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The paper reports a spectral method for generating an approximate solution for the space-time fractional PDEs with variable coefficients based on the spectral shifted Jacobi collocation method in conjunction with the shifted Jacobi operational matrix of fractional derivatives. The spectral collocation method investigates both temporal and spatial discretizations. By applying the shifted Jacobi collocation method, the problem reduces to a system of algebraic equations, which greatly simplifies the problem. Numerical results are given to establish the validity and accuracy of the presented procedure for space-time fractional PDE. [ABSTRACT FROM AUTHOR]

In this article, a class of fractional optimal control problems (FOCPs) are solved using a direct method. We present a new operational matrix of the fractional derivative in the sense of Caputo based on the B-spline functions. Then we reduce the solution of fractional optimal control problem to a nonlinear programming (NLP) one, where some existing well-developed algorithms may be applied. Numerical results demonstrate the efficiency of the presented technique. [ABSTRACT FROM AUTHOR]

An effective method based upon cardinal Hermite interpolant multiscaling functions is proposed for the solution of the one-dimensional parabolic partial differential equation with given initial condition and known boundary conditions and subject to overspecification at a point in the spatial domain. The properties of multiscaling functions are first presented. These properties together with a collocation method are then utilized to reduce the parabolic inverse problem to the solution of algebraic equations. The scheme described is efficient. The numerical results obtained using the present algorithms for test problems show that this method can solve the model effectively. [ABSTRACT FROM AUTHOR]

Three numerical techniques based on cubic Hermite spline functions are presented for the solution of Lane-Emden equation. Some properties of Hermite splines are presented and are utilized to reduce the solution of Lane-Emden equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of these techniques. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: A numerical technique based on the finite difference and collocation methods is presented for the solution of generalized Kuramoto–Sivashinsky (GKS) equation. The derivative matrices between any two families of B-spline functions are presented and are utilized to reduce the solution of GKS equation to the solution of linear algebraic equations. Numerical simulations for five test examples have been demonstrated to validate the technique proposed in the current paper. It is found that the simulating results are in good agreement with the exact solutions. [Copyright &y& Elsevier]

To numerically solve the Burgers’ equation, in this paper we propose a general method for constructing wavelet bases on the interval [ 0 , 1 ] derived from symmetric biorthogonal multiwavelets on the real line. In particular, we obtain wavelet bases with simple structures on the interval [ 0 , 1 ] from the Hermite cubic splines. In comparison with all other known constructed wavelets on the interval [ 0 , 1 ] , our constructed wavelet bases on the interval [ 0 , 1 ] from the Hermite cubic splines not only have good approximation and symmetry properties with extremely short supports, but also employ a minimum number of boundary wavelets with a very simple structure. These desirable properties make them to be of particular interest in numerical algorithms. Our constructed wavelet bases on the interval [ 0 , 1 ] are then used to solve the nonlinear Burgers’ equation. Our method is based on the finite difference formula combined with the collocation method. Therefore, our proposed numerical scheme in this paper is abbreviated as MFDCM (Mixed Finite Difference and Collocation Method). Some numerical examples are provided to demonstrate the validity and applicability of our proposed method which can be easily implemented to produce a desired accuracy. [ABSTRACT FROM AUTHOR]