This study investigates a time-domain method for modeling general transient elastodynamic problems using the Spectral-based finite element method (SEM) which is based upon a conforming mesh of two-dimensional quadrilaterals. Employing the Galerkin weighted residual method, detailed formulation of the SEM is derived in which various aspects involving in elastodynamic problems are discussed. The accuracy and efficiency of the method is fully demonstrated by comparing results obtained from the SEM with those reported in other studies. For this purpose, a set of wave propagation and structural dynamic problems, subjected to various load forms such as triangular load, Heaviside step load, sinusoidal impulsive load, and ramped load are modeled using the SEM. Furthermore, support motion boundary conditions are examined using the SEM. Each problem is successfully modeled using a very small number of degrees of freedom in comparison with other numerical methods. The numerical results agree very well with the analytical solutions as well the results from other numerical methods.

Herein, we have proposed a scheme for numerically solving hyperbolic partial differential equations (HPDEs) with given initial conditions. The operational matrix of differentiation for exponential Jacobi functions was derived, and then a collocation method was used to transform the given HPDE into a linear system of equations. The preferences of using the exponential Jacobi Spectral collocation method over other techniques were discussed. The convergence and error analyses were discussed in detail. The validity and accuracy of the proposed method are investigated and checked through numerical experiments.

This work presented here is the solution of the one-dimensional Bratus problem. The nonlinear Bratus problem is first linearised using the quasi-linearization method and then solved by the Spectral element method. We use the Legendre polynomials for interpolation. Finally we show the results with a numerical example.

In this paper, we intend to introduce the Sturm-Liouville fractional problem and solve it using the collocation method based on Chebyshev cardinal polynomials. To this end, we first provide an introduction to the Sturm-Liouville fractional equation. Then the Chebyshev cardinal functions are introduced along with some of their properties and the operational matrices of the derivative, fractional integral, and Caputo fractional derivative are obtained for it. Here, for the first time, we solve the equation using the operational matrix of the fractional derivative without converting it to the corresponding integral equation. In addition to efficiency and accuracy, the proposed method is simple and applicable. The convergence of the method is investigated, and an example is presented to show its accuracy and efficiency.

Optimal control theory is a branch of mathematics. It is developed to find optimal ways to control a dynamic system. In 1957, R.Bellman applied dynamic programming to solve optimal control of discrete-time systems. His procedure resulted in closed-loop,  generally nonlinear, and feedback schemes. Optimal control problems which will be tackled involve the minimization of a cost function subject to constraints on the state vector and the control. Lagrange multipliers provide a method of converting a constrained minimization problem into an unconstrained minimization problem of higher order. The necessary condition for optimality can be obtained as the solution of the unconstrained optimization problem of the Lagrange function and the bordered Hessian matrix is used for the second-derivative test. A Spectral method for solving optimal control problems is presented. The method is based on Bernoulli polynomials approximation. By using the Bernoulli operational matrix of integration and the Lagrangian function, stochastic optimal control is transformed into an optimisation problem, where the unknown Bernoulli coefficients are determined by using Newton's iterative method. The convergence analysis of the proposed method is given. The simulation results based on the Monte-Carlo technique prove the performance of the proposed method. Some error estimations are provided and illustrative examples are also included to demonstrate the efficiency and applicability of the proposed method.