Numerical Solution of  Nonlinear Partial Differential Equations Using Adaptive Sinc Collocation

Jinan Salman

Jinan Hraiga Salman Islamic Azad university, South Tehran Branch, Faculty of Technical Engineering

Keywords: Adaptive Sinc Collocation Method (ASCM), Nonlinear Partial Differential Equations (PDEs), Sinc Function, Nonlinear Schrödinger Equations, Burgers’ Equations, Reaction-Diffusion Equations, Computational Efficiency, Finite Difference Method (FDM), Finite Element Method (FEM), High-Dimensional Matrices

Abstract

This work examines the use of ASCM to approximate the solutions for numerous PDEs that are nonlinear in nature. The ASCM, is a new approach formulated to solve complex nonlinear PDEs which makes use of the sinc function as it provides small computational overhead although the solution is very complex. The method is also able to adjust the position of the collocation points with respect to the solution and the current phase, to save computational costs, while increasing the accuracy. For this reason, the ASCM is well suited to solve nonlinear problems that present steep slopes, singularities or difficulties in behavior. The proposed ASCM has been tested with several nonlinear PDEs, for example, the nonlinear Schrödinger equations, Burgers’ equations, and reaction-diffusion equations. Computational cessions from this study show that ASCM gives approximate solutions to partial differential equation models as appropriately as or even better than the most conventional solution techniques including FDM and FEM while employing significantly less computation time. The results established show the method’s effectiveness in capturing problems with steep gradients and non-linearities, providing accurate solutions in less time and from fewer collocation points as compared to more traditional approaches. Some advantages of ASCM, such as high accuracy, time-saving and versatility are articulated and its performance is compared with other numerical approaches. Furthermore, the study focuses on matrices in high dimensions as well as geometries of varying complexity with respect to ASCM. Last but not least, possible applications to future developments of the method are presented, such as time-dependent and multiscale method and combining this method with other methods to enrich the method to solve more realistic non-linear PDEs. The results point out that further investigations of ASCM as a tool for solving complicated nonlinear PDEs will be fruitful for a wide range of applied science and engineering disciplines.

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