Abstract
A table lookup method for solving nonlinear fractional partial differential equations (fPDEs) is proposed in this paper. Looking up the corresponding tables, we can quickly obtain the exact analytical solutions of fPDEs by using this method. To illustrate the validity of the method, we apply it to construct the exact analytical solutions of four nonlinear fPDEs, namely, the time fractional simplified MCH equation, the space-time fractional combined KdV-mKdV equation, the (2+1)-dimensional time fractional Zoomeron equation, and the space-time fractional ZKBBM equation. As a result, many new types of exact analytical solutions are obtained including triangular periodic solution, hyperbolic function solution, singular solution, multiple solitary wave solution, and Jacobi elliptic function solution.
Highlights
Fractional partial differential equations are the generalized form of the integer order differential equations. fPDEs can more accurately describe the complex physical phenomena occurring in fluid dynamics, high-energy physics, plasma physics, elastic media, optical fibers, chemical kinematics, chemical physics, acoustic waves, biomathematics, and many other areas [1, 2]
We propose a table lookup method in this paper. This method is straightforward and has small computational cost. We apply it to solve nonlinear fractional order partial differential equations with using the fractional complex transform and the modified RiemannLiouville derivative defined by Jumarie [33]
We use the proposed methods to construct the exact solutions of the following nonlinear (2 + 1)dimensional time fractional Zoomeron equation [39]: Dt2tα
Summary
Fractional partial differential equations (fPDEs) are the generalized form of the integer order differential equations. fPDEs can more accurately describe the complex physical phenomena occurring in fluid dynamics, high-energy physics, plasma physics, elastic media, optical fibers, chemical kinematics, chemical physics, acoustic waves, biomathematics, and many other areas [1, 2]. Fractional partial differential equations (fPDEs) are the generalized form of the integer order differential equations. We propose a table lookup method in this paper. This method is straightforward and has small computational cost. We apply it to solve nonlinear fractional order partial differential equations with using the fractional complex transform and the modified RiemannLiouville derivative defined by Jumarie [33]. Jumarie’s modified Riemann-Liouville derivative of order α is defined by the following expression [34]: Dxαf (x) = {{{{{{{{{{{{{{{[ΓΓf(((11α11−−−n)αα())xd∫)d]0xx(n(∫)x0,x−(xξ)−−αξ−)1−α[f[f(ξ()ξ−) −ff(0()0])d] ξdξ α < 0, 0 < α < 1, n ≤ α < n + 1, n ≥ 1,.
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