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In this article, Ritz approximation have been employed to obtain numerical solutions of fractional partial differential equations (FPDEs) based on the Caputo fractional derivative. Transforming fractional partial differential equations into optimization problem and using polynomial basis functions, we obtain the system of algebraic equation. Then, we solve the system of nonlinear algebraic equation using Mathematica7 and we have the coefficients of polynomial expansion. We extensively discuss the convergence of the method. Some numerical examples are presented which illustrate the theoretical results and the performance of the method. [ABSTRACT FROM AUTHOR]

This paper presents a numerical solution for solving a nonlinear 2-D optimal control problem (2DOP). The performance index of a nonlinear 2DOP is described with a state and a control function. Furthermore, dynamic constraint of the system is given by a classical diffusion equation. It is preferred to use the Ritz method for finding the numerical solution of the problem. The method is based upon the Legendre polynomial basis. By using this method, the given optimisation nonlinear 2DOP reduces to the problem of solving a system of algebraic equations. The benefit of the method is that it provides greater flexibility in which the given initial and boundary conditions of the problem are imposed. Moreover, compared with the eigenfunction method, the satisfactory results are obtained only in a small number of polynomials order. This numerical approach is applicable and effective for such a kind of nonlinear 2DOP. The convergence of the method is extensively discussed and finally two illustrative examples are included to observe the validity and applicability of the new technique developed in the current work. [ABSTRACT FROM AUTHOR]

Abstract: A numerical direct method for solving a general class of fractional optimal control problems (FOCPs) is presented. In the discussed FOCP, the fractional derivative in the dynamical system is considered in the Caputo sense. To solve the problem, first the FOCP is transformed into an equivalent variational problem, then using the Legendre orthonormal basis, the problem is reduced to the problem of solving a system of algebraic equations. With the aid of an operational matrix of Riemann–Liouville fractional integration, Gauss quadrature formula and Newton’s iterative method for solving a system of algebraic equations, the problem is solved approximately. Approximations achieved by this method satisfy all the initial conditions of the problem which is an important property. The convergence of the method is extensively discussed and finally some illustrative examples are included to demonstrate the applicability of the new technique. [Copyright &y& Elsevier]

Abstract: This paper presents a numerical method for solving a class of fractional variational problems (FVPs) with multiple dependent variables, multi order fractional derivatives and a group of boundary conditions. The fractional derivative in the problem is in the Caputo sense. In the presented method, the given optimization problem reduces to a system of algebraic equations using polynomial basis functions. An approximate solution for the FVP is achieved by solving the system. The choice of polynomial basis functions provides the method with such a flexibility that initial and boundary conditions can be easily imposed. We extensively discuss the convergence of the method and finally present illustrative examples to demonstrate validity and applicability of the new technique. [Copyright &y& Elsevier]

Abstract: In this paper a numerical method for solving the nonlinear age-structured population models is presented which is based on Bernstein polynomials approximation. Operational matrices of integration, differentiation, dual and product are introduced and are utilized to reduce the age-structured population problem to the solution of algebraic equations. The method in general is easy to implement, and yields good results. Illustrative examples are included to demonstrate the validity and applicability of the new technique. [Copyright &y& Elsevier]

Abstract: In this article we are going to present necessary conditions which must be satisfied to make the fractional variational problems (FVPs) with completely free boundary conditions have an extremum. The fractional derivatives are defined in the Caputo sense. First we present the necessary conditions for the problem with only one dependent variable, and then we generalize them to problems with multiple dependent variables. We also find the transversality conditions for when each end point lies on a given arbitrary curve in the case of a single variable or a surface in the case of multiple variables. It is also shown that in special cases such as those with specified and unspecified boundary conditions and problems with integer order derivatives, the new results reduce to the known necessary conditions. Some examples are presented to demonstrate the applicability of the new formulations. [Copyright &y& Elsevier]

Abstract: In this paper, we develop a framework to obtain approximate numerical solutions to ordinary differential equations (ODEs) involving fractional order derivatives using Legendre wavelet approximations. The properties of Legendre wavelets are first presented. These properties are then utilized to reduce the fractional ordinary differential equations (FODEs) to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. Results show that this technique can solve the linear and nonlinear fractional ordinary differential equations with negligible error compared to the exact solution. [Copyright &y& Elsevier]

Abstract: Some physical problems in science and engineering are modelled by the parabolic partial differential equations with nonlocal boundary specifications. In this paper, a numerical method which employs the Bernstein polynomials basis is implemented to give the approximate solution of a parabolic partial differential equation with boundary integral conditions. The properties of Bernstein polynomials, and the operational matrices for integration, differentiation and the product are introduced and are utilized to reduce the solution of the given parabolic partial differential equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the new technique. [Copyright &y& Elsevier]

In this paper, the variational iteration method is implemented to give approximate solution of the Euler-Lagrange, Euler-Poisson and Euler-Ostrogradsky equations as ordinary (or partial) differential equations which arise from the variational problems. In this method, general Lagrange multipliers are introduced to construct correction functional for the variational problems. The multipliers in the functionals can be identified optimally via the variational theory. The initial approximations can be freely chosen with possible unknown constant, which can be determined by imposing the boundary or initial conditions. Illustrative examples are included to demonstrate the validity and applicability of the variational iteration method. [ABSTRACT FROM AUTHOR]

In this article, He's variational iteration method has been applied to find the optimal control of linear systems, approximately. Numerical results are given for several test examples involving scalar and second-order systems to demonstrate the applicability and efficiency of the method. [ABSTRACT FROM AUTHOR]

A numerical method for solving Abel's integral equation as singular Volterra integral equations is presented. The method is based upon Bernstein polynomial (B-polynomial) multiwavelet basis approximations. The properties of B-polynomial multiwavelets are first presented. These properties are then utilized to reduce the singular Volterra integral equations to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. [ABSTRACT FROM AUTHOR]

Abstract: In this paper He’s variational iteration method is used to give the approximate solution of nonlinear mixed Volterra–Fredholm integral equations. The method constructs a convergent sequence of functions, which approximates the exact solution with few iterations. Numerical results and a comparison with the exact solution are given, which reveal its efficiency. [Copyright &y& Elsevier]

An inverse problem concerning a diffusion equation with source control parameter is considered. The approximation of the problem is based on the Bernstein polynomial basis. The properties of Bernstein polynomials are first presented. The Bernstein polynomial basis vanishes except the first polynomial at x = 0, which is equal to 1 and the last polynomial at x = R, which is also equal to 1 over the interval [0, R]. This provides greater flexibility in which the boundary conditions are imposed at the end points of the interval. These properties together with the Galerkin method are then utilized to reduce the inverse problem to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a numerical method for solving a class of fractional optimal control problems (FOCPs). The fractional derivative in these problems is in the Caputo sense. The method is based upon the Legendre orthonormal polynomial basis. The operational matrices of fractional Riemann–Liouville integration and multiplication, along with the Lagrange multiplier method for the constrained extremum are considered. By this method, the given optimization problem reduces to the problem of solving a system of algebraic equations. By solving this system, we achieve the solution of the FOCP. Illustrative examples are included to demonstrate the validity and applicability of the new technique. [Copyright &y& Elsevier]