ELSIM—A PC program for electrochemical kinetic simulations. Version 2.0—solution of the sets of kinetic partial differential equations in one-dimensional geometry, using finite difference and orthogonal collocation methods

Background: In this paper, a modified Taylor wavelet Galerkin method (MTWGM) based on approximation scheme is used to solve partial differential equations (PDEs), which is play an important role in electrical circuit models. Objective: The objective of this work is to give fine and accurate implementation of proposed method for the solution of PDEs, which is the best tool for the analysis of electric circuit problems. Methods: In this work, we used an effective, modified Taylor wavelet Galerkin method with its residual technique and we obtained more accurate numerical solution of the one dimensional PDEs. The Introduced wavelet method is more efficiently applicable in the comparison of some existing numerical methods such as, finite difference method, finite element method, finite volume method, spectral method etc. This method is the best tool for solving PDEs. Therefore, it has significance in the field of electrical engineering and others. Results: The experimentally four numerical problems are given which are showing the numerical results extractive by introduced method and those results compared with exact solution and other available numerical methods i.e., Hermite wavelet Galarkin method (HWGM), Finite difference method (FDM) and spectral procedures which shows that proposed method is more effective. Conclusion: This work is significantly helpful for the electrical circuits in which the governing models are available in the form of PDEs.

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Existence of weak solutions to SPDEs with fractional Laplacian and non-Lipschitz coefficients

Stochastic Partial Differential Equations Analysis and Computations ◽

10.1007/s40072-021-00199-6 ◽

2021 ◽

Author(s):

Shohei Nakajima

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Weak Solutions ◽

Stochastic Partial Differential Equations ◽

Fractional Laplacian ◽

Existence Of Solutions ◽

Partial Differential ◽

Continuity Theorem ◽

Existence Of Weak Solutions ◽

One Dimensional

AbstractWe prove existence of solutions and its properties for a one-dimensional stochastic partial differential equations with fractional Laplacian and non-Lipschitz coefficients. The method of proof is eatablished by Kolmogorov’s continuity theorem and tightness arguments.

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

Advances in Difference Equations ◽

10.1186/s13662-021-03472-z ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Raheel Kamal ◽

Kamran ◽

Gul Rahmat ◽

Ali Ahmadian ◽

Noreen Izza Arshad ◽

...

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Laplace Transform ◽

Fractional Derivative ◽

Meshless Method ◽

Inverse Laplace Transform ◽

Contour Integral ◽

Partial Differential ◽

One Dimensional ◽

The Laplace Transform

AbstractIn 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.

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