# Effect of the Arbitrary Coefficients on the convergence of numerical solution of General Second Order Linear Homogeneous Partial Differential Equation

## Authors

• Liaquat Ali Zardari Department of Mathematics and Statistics, Quaid-e-Awam University of Engineering, Science and Technology, Nawabshah, Pakistan
• Shakeel Ahmed Kamboh Department of Mathematics and Statistics, QUEST, Quaid-e-Awam University of Engineering, Science and Technology, Nawabshah, Pakistan
• Abbas Ali Ghoto Department of Mathematics and Statistics, QUEST, Quaid-e-Awam University of Engineering, Science and Technology, Nawabshah, Pakistan
• Dr. Kirshan Kumar Luhana Department of Computer Science, Laar Campus, Sindh University Jamshororo, Pakistan
• Dr. Shah Zaman Nizamani Department of Information Technology, Quaid-e-Awam University, Pakistan

## Abstract

In this study the eﬀect of the coecients on the convergence of numerical solution of general second order linear homogeneous partial diﬀerential equation has been investigated. The main objective was to determine the sensitivity of the coecients of the PDE in relation to the domain and mesh size. The nite diﬀerence method was used to discretize the PDE and numerical solution was obtained by implementing the algorithm on MATLAB. The outcomes of the research have provided interesting facts about the stable values of the coecients. From the results it is found that the arbitrary coecients d and e are more sensitive as compared to a, b, c and f. The outcomes of this research study are expected to provide the ways to predict and control the numerical solution convergence behavior obtained by the general second order PDE based on the variable coecients of the PDE.

## References

Agarwal, P., Deniz, S., Jain, S., Alderremy, A. A. and Aly, S. [2020], ‘A new analysis of a partial diﬀerential

equation arising in biology and population genetics via semi analytical techniques’, Physica A: Statistical

Mechanics and its Applications 542, 122769.

Ang, S., Yeo, K., Chew, C. and Shu, C. [2008], ‘A singular-value decomposition (svd)-based generalized finite diﬀerence (gfd) method for close-interaction moving boundary flow problems’, International journal for numerical methods in engineering 76(12), 1892–1929.

Bar-Sinai, Y., Hoyer, S., Hickey, J. and Brenner, M. P. [2019], ‘Learning data-driven discretizations for partial diﬀerential equations’, Proceedings of the National Academy of Sciences 116(31), 15344–15349.

Benito, J., Urena, F. and Gavete, L. [2001], ‘Influence of several factors in the generalized nite diﬀerence

method’, Applied Mathematical Modelling 25(12), 1039–1053.

Cai, Z., Chen, J., Liu, M. and Liu, X. [2020], ‘Deep least-squares methods: An unsupervised learning based numerical method for solving elliptic pdes’, Journal of Computational Physics 420, 109707.

Chan, H.-F., Fan, C.-M. and Kuo, C.-W. [2013], ‘Generalized finite diﬀerence method for solving two dimensional non-linear obstacle problems’, Engineering Analysis with Boundary Elements 37(9), 1189–1196.

Chen, C. S., Ganesh, M., Golberg, M. A. and Cheng, A.-D. [2002], ‘Multilevel compact radial functions based computational schemes for some elliptic problems’, Computers & Mathematics with Applications

(3-5), 359–378.

Gavete, L., Gavete, M. and Benito, J. [2003], ‘Improvements of generalized finite diﬀerence method and comparison with other meshless method’, Applied Mathematical Modelling 27(10), 831–847.

Gu, Y., Chen, W., Gao, H. and Zhang, C. [2016], ‘A meshless singular boundary method for threedimensional elasticity problems’, International Journal for Numerical Methods in Engineering 107(2), 109–126.

Gu, Y., Wang, L., Chen, W., Zhang, C. and He, X. [2017], ‘Application of the meshless generalized finite diﬀerence method to inverse heat source problems’, International Journal of Heat and Mass Transfer 108, 721–729.

Jiao, X. and Zha, H. [2008], Consistent computation of first-and second-order diﬀerential quantities for surface meshes, in ‘Proceedings of the 2008 ACM symposium on Solid and physical modeling’, pp. 159–170.

Karageorghis, A., Lesnic, D. and Marin, L. [2015], ‘The method of fundamental solutions for solving direct and inverse signorini problems’, Computers & Structures 151, 11–19.

Liu, G., Nguyen-Thoi, T., Nguyen-Xuan, H. and Lam, K. [2009], ‘A node-based smoothed finite element method (ns-fem) for upper bound solutions to solid mechanics problems’, Computers & structures 87(1-2), 14–26.

Marin, L. [2010], ‘An alternating iterative mfs algorithm for the cauchy problem for the modified helmholtz equation’, Computational Mechanics 45(6), 665–677.

Mickens, R. E. [2005], ‘Dynamic consistency: a fundamental principle for constructing nonstandard finite diﬀerence schemes for diﬀerential equations’, Journal of diﬀerence equations and Applications 11(7), 645–653.

Nakao, M. T., Plum, M. and Watanabe, Y. [2019], Numerical verification methods and computer-assisted proofs for partial diﬀerential equations, Springer.

Robertsson, J. O. and Blanch, J. O. [2020], ‘Numerical methods, finite diﬀerence’, Encyclopedia of solid earth geophysics pp. 1–9.

Rudy, S., Alla, A., Brunton, S. L. and Kutz, J. N. [2019], ‘Data-driven identification of parametric partial diﬀerential equations’, SIAM Journal on Applied Dynamical Systems 18(2), 643–660.

Rudy, S. H., Brunton, S. L., Proctor, J. L. and Kutz, J. N. [2017], ‘Data-driven discovery of partial diﬀerential equations’, Science advances 3(4), e1602614.

Salsa, S. [2016], Partial diﬀerential equations in action: from modelling to theory, Vol. 99, Springer.

Šarler, B. and Vertnik, R. [2006], ‘Meshfree explicit local radial basis function collocation method for diﬀusion problems’, Computers & Mathematics with applications 51(8), 1269–1282.

Simos, T. and Tsitouras, C. [2017], ‘Evolutionary generation of high-order, explicit, two-step methods for second-order linear ivps’, Mathematical Methods in the Applied Sciences 40(18), 6276–6284.

2022-12-09

## How to Cite

Zardari, L. A., Kamboh, S. A., Ghoto, A. A., Kumar Luhana, D. K., & Nizamani, D. S. Z. (2022). Effect of the Arbitrary Coefficients on the convergence of numerical solution of General Second Order Linear Homogeneous Partial Differential Equation. VFAST Transactions on Mathematics, 10(2), 102–117. https://doi.org/10.21015/vtm.v10i2.1211

Articles