The integral inequalities play a fundamental role in the study of existence, uniqueness, boundedness, stability, invariant manifolds, and other qualitative properties of solutions of the theory of differential and integral equations. There are a lot of papers investigating them such as [1–8]. In particular, Pachpatte [2] discovered some new integral inequalities involving functions of two varia...

This work is devoted to prove the existence and uniqueness of solution BVP with non-local assumptions on boundary integral gluing conditions for parabolic-hyperbolic type equation involving Caputo derivatives. Using method energy, have been proved. Existence was proved by equations

In this paper, a reliable iterative approach, for solving a wide range of linear and nonlinear Volterra-Fredholm integral equations is established. First the approach considers a discretized form of the integral terms where considering some conditions on the kernel of the integral equation it is proved that solution of the discretized form converges to the exact solution of the problem. Then th...

The numerical contour integral method with hyperbolic contour is exploited to solve space-fractional diffusion equations. By making use of the Toeplitzlike structure of spatial discretized matrices and the relevant properties, the regions that the spectra of resulting matrices lie in are derived. The resolvent norms of the resulting matrices are also shown to be bounded outside of the regions. ...

Keywords: boundary control, parabolic-hyperbolic system, integral time delays, Neumann problem, quadratic programming method, Hilbert space In this paper, we consider optimal boundary control problems for parabolic-hyperbolic systems in which time delays appear in the integral form in the state equations. The right-hand side of the state equation and the initial and boundary conditions are not ...

In [16], the author proved a number of multivariate elliptic hypergeometric integrals. The purpose of the present note is to explore more carefully the various limiting cases (hyperbolic, trigonometric, rational, and classical) that exist. In particular, we show (using some new estimates of generalized gamma functions) that the hyperbolic integrals (previously treated as purely formal limits) a...