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2 Fundamenta Informaticae

2 2017

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An Optimization Wavelet Method for Multi Variable-order Fractional Differential Equations

Heydari M. H., Hooshmandasl M. R., Cattani C., Hariharan G.

Fundamenta Informaticae

2017

Vol. 151, nr 1/4

255--273

In this paper, a new operational matrix of variable-order fractional derivative (OMV-FD) is derived for the second kind Chebyshev wavelets (SKCWs). Moreover, a new optimization wavelet method based on SKCWs is proposed to solve multi variable-order fractional differential equations (MV-FDEs). In the proposed method, the solution of the problem under consideration is expanded in terms of SKCWs. Then, the residual function and its errors in 2-norm are employed for converting the problem under study to an optimization one, which optimally chooses the unknown coefficients. Finally, the method of constrained extremum is applied, which consists of adjoining the constraint equations obtained from the given initial conditions to the object function obtained from residual function by a set of unknown Lagrange multipliers. The main advantage of this approach is that it reduces such problems to those optimization problems, which greatly simplifies them and also leads to obtain a good approximate solution for them.

A New Optimization Method Based on Generalized Polynomials for Fractional Differential Equations

Hassani H., Dahaghin M. S., Heydari M. H., Eshkaftaki A. B.

Fundamenta Informaticae

2017

Vol. 151, nr 1/4

443--457

In this paper, we present a new optimization method based on a new class of functions, namely generalized polynomials (GPs) for solving linear and nonlinear fractional differential equations (FDEs). In the proposed method, the solution of the problem under study is expanded in terms of the GPs with fixed coefficients, free coefficients and control parameters. The initial conditions are employed to compute the fixed coefficients. The residual function and its ǁ.ǁ2 are employed for converting the problem under consideration to an optimization one and then choosing the unknown free coefficients and control parameters optimally. As a useful result, the necessary conditions of optimality are derived as a system of nonlinear algebraic equations with unknown free coefficients and control parameters. The validity and accuracy of the approach are illustrated by some numerical examples. The obtained results show that the proposed method is very efficient and accurate.