Journal of Applied Mathematics and Computation

DOI：http://dx.doi.org/10.26855/jamc.2022.12.003

Date: November 3,2022 Hits: 503

In this study, numerical solutions are obtained for the time-dependent two-dimensional nonlinear parabolic partial differential equations (PDEs) with initial and Dirichlet boundary conditions. In assessing spatial derivatives, we employ the modified Galerkin method with the aid of Green’s theorem, which minimizes the derivatives’ order and incorporates boundary conditions. In the trial function, we use bivariate Bernstein polynomial bases. All the initial and boundary conditions are handled carefully by suitable transformation. Further, we exploit an iterative α-family approximation, especially the Crank Nicolson scheme, to take care the time derivative. Applying the proposed technique to a variety of nonlinear 2D parabolic PDEs, such as the 2D Burger’s equation and the 2D Convection-Diffusion Reaction equation, the numerical results are presented in the form of tables and figures. The numerical results provide conclusive evidence that the technique being proposed is accurate and effective.

[1] Ureña, F., Gavete, L., Garcia, A., Benito, J. J., & Vargas, A. M. (2019). Solving second order non- linear parabolic PDEs using generalized finite difference method (GFDM). Journal of Computational and Applied Mathematics, 354, 221-241.

[2] Saleem, S., and Aziz, I., & Hussain, M. Z. (2020). A simple algorithm for numerical solution of nonlinear parabolic partial differential equations. Engineering with Computers, 36(4), 1763-1775.

[3] Akter, S. I., Mahmud, M. S., Kamrujjaman, M., & Ali, H. (2020). Global spectral collocation method with Fourier transform to solve differential equations. GANIT: Journal of Bangladesh Mathematical Society, 40(1), 28-42.

[4] Polyanin, A. D., & Zaitsev, V. F. (2003). Handbook of Nonlinear Partial Differential Equations: Exact Solutions, Methods, and Problems. Chapman and Hall/CRC.

[5] Friedman, A. (2008). Partial differential equations of parabolic type. Courier Dover Publications.

[6] Lieberman, G. M. (1996). Second order parabolic differential equations. World scientific.

[7] Ladyženskaja, O. A., Solonnikov, V. A., & Uralceva, N. N. (1988). Linear and quasi-linear equations of parabolic type (Vol. 23). American Mathematical Soc.

[8] Ali, H., Kamrujjaman, M., & Islam, M. S. (2020). Numerical computation of FitzHugh-Nagumo equation: Anovel Galerkin finite element approach. International Journal of Mathematical Research, 9(1), 20-27.

[9] Jain, M. K., Jain, R., & Mohanty, R. (1992). Fourth-order finite difference method for 2D parabolic partial differential equations with nonlinear first-derivative terms. Numerical Methods for Partial Differential Equations, 8(1), 21-31.

[10] Kim, D., & Proskurowski, W. (2004). An efficient approach for solving a class of nonlinear 2D parabolic PDEs. International-Journal of Mathematics and Mathematical Sciences, 17, 881-889.

[11] Dehghan, M. (2004). Application of the Adomian decomposition method for two-dimensional parabolic equation subject to non-standard boundary specifications. Applied mathematics and computation, 157(2), 549-560.

[12] Nurwidiyanto, N., Ghani, M. (2022). Numerical results and stability of ADI method to two- dimensional advection-diffusion equations with half step of time. PRISMA, Prosiding Seminar Nasional Matematika, 5, 773-780.

[13] Borcea, L., Druskin, V., Mamonov, A. V., & Zaslavsky, M. (2014). A model reduction approach to numerical inversion for aparabolic partial differential equation. Inverse Problems, 30(12), 125011.

[14] Bhrawy, A. H., Abdelkawy, M. A., & Mallawi, F. (2015). An accurate Chebyshev pseudospectral scheme for multi-dimensional parabolic problems with time delays. Boundary Value Problems, 2015(1), 1-20.

[15] Beneš, M., & Kruis, J. (2018). Multi-time-step domain decomposition and coupling methods for nonlinear parabolic problems. Applied Mathematics and Computation, 319, 444-460.

[16] Guo, X., Shi, B., & Chai, Z. (2018). General propagation lattice Boltzmann model for nonlinear advection-diffusion equations. Physical Review E, 97(4), 043310.

[17] Fu, Z.-J., Tang, Z.-C., Zhao, H.-T., Li, P.-W., & Rabczuk, T. (2019). Numerical solutions of the coupled unsteady nonlinear convection-diffusion equations based on generalized finite difference method. The European Physical Journal Plus, 134(6), 1-20.

[18] Ali, I., Haq, S., Nisar, K. S., & Arifeen, S. U. (2021). Numerical study of 1D and 2D advection-diffusion-reaction equations using Lucas and Fibonacci polynomials. Arabian Journal of Mathematics, 10(3), 513-526.

[19] Lima, S. A., Kamrujjaman, M., & Islam, M. S. (2020). Direct approach to compute a class of reaction-diffusion equation by a finite element method. Journal of Applied Mathematics and Computing, 4(2), 26-33.

[20] Yousefi, S. (2009). Finding a control parameter in a one-dimensional parabolic inverse problem by using the Bernstein Galerkin method. Inverse problems in science and engineering, 17(6), 821-828.

[21] Yousefi, S., Barikbin, Z., Dehghan, M. (2012). Ritz-Galerkin method with Bernstein polynomial basis for finding the product solution form of heat equation with non-classic boundary conditions. International Journal of Numerical Methods for Heat & Fluid Flow, 2012.

