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.

The collocation methods based on cubic B-spline, are developed to approximate solution of the second and first kind Fredholm Integral equations. First we collocate the solution by B-spline and the Newton-Cotes formula is used to approximate Integral. Convergence analysis has been investigated and proved that the quadratur rule is fourth order convergent. The presented methods are tested to the problem, and the absolute error in the solution are compared with existing methods [1, 2, 7, 10] to verify the accuracy and convergent nature of proposed methods.

In this research paper, we introduce a new transform belonging to the class of Laplace transforms, called the Yasser-Jassim Transform (YJ Transform). We explore its properties and compare it to the classical Laplace transform. Furthermore, we provide proofs for the key properties associated with this transform and demonstrate its application in solving differential and Integral equations. By employing this new transform, we can reduce the original problem to an algebraic equation that can be solved directly, followed by applying the inverse transform to obtain the solution to the original problem.

These days, Integral equations govern a wide range of physical processes, including those related to electricity, mechanics, fluid dynamics, ecology, soil moisture dynamics, shallow water wave propagation, and chemical science. As a result, creating new techniques and putting them into practice for solving Integral equations becomes more crucial. In this paper, the Upadhyaya Integral transform is employed for determining the analytical solutions of the non-linear Volterra Integral equations (NVIEs) of the first kind. Four examples suggest the effectiveness of the Upadhyaya Integral transform, particularly for NVIEs of the first kind. The calculation results suggest that the proposed method provides accurate solutions to the original problem and this method is a valuable tool for researchers and scientists working on the broader range of problems involving NVIEs of the first kind.