We consider the numerical solution of one type of integro-differential equation by a probability method based on the fundamental martingale of mixed Gaussian processes. As an application, we try to simulate the estimation of ruin probability with an unknown parameter driven not by the classical Lévy process, but by the mixed fractional Brownian motion.

We study the solvability of integral equations using Darbo’s fixed point theorem in Banach algebra. We prove the existence of the solution for nonlinear functional integral equations, which contain various functional integral equations in the nonlinear analysis. A numerical example showing the specified existence of our main results.

The stability and approximation of solutions of Nicholson’s blowflies model for the stochastic case on time scales is discussed. Using various tools of analysis, sufficient conditions for the existence of a square mean almost automorphic solution are derived. The randomness and time scales make the model a hybrid model, which is more realistic and useful. The analysis works for both discrete and continuous cases, as well as for several other cases such as quantum and Cantor sets. We establish appropriate conditions and results to explore the Ulam–Hyers–Rassias stability. Furthermore, the model with piecewise constant argument is analyzed. Then the approximate solution and a nicer bound of this model using the discretization method is established. We conclude with an example to demonstrate our analytical results.

We propose a method to solve a nonlinear boundary value problem for a Fredholm integro-differential equation on a finite interval. By introducing additional parameters chosen as the values of the solution at the left-end points of the partition subintervals, the problem under consideration is transformed into an equivalent boundary value problem for a system of nonlinear integro-differential equations with parameters on the subintervals. For fixed parameters, we obtain a special Cauchy problem for this system, which is represented as a nonlinear operator equation and solved by an iterative method. By substitution of the solution to the special Cauchy problem with parameters into the boundary condition and the continuity conditions of the solution to the original problem at the interior partition points, we construct a system of nonlinear algebraic equations in parameters. It is proved that the solvability of this system provides the existence of a solution to the original boundary value problem.

The algorithm for solving the special Cauchy problem includes two auxiliary problems: the Cauchy problems for ordinary differential equations and the evaluation of definite integrals. The accuracy of the method that we propose to solve the boundary value problem depends on the accuracy of methods applied to the auxiliary problems and does not depend on the number of the partition subintervals. Since iterative methods are used to solve both the constructed system of algebraic equations and the special Cauchy problem, we offer an approach to find an initial guess for the solutions to these problems.

We study the quasinormality of singular integral operators with Cauchy kernel on $L^2$. Moreover, we give characterizations for singular integral operators to be the square root of a self-adjoint operator and an isometry, respectively. Furthermore, we consider the condition for singular integral operators to be $D$-operators. We provide several results and examples of such operators as applications.

Asymptotically convolution Volterra equations are characterized by kernel functions which exponentially decay to convolution ones. Their importance in the applications motivates a numerical analysis of the asymptotic behavior of the solution. Here the quasi-convolution nature of the kernel is exploited in order to investigate the stability of $\left(\rho ,\sigma \right)$ methods for general systems and in some particular cases.

In this paper, we consider an initial boundary value problem for a nonlinear Love equation with infinite memory. By combining the linearization method, the Faedo–Galerkin method and the weak compactness method, we prove the local existence and uniqueness of a weak solution. The finite time blow-up of the weak solution is considered.