Approximation of linear one dimensional partial differential equations including fractional derivative with non-singular kernel

Advances in Difference Equations

article

Kamal, Raheel¹  Kamran¹  Rahmat, Gul¹  Ahmadian, Ali²  Arshad, Noreen Izza³  Salahshour, Soheil⁴ 

[1] Department of Mathematics, Islamia College Peshawar;Institute of IR 4.0, The National University of Malaysia;Positive Computing Research Group, Institute of Autonomous Systems, Department of Computer & Information Sciences, Universiti Teknologi Petronas;Young Researchers and Elite Club, Mobarakeh Branch, Islamic Azad University;Faculty of Engineering and Natural Sciences, Bahcesehir University

关键词: Local meshless method;    Linear partial differential equations;    Laplace transform;    Caputo Fabrizio fractional derivative;   

DOI  :  10.1186/s13662-021-03472-z

学科分类：航空航天科学

来源: SpringerOpen

【 摘 要 】

In this article we propose a hybrid method based on a local meshless method and the Laplace transform for approximating the solution of linear one dimensional partial differential equations in the sense of the Caputo–Fabrizio fractional derivative. In our numerical scheme the Laplace transform is used to avoid the time stepping procedure, and the local meshless method is used to produce sparse differentiation matrices and avoid the ill conditioning issues resulting in global meshless methods. Our numerical method comprises three steps. In the first step we transform the given equation to an equivalent time independent equation. Secondly the reduced equation is solved via a local meshless method. Finally, the solution of the original equation is obtained via the inverse Laplace transform by representing it as a contour integral in the complex left half plane. The contour integral is then approximated using the trapezoidal rule. The stability and convergence of the method are discussed. The efficiency, efficacy, and accuracy of the proposed method are assessed using four different problems. Numerical approximations of these problems are obtained and validated against exact solutions. The obtained results show that the proposed method can solve such types of problems efficiently.

【 授权许可】

CC BY