Toeplitz Matrix Method and Volterra-Hammerstien Integral Equation with a Generalized Singular Kernel

In this work, the existence of a unique solution of Volterra-Hammerstein integral equation of the second kind (V-HIESK) is discussed. The Volterra integral term (VIT) is considered in time with a continuous kernel, while the Fredholm integral term (FIT) is considered in position with a generalized singular kernel. Using a numerical technique, V-HIESK is reduced to a nonlinear system of Fredholm integral equations (SFIEs). Using Toeplitz matrix method we have  a nonlinear algebraic system of equations. Also, many important theorems related to the existence and uniqueness of the produced algebraic system are derived. Finally, some numerical examples when the kernel takes the logarithmic, Carleman, and Cauchy forms, are considered

Backstepping PDE Design: A Convex Optimization Approach

Abstract\u2014Backstepping design for boundary linear PDE is formulated as a convex optimization problem. Some classes of parabolic PDEs and a first-order hyperbolic PDE are studied, with particular attention to non-strict feedback structures. Based on the compactness of the Volterra and Fredholm-type operators involved, their Kernels are approximated via polynomial functions. The resulting Kernel-PDEs are optimized using Sumof- Squares (SOS) decomposition and solved via semidefinite programming, with sufficient precision to guarantee the stability of the system in the L2-norm. This formulation allows optimizing extra degrees of freedom where the Kernel-PDEs are included as constraints. Uniqueness and invertibility of the Fredholm-type transformation are proved for polynomial Kernels in the space of continuous functions. The effectiveness and limitations of the approach proposed are illustrated by numerical solutions of some Kernel-PDEs

Solutions for a class of nonlinear Volterra integral and integro-differential equation using cyclic -contraction

We establish the existence and uniqueness of solutions for a class of nonlinear Volterra integral and integro-differential equations using fixed-point theorems for a new variant of cyclic -contractive mappings. Nontrivial examples are given to support the usability of our results