Gurpreet Singh Bhatia ¹, Geeta Arora ¹

Affiliations
1 Department of Mathematics, Lovely Professional University, Punjab, 144411,, IN

Abstract

Background/Objectives: The approximation using radial basis function (RBF) is an extremely powerful method to solve partial differential equations (PDEs). This paper presents different types of RBF methods to solve PDEs. Methods/ Statistical Analysis: Due to their meshfree nature, ease of implementation and independence of dimension, RBF methods are popular to solve PDEs. In this paper we examine different generalized RBF methods, including Kansa method, Hermite symmetric approach, localized and hybrid methods. We also discussed the preference of using meshfree methods like RBF over the mesh based methods. Findings: This paper presents a state-of-the-art review of the RBF methods. Some recent development of RBF approximation in solving PDEs is also discussed. The mathematical formulation of different RBF methods are discussed for better understanding. RBF methods have been actively developed over the years from global to local approximation and then to hybrid methods. Hybrid RBF methods help in reduction of computational cost and become very effective in solving large scale problems. Application/Improvements: RBF methods have been applied to various diverse fields like image processing, geo-modeling, pricing option and neural network etc.

Keywords

Differential Quadrature Radial Basis Function, Kansa Collocation Method, Partition of unity, Partial Differential Equation, Radial Basis Function.

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Geeta Arora ¹, Varun Joshi ¹

Affiliations
1 Department of Mathematics, Lovely Professional University, Punjab − 144411, IN

Abstract

Objectives: This paper aims to compute the approximate solution of one dimensional (1D) hyperbolic telegraph equation with appropriate primary and limiting conditions. Methods/Statistical Analysis: To find the approximate solution, two different modified spline basis function are used with the differential quadrature method and splines are used to compute the weighting coefficients and thus the equation is transformed to a set of first order conventional differential equations which is further solved by the SSP-RK43 method. Three test problems of this equation are simulated to establish the precision and usefulness of the proposed scheme. Findings: The obtained numerical results are found to be good in terms of accuracy, efficiency and simplicity. To validate the computed results using proposed scheme, various comparisons at different time levels has been done in the form of and errors. These errors are compared and enlisted in the form of tables with computed errors enlisted in literature. Application/Improvements: Being an important equation of nuclear material science, one dimensional (1D) hyperbolic telegraphs equation needs to take care in the sense of better numerical solution. In this context, a successful effort has been done in this research article by proposing a hybrid numerical scheme.

Keywords

35K57, Differential Quadrature Method, Hyperbolic Telegraphs Equation, Mathematics Subject Classification (2010): 65M06, Modified Cubic B-Spline, Modified Trigonometric B-Spline, SSP-RK43.

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