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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]

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]