Solving Singularly Perturbed Quasilinear Boundary Value Problems on Shishkin Mesh. TJMCS. 2022;14:145–154.
MLA Duru, Hakkı and Mutlu Demirbaş. “A Numerical Method for Solving Singularly Perturbed Quasilinear Boundary Value Problems on Shishkin Mesh”. Turkish Journal of Mathematics and Computer Science, vol. 14, no. 1, 2022, pp. 145-54, doi:10.47000/tjmcs.1010528.
Vancouver Duru H, Demirbaş M. A Numerical Method for Solving Singularly Perturbed Quasilinear Boundary Value Problems on Shishkin Mesh. TJMCS. 2022;14(1):145-54.