Abstract
The purpose of this paper is to present a new kind of analytical method, the so-called residual power series, to predict and represent the multiplicity of solutions to nonlinear boundary value problems of fractional order. The present method is capable of calculating all branches of solutions simultaneously, even if these multiple solutions are very close and thus rather difficult to distinguish even by numerical techniques. To verify the computational efficiency of the designed proposed technique, two nonlinear models are performed, one of them arises in mixed convection flows and the other one arises in heat transfer, which both admit multiple solutions. The results reveal that the method is very effective, straightforward, and powerful for formulating these multiple solutions.
Highlights
Multiple or dual solutions of nonlinear boundary value problems (BVPs) of fractional order are an interesting subject in the area of mathematics, physics, and engineering
It is more consequential not to lose any solution of nonlinear BVPs of fractional order due to their wide application in scientific and engineering research. Based on this important fact, the present paper is going to present an analytical method, the so-called residual power series (RPS), that enables us to predict the multiplicity of solutions which nonlinear BVP of fractional order admits and to calculate the multiple solutions analytically at the same time
In order to verify the computational efficiency of the designed RPS method, two nonlinear models are performed, one of them arises in mixed convection flows and the other one arises in heat transfer, which both admit multiple solutions
Summary
Multiple or dual solutions of nonlinear boundary value problems (BVPs) of fractional order are an interesting subject in the area of mathematics, physics, and engineering. The advent of computers, it has, become more common to treat differential equations directly, using different approximation method instead of series expansions, but in connection with the development of automatic methods for formula manipulation, one can anticipate renewed interest in series methods These methods have some advantages, especially in multidimensional and multiple solutions for BVPs of fractional order. The RPS method was developed by the first author [28] as an efficient method for determining values of coefficients of the power series solution for first and the second-order fuzzy differential equations.
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