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

In this paper, a new operational matrix of variable-order fractional derivative (OMVFD) 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.

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