in this paper, an effective direct method to determine the numerical solution of linear and nonlinear fredholm and volterra integral and integro-differential equations is proposed. the method is based on expanding the required approximate solution as the elements of chebyshev cardinal functions. the operational matrices for the integration and product of the chebyshev cardinal functions are des...

Delay-Differential Equations (DDEs) are the most common representation for systems with delay. However, DDE is limited. In network models delay, delayed channels low-dimensional and accounting this heterogeneity not possible in framework. addition, DDEs cannot be used to model difference equations. Furthermore, estimation control of format has proven challenging, despite decades study. paper, w...

Differential equations involving causal operators have gained much attention of late and some results are assembled in a recent monograph [1]. The term causal is adopted from the engineering literature. Basically, a causal operator is a nonanticipative operator. The theory of these operators has the powerful quality of unifying ordinary differential equations, integrodifferential equations, dif...

One of the open problems in numerical analysis solutions to high-dimensional nonlinear integral equations with memory kernel and proportional delay is how preserve high-order accuracy for nonsmooth solutions. It well-known that these display a typical weak singularity at initial time, which causes challenges developing efficient algorithms. The key idea proposed approach adopt smoothing transfo...

Lyapunov functions/functionals have found their footing in Volterra integro-differential equations. This is not the case for integral equations, and it therefore further explored this paper. In manuscript, we utilize functionals combined with Laplace transform to qualitatively analyze solutions of equation addition, extend our method nonlinear equations infinite delay, several kernels. We menti...