Abstract
AbstractIn this paper, a novel method based on Bessel functions (BF), generalized Bessel functions (GBF), the collocation method and the Jacobian free Newton-Krylov sub-space (JFNK) will be introduced to solve the nonlinear time-fractional Burger equation. In this paper, an implicit formula is introduced to calculate Riemann–Liouville fractional derivative of GBFs, that can be very useful in spectral methods. In this work, the nonlinear time-fractional Burger equation is converted to a nonlinear system of algebraic equations via collocation algorithm based on BFs and GBFs without any linearization and descretization methods. Finally, by using JFNK, the solution of this nonlinear system will be achieved. To show the reliability and applicability of the proposed method, we solve some examples of time-fractional Burger equation and compare our results with others.
Highlights
Solving the nonlinear fractional partial differential equations (FPDE) with high accuracy and high convergent rate is a big challenge among engineer, numerical and mathematic researchers; such that in the last few decades, many mathematicians, numerical analysts and computer scientists have tried to solve these problems by differ-The Burger equation has many application in traffic flow, shock waves in a viscous medium, gas dynamics, etc. [2, 5]
In this paper, a novel method based on Bessel functions (BF), generalized Bessel functions (GBF), the collocation method and the Jacobian free Newton-Krylov sub-space (JFNK) will be introduced to solve the nonlinear time-fractional Burger equation
In this paper, we introduce a new method to solve nonlinear time-fractional Burger equation by using the spectral collocation method based on the Bessel functions and generalized Bessel function of the first kind and JFNK method with adaptive preconditioner
Summary
Solving the nonlinear fractional partial differential equations (FPDE) with high accuracy and high convergent rate is a big challenge among engineer, numerical and mathematic researchers; such that in the last few decades, many mathematicians, numerical analysts and computer scientists have tried to solve these problems by differ-. Solving and finding the solution of the time-fractional Burger equation (1) has been studied for the last half century and still it is an active area of research to develop some better numerical algorithms and methods to approximate its solution [3, 5, 14, 15, 18, 26].
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