Abstract
The linear non-polynomial spline is used here to solve the fractional partial differential equation (FPDE). The fractional derivatives are described in the Caputo sense. The tensor products are given for extending the one-dimensional linear non-polynomial spline to a two-dimensional spline to solve the heat equation. In this paper, the convergence theorem of the method used to the exact solution is proved and the numerical examples show the validity of the method. All computations are implemented by Mathcad15.
Highlights
Fractional calculus is the study of fractional order derivatives and integrals
In [4], a method based the fractional shifted Legendre polynomials was applied to solve non-homogeneous space and time fractional partial differential equations (FPDEs), in which space and time fractional derivatives are described in the Caputo sense
4- Solution of Fractional Heat Equation Using Tensor Product of Linear Non-Polynomial Spline The non-polynomial spline method will be used here to approximate the solution of the failure probability density function (FPDF):
Summary
Fractional calculus is the study of fractional order derivatives and integrals. It gained extensive attention form the researchers in the last few decades. In [1], Saeah solved linear Volterra integral equations using non-polynomial spline function. In [4], a method based the fractional shifted Legendre polynomials was applied to solve non-homogeneous space and time fractional partial differential equations (FPDEs), in which space and time fractional derivatives are described in the Caputo sense. The main objective of the present paper lies on introducing a new approximate solution of time-space fractional heat equations by using the linear non-polynomial spline method. The treatment of high-dimensional problems, such as heat equation, can be approached by concepts of tensor product approximation
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