Using Darbo’s fixed point theorem, we present an existence result for the solution of a nonlinear integral equation of Fredholm type in the space of functions satisfying the Hölder condition. We also give an example to illustrate the mentioned existence result.

This paper concerns two classes of initial-boundary value problems of semilinear time fractional wave equations in an exterior domain. Based on a special technique of test functions, we determine some critical exponents in the sense of Fujita for a corresponding problem. We also obtain the existence and nonexistence of global weak solutions with certain appropriate conditions.

We examine a class of fractional-order Volterra functional integral equations, where the fractional integral is viewed in the Hadamard type. By using Petryshyn’s fixed point theorem for Banach spaces, we investigate the existence solutions for fractional integral equations. Also, we introduce an iterative method using the sinc quadrature rule to find the approximate solutions of Hadamard fractional integral equations. Several examples are presented to support the theoretical and numerical results.

This paper fills a gap in the convergence analysis of collocation solutions in continuous piecewise polynomial space for generalized auto-convolution Volterra integral equations (AVIEs). The solvability of continuous collocation methods is discussed and the uniform boundedness of the collocation solution is provided by a discrete weighted exponential norm. The necessary and sufficient conditions for the optimal global and local (super-) convergence properties of continuous collocation solutions are obtained. Some numerical experiments are given to illustrate the theoretical results.

Functional reproducing kernel Hilbert spaces (FRKHSs) are appropriate function spaces in which we seek a target function from a finite number of non-point-evaluation functional data. We consider reconstructing a function from a finite number of generalized moment data via regularization in an FRKHS with respect to the generalized moment functionals. We construct specific FRKHSs and their associated FRKHS kernels with respect to two classes of generalized moment functionals, the Hamburger moment functionals and the trigonometric moment functionals. We solve the regularization problem in the resulting FRKHSs by the representer theorem. Numerical examples are presented to illustrate the better performance of regularization in an FRKHS than regularization in the square integrable functions space.

This work addresses the regularity of solutions for a nonlocal diffusion equation over the space of periodic distributions. The spatial operator for the nonlocal diffusion equation is given by a nonlocal Laplace operator with a compactly supported integral kernel. We follow a unified approach based on the Fourier multipliers of the nonlocal Laplace operator, which allows the study of regular as well as distributional solutions of the nonlocal diffusion equation, with integrable as well as singular kernels, in any spatial dimension. In addition, the results extend beyond operators with singular kernels to nonlocal superdiffusion operators. We present results on the spatial and temporal regularity of solutions in terms of regularity of the initial data or the diffusion source term. Moreover, solutions of the nonlocal diffusion equation are shown to converge to the solution of the classical diffusion equation for two types of limits: as the spatial nonlocality vanishes or as the singularity of the integral kernel approaches a certain critical singularity that depends on the spatial dimension. Furthermore, we show that, for the case of integrable kernels, discontinuities in the initial data propagate and persist in the solution of the nonlocal diffusion equation. The magnitude of a jump discontinuity is shown to decay over time.

The main purpose of this paper is to investigate the existence, uniqueness, and regularity of solutions for systems of linear and nonlinear weakly singular Volterra integral equations of the first kind, which have been stated as research problems (for the linear case) by Brunner (2017). To get these results, we first state sufficient conditions for the existence and uniqueness of solutions for a system of weakly singular Volterra integral equations of the second kind. We demonstrate the results by giving an example.

Based on the principle of maximum entropy and piecewise linear basis functions, a numerical method for approximating a nonnegative solution of Volterra integral equations is proposed. Then a discrete form of the maximum entropy method is given to reduce the cost. The error analysis and the convergence rate are provided. Numerical experimental results are consistent with the theoretical analysis of errors and the convergence rate.