In this paper, the continuse Legendre wavelets constructed on the interval [0, 1] are used to solve the nonlinear Fredholm integro-DIFFERENTIAL equation. The nonlinear part of integro-DIFFERENTIAL is ap- proximated by Legendre wavelets, and the nonlinear integro-DIFFERENTIAL is reduced to a system of nonlinear EQUATIONS. We give some numerical examples to show applicability of the proposed method.

We study the calculus of variations problem in the presence of a system of DIFFERENTIAL-integral (D-I) EQUATIONS. In order to identify the necessary optimality conditions for this problem, we derive the so-called D-I Euler–Lagrange EQUATIONS. We also generalize this problem to other cases, such as the case of higher orders, the problem of optimal control, and we derive the so-called D-I Pontryagin EQUATIONS. In special cases, these formulations lead to classical Euler–Lagrange EQUATIONS. To illustrate our results, we provide simple examples and applications such as obtaining the minimumpower for an RLC circuit.

In this paper, we prove the existence and uniqueness results to the random fractional functional DIFFERENTIAL EQUATIONS under assumptions more general than the Lipschitz type condition. Moreover, the distance between exact solution and appropriate solution, and the existence extremal solution of the problem is also considered.