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In this paper, the one- and two-dimensional stochastic multi-order fractional diffusion-wave equations are introduced and a collocation procedure based on the shifted Jacobi polynomials is established to find their numerical solutions. Through this way, some operational matrices regarding classical and stochastic integrals as well as fractional and classical differentiations of these polynomials, are obtained. By representing the problem solution using an expansion of these polynomials (in which the coefficients of the expansion are unknown) and substituting it into the first problem, as well as by employing the obtained operational matrices, a system containing algebraic equations is obtained. Eventually, the coefficients of expansion and subsequently the solution of the original problem are found by solving this system. The correctness of the procedure is studied by solving four examples. [ABSTRACT FROM AUTHOR]

In this paper, a new numerical method for solving fractional optimal control problems by using hybrid functions is presented. The Riemann–Liouville fractional integral operator for hybrid functions is utilized to reduce the solution of optimal control problems to a nonlinear programming one, to which existing, well-developed algorithms may be applied. The method is computationally very attractive and gives very accurate results. [ABSTRACT FROM AUTHOR]

This paper examines a time fractional version of the 2D Fokker-Planck equation involved with the Atangana-Baleanu-Caputo fractional derivative, under the Dirichlet boundary conditions. A fully discretization approach based on the meshless local Petrov-Galerkin method and (3 − β) -order approximation is proposed for this equation. More precisely, we apply the meshless local Petrov-Galerkin method based on the Moving Kriging interpolation to discretize the space domain, and utilize the (3 − β) -order approximation together with the θ -weighted finite difference method to discretize the temporal domain. By implementing this method, we get a solution for the problem by solving a system of algebraic equations. The validity of the method is investigated by solving four examples. [ABSTRACT FROM AUTHOR]

In this research, by utilizing the concept of the mixed Caputo fractional derivative and left-sided mixed Riemann–Liouville fractional integral, we approximate the solution of generalized fractional Benjamin–Bona–Mahony–Burgers equations (GF-BBMBEs). In addition, using Genocchi polynomial properties, we obtain a new formula to approximate the functions by Genocchi polynomials. In the process of computation, we discuss a method of obtaining the operational matrix of integration and pseudo-operational matrices of the fractional order of derivative. Also, an algorithm of obtaining the mixed fractional integral operational matrix is presented. Using the collocation method and matrices introduced, the proposed equations are converted to a system of nonlinear algebraic equations with unknown Genocchi coefficients. In addition, we discuss the upper bound of the error for the proposed method. Finally, we examine several problems to demonstrate the validity and applicability of the proposed method. [ABSTRACT FROM AUTHOR]

This paper presents a new numerical method for solving fractional optimal control problems (FOCPs). The fractional derivative in the dynamic system is described in the Caputo sense. The method is based upon Bernoulli polynomials. The operational matrices of fractional Riemann–Liouville integration and multiplication for Bernoulli polynomials are derived. The error upper bound for the operational matrix of the fractional integration is also given. The properties of Bernoulli polynomials are utilized to reduce the given optimization problems to the system of algebraic equations. By using Newton’s iterative method, this system is solved and the solution of FOCPs are achieved. Illustrative examples are included to demonstrate the validity and applicability of the technique. [ABSTRACT FROM AUTHOR]

In this paper, a new numerical method for solving fractional differential equations is presented. The fractional derivative is described in the Caputo sense. The method is based upon Bernoulli wavelet approximations. The Bernoulli wavelet is first presented. An operational matrix of fractional order integration is derived and is utilized to reduce the initial and boundary value problems to system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. [ABSTRACT FROM AUTHOR]

We present an efficient computational procedure for the solution of bang–bang optimal control problems. The method is based on a well-known adaptive control parametrization method, which is one of the direct methods for numerical solution of optimal control problems. First, the adaptive control parametrization method is reviewed and then its advantages and disadvantages are illustrated. In order to resolve the need for a priori knowledge about the structure of optimal control and for resolving the sensitivity to an initial guess, a homotopy continuation technique is combined with the adaptive control parametrization method. The present combined method does not require any assumptions on the control structure and the number of switching points. In addition, the switching points are captured accurately; also, efficiency of the method is reported through illustrative examples. [ABSTRACT FROM PUBLISHER]