[22] Sun, Z., Carrillo, J. A., & Shu, C.-W. (2018). A discontinuous Galerkin method for nonlinear parabolic equations and gradient flow problems with interaction potentials. Journal of Computational Physics, 352, 76-104.

[23] Ali, H., & Kamrujjaman, M. (2022). Numerical solutions of nonlinear parabolic equations with Robin condition: Galerkin approach. TWMS Journal of Applied and Engineering Mathematics, 12(3), 851-863.

[24] Kestler, S., Steih, K., & Urban, K. (2016). An efficient space-time adaptive wavelet Galerkin method for time-periodic parabolic partial differential equations. Mathematics of Computation, 85(299), 1309-1333.

[25] Alam, M., & Islam, M. S. (2019). Numerical solutions of time dependent partial differential equations using weighted residual method with piecewise polynomials. Dhaka University Journal of Science, 67(1), 5-12.

[26] Arora, G., & Joshi, V. (2018). A computational approach for solution of one dimensional parabolic partial differential equation with application in biological processes. Ain Shams Engineering Journal, 9(4), 1141-1150.

[27] Lima, S. A., Kamrujjaman, M., & Islam, M. S. (2021). Numerical solution of convection–diffusion– reaction equations by a finite element method with error correlation. AIP Advances, 11(8), 085225.

[28] Ali, H., & Islam, M. S. (2017). Generalized Galerkin finite element formulation for the numerical solutions of second order non-linear boundary value problems. GANIT: Journal of Bangladesh Mathematical Society, 37, 147-159.

[29] Busch, K., Koenig, M., & Niegemann, J. (2011). Discontinuous Galerkin methods in nanophotonics. Laser & Photonics Reviews, 5(6), 773-809.

[30] Li, X., & Li, S. (2021). A fast element-free Galerkin method for the fractional diffusion-wave equation. Applied Mathematics Letters, 122, 107529.

[31] Wu, Q., Liu, F., & Cheng, Y. (2020). The interpolating element-free Galerkin method for three- dimensional elastoplasticity problems. Engineering Analysis with Boundary Elements, 115, 156-167.

[32] Ali, H., Kamrujjaman, M., & Islam, M. S. (2022). An Advanced Galerkin Approach to Solve the Nonlinear Reaction-Diffusion Equations with Different Boundary Conditions. Journal of Mathematics Research, 14(1).

[33] Kanwal, A., Phang, C., & Iqbal, U. (2018). Numerical solution of fractional diffusion wave equation and fractional Klein–Gordon equation via two-dimensional Genocchi polynomials with a Ritz–Galerkin method. Computation, 6(3), 40.

[34] Nadukandi, P., Oñate, E., & Garcia, J. (2010). A high-resolution Petrov–Galerkin method for the 1 Dconvection–diffusion–reaction problem. Computer methods in applied mechanics and engineering, 199, 525-546.

[35] Lai, W., & Khan, A. (2012). Discontinuous Galerkin method for 1D shallow water flows in natural rivers. Engineering Applications of Computational Fluid Mechanics, 6(1), 74-86.

[36] Li, X., & Dong, H. (2021). An element-free Galerkin method for the obstacle problem. Applied Mathematics Letters, 112, 106724.

[37] Abo-Bakr, R. M., Mohamed, N. A., & Mohamed, S. A. (2022). Meta-heuristic algorithms for solving nonlinear differential equations based on multivariate Bernstein polynomials. Soft Computing, 26(2), 605-619.

[38] Lewis, P. E. & Ward, J. P. (1991). The finite element method: principles and applications. Addison-Wesley Wokingham.

[39] Anton, H., Bivens, I. C., & Davis, S. (2021). Calculus: Early Transcendentals. John Wiley & Sons.

[40] Reddy, J. N. (2014). An Introduction to Nonlinear Finite Element Analysis Second Edition: with applications to heat transfer, fluid mechanics, and solid mechanics. OUP Oxford.

[41] Mohamed, N. (2019). Solving one-and two-dimensional unsteady Burgers’ equation using fully implicit finite difference schemes. Arab Journal of Basic and Applied Sciences, 26(1), 254-268.

[42] Mittal, R., & Tripathi, A. (2015). Numerical solutions of two-dimensional Burgers’ equations using modified Bi-cubic B-spline finite elements. Engineering Computations.

[43] Jianchun, L., Pope, G. A., & Sepehrnoori, K. (1995). A high-resolution finite-difference scheme for nonuniform grids. Applied mathematical modelling, 19(3), 162-172.

[44] Ngondiep, E. (2022). A novel three-level time-split approach for solving two-dimensional nonlinear unsteady convection-diffusion-reaction equation. J.Math.Comput.Sci., 26(3), 222-248.

**On the Numerical Treatment of 2D Nonlinear Parabolic PDEs by the Galerkin Method with Bivariate Bernstein Polynomial Bases**

**How to cite this paper:** Shovan Sourav Datta Pranta, Hazrat Ali, Md. Shafiqul Islam, Md. Shariful Islam. (2022) On the Numerical Treatment of 2D Nonlinear Parabolic PDEs by the Galerkin Method with Bivariate Bernstein Polynomial Bases. *Journal of Applied Mathematics and Computation*, **6**(**4**), 410-422.

**DOI: http://dx.doi.org/10.26855/jamc.2022.12.003**

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