Numerical Approximation of Fractional-Order Volterra Integrodifferential Equation

Laplace transform is a powerful tool for solving differential and integrodifferential equations in engineering sciences. The use of Laplace transform for the solution of differential or integrodifferential equations sometimes leads to the solutions in the Laplace domain that cannot be inverted to the real domain by analytic methods. Therefore, we need numerical methods to invert the solution to the real domain. In this work, we construct numerical schemes based on Laplace transform for the solution of fractional-order Volterra integrodifferential equations in the sense of the Atangana-Baleanu Caputo derivative. We propose two numerical methods for approximating the solution of fractional-order linear and nonlinear Volterra integrodifferential equations. In our scheme, the inverse Laplace transform is approximated using a contour integration method and Stehfest method. Some numerical experiments are performed to check the accuracy and efficiency of the methods. The results obtained using these methods are compared.

ON FRACTIONAL VOLTERRA INTEGRODIFFERENTIAL EQUATIONS WITH FRACTIONAL INTEGRABLE IMPULSES

We consider a class of nonlinear fractional Volterra integrodifferential equation with fractional integrable impulses and investigate the existence and uniqueness results in the Bielecki’s normed Banach spaces. Further, Bielecki-Ulam type stabilities have been demonstrated on a compact interval. A concrete example is provided to illustrate the outcomes we acquired.

On Minimizing the Ultimate Ruin Probability of an Insurer by Reinsurance

We consider an insurance company whose reserves dynamics follow a diffusion-perturbed risk model. To reduce its risk, the company chooses to reinsure using proportional or excess-of-loss reinsurance. Using the Hamilton-Jacobi-Bellman (HJB) approach, we derive a second-order Volterra integrodifferential equation (VIDE) which we transform into a linear Volterra integral equation (VIE) of the second kind. We then proceed to solve this linear VIE numerically using the block-by-block method for the optimal reinsurance policy that minimizes the ultimate ruin probability for the chosen parameters. Numerical examples with both light- and heavy-tailed distributions are given. The results show that proportional reinsurance increases the survival of the company in both light- and heavy-tailed distributions for the Cramér-Lundberg and diffusion-perturbed models.

Stochastic Volterra integro-differential equations driven by a fractional Brownian motion with delayed impulses

In this paper, the problem of existence of mild solutions for a stochastic Volterra integrodifferential equation with delayed impulses and driven by a fractional Brownian motion (Hurst parameter H ? (1/2,1)) is investigated. Here, we assume that the delayed impulses are linear and impulsive transients depend on not only their current but also historical states of the system. Utilizing the fixed point theorem combine with fractional power of operators and the semi-group theory, sufficient conditions that guarantee the existence and uniqueness of mild solutions for such equation are obtained. Finally, an example is presented to demonstrate the effectiveness of the proposed results.

Existence, Uniqueness, and Characterization Theorems for Nonlinear Fuzzy Integrodifferential Equations of Volterra Type

Existence and uniqueness theorem are the tool which makes it possible for us to conclude that there exists only one solution to a given problem which satisfies a constraint condition. How does it work? Why is it the case? We believe it, but it would be interesting to see the main ideas behind this. To this end, in this paper, we investigate existence, uniqueness, and other properties of solutions of a certain nonlinear fuzzy Volterra integrodifferential equation under strongly generalized differentiability. The main tools employed in the analysis are based on the applications of the Banach fixed point theorem and a certain integral inequality with explicit estimate. Also, some results for characterizing solution by an equivalent system of crisp Volterra integrodifferential equations are presented. In this way, a new direction for the methods of analytic and approximate solutions is proposed.

Bernstein Collocation Method for Solving Nonlinear Fredholm-Volterra Integrodifferential Equations in the Most General Form

A collocation method based on the Bernstein polynomials defined on the interval[a,b]is developed for approximate solutions of the Fredholm-Volterra integrodifferential equation (FVIDE) in the most general form. This method is reduced to linear FVIDE via the collocation points and quasilinearization technique. Some numerical examples are also given to demonstrate the applicability, accuracy, and efficiency of the proposed method.