An efficient numerical scheme based on Lucas polynomials for the study of multidimensional Burgers-type equations

Advances in Difference Equations

article

Ali, Ihteram¹  Haq, Sirajul¹  Nisar, Kottakkaran Sooppy²  Baleanu, Dumitru³ 

[1] Faculty of Engineering Sciences, GIK Institute;Department of Mathematics, College of Arts and Sciences, Prince Sattam bin Abdulaziz University;Department of Mathematics, Cankaya University;Institute of Space Sciences;Department of Medical Research, China Medical University Hospital, China Medical University

关键词: Lucas polynomials;    Fibonacci polynomials;    Finite differences;    Stability analysis;   

DOI  :  10.1186/s13662-020-03160-4

学科分类：航空航天科学

来源: SpringerOpen

【 摘 要 】

We propose a polynomial-based numerical scheme for solving some important nonlinear partial differential equations (PDEs). In the proposed technique, the temporal part is discretized by finite difference method together with θ-weighted scheme. Then, for the approximation of spatial part of unknown function and its spatial derivatives, we use a mixed approach based on Lucas and Fibonacci polynomials. With the help of these approximations, we transform the nonlinear partial differential equation to a system of algebraic equations, which can be easily handled. We test the performance of the method on the generalized Burgers–Huxley and Burgers–Fisher equations, and one- and two-dimensional coupled Burgers equations. To compare the efficiency and accuracy of the proposed scheme, we computed$L_{\infty }$ ,$L_{2}$ , and root mean square (RMS) error norms. Computations validate that the proposed method produces better results than other numerical methods. We also discussed and confirmed the stability of the technique.

【 授权许可】

CC BY