Abstract
In this paper a new method called the least squares differential quadrature method (LSDQM) is introduced as a straightforward and efficient method to compute analytical approximate polynomial solutions for nonlinear partial differential equations with fractional time derivatives. LSDQM is a combination of the differential quadrature method and the least squares method and in this paper it is employed to find approximate solutions for a very general class of nonlinear partial differential equations, wherein the fractional derivatives are described in the Caputo sense. The paper contains a clear, step-by-step presentation of the method and a convergence theorem. In order to emphasize the accuracy of LSDQM we included two test problems previously solved by means of other, well-known methods, and observed that our solutions present not only a smaller error but also a much simpler expression. We also included a problem with no known exact solution and the solutions computed by LSDQM are in good agreement with previous ones.
Highlights
Fractional calculus has its beginnings in the late part of the 17th century, when the concept of fractional derivative was developed in a letter between the mathematicians Leibniz and L’Hospital regarding the derivative of order 12 [1]
For several particular problems modeled by fractional order partial differential equations it is possible to find exact solutions
In the cases of most of the problems modeled by fractional order partial differential equations it may be not possible to find exact solutions for the problems, the need to apply numerical or approximate analytical methods
Summary
Fractional calculus has its beginnings in the late part of the 17th century, when the concept of fractional derivative was developed in a letter between the mathematicians Leibniz and L’Hospital regarding the derivative of order 12 [1]. For several particular problems modeled by fractional order partial differential equations it is possible to find exact solutions. In the cases of most of the problems modeled by fractional order partial differential equations it may be not possible to find exact solutions for the problems, the need to apply numerical or approximate analytical methods. In the following we present a few of the recently employed methods for obtaining numerical and analytical approximate solutions of fractional partial differential equations (in no particular order; some overlapping is present since several methods may be included in more than one category):. The LSDQM can be used to solve nonlinear partial differential equations of fractional order of the type: D∗αt u( x, t) = f (u, u x , u xx ), t > 0, ( x, t) ∈ D (1).
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