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.

Hybrid system is a dynamic system that exhibits both continuous and discrete dynamic behavior. The hybrid DIFFERENTIAL EQUATIONS have a wide range of applications in science and engineering. The hybrid systems are devoted to modeling, design, and validation of interactive systems of computer programs and continuous systems. Hybrid fuzzy DIFFERENTIAL EQUATIONS (HFDEs) is considered by Kim et al. [11]. In the present paper it is shown that the example presented by Kim et al. in the Case I is not very accurate and in the Case II, is incorrect. Namely, the exact solution proposed by the authors in the Case II are not solutions of the given HFDE. The correct exact solution is also presented here, together with some results for characterizing solutions of FDEs under Hukuhara differentiability by an equivalent system of ODEs. Then, the homotopy analysis method (HAM) is applied to obtained the series solution of the HFDEs. Finally, we illustrate our approach by a numerical example.

In this paper, we propose a new type of fuzzy fractional DIFFERENTIAL EQUATIONS under fuzzy Kolwankar- Gangal local fractional derivatives (for short, fuzzy KG-LFD) using fuzzy Riemann-Liouville differentiability. Then, we prove some basic results in this area like, the relation between different types of fuzzy KG-LFD and their r-cut representations, composition of the fuzzy KG-LFD and the fuzzy local fractional integral. Also, an application is provided in details such that the explicit solutions are expressed through Mittag-Leffler function.

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.

We expand a new generalization of the two-dimensional DIFFERENTIAL trans-form method. The new generalization is based on the two-dimensional differ-ential transform method, fractional power series expansions, and conformable fractional derivative. We use the new method for solving a nonlinear con-formable fractional partial DIFFERENTIAL equation and a system of conformable fractional partial DIFFERENTIAL equation. Finally, numerical examples are pre-sented to illustrate the preciseness and effectiveness of the new technique.