Types Of Integro-differential Equations Research Articles

On Fuzzy Conformable Double Laplace Transform with Applications to燩artial Differential Equations

The Laplace transformation is a very important integral transform, and it is extensively used in solving ordinary differential equations, partial differential equations, and several types of integro-differential equations. Our purpose in this study is to introduce the notion of fuzzy double Laplace transform, fuzzy conformable double Laplace transform (FCDLT). We discuss some basic properties of FCDLT. We obtain the solutions of fuzzy partial differential equations (both one-dimensional and two-dimensional cases) through the double Laplace approach. We demonstrate through numerical examples that our proposed method is very successful and convenient for resolving partial differential equations.

On a New Class of Singular Integro-differential Equations

In this paper for a new class of model and non-model partial integro-differential equations with singularity in the kernel, we obtained integral representation of family of solutions by aid of arbitrary functions. Such type of integro-differential equations are different from Cauchy-type singular integro-differential equations. Cauchy-type singular integro-differential equations are studied by the methods of the theory of analytic functions. In the process of our research the new types of singular integro-differential operators are introduced and main property of entered operators are learned. It is shown that the solution of studied equation is equivalent to the solution of system of two equations with respect to x and y, one of which is integral equation and the other is integro-differential equation. Further, non-model integro-differential equations are studied by regularization method. This regularization method for non-model equation is based on selecting and analysis of a model part of the equation and reduced to the solution of two second kind Volterra type integral equations with weak singularity in the kernel. It is shown that the presence of a non-model part in the equation does not affect to the general structure of the solutions. From here investigation of the model equations for given class of the integro-differential equations becomes important. In the cases, when the solution of given integro-differential equation depends on any arbitrary functions, a Cauchy type problems are investigated.

Simulation of an integro-differential equation and application in estimation of ruin probability with mixed fractional Brownian motion

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 Levy process, but by the mixed fractional Brownian motion.

New Type Super Singular Integro-Differential Equation and Its Conjugate Equation

In this paper for a class of model partial integro-differential equation with super singularity in the kernels, is obtained an integral representation of manifold solutions by arbitrary constants. The conjugate equation for the above-mentioned type of equations is also investigated. Such types of integro-differential equations are different from Cauchy-type singular integro-differential equations. Cauchy-type singular integro-differential equations are studied by the methods of theory of analytical functions. However, the method of analytical functions is not applicable for our case of super singular equations with integrals understanding in Riemann–Stieltjes sense. Here, we have used the method of representation the considering equation as a product of two one-dimensional singular first order integro-differential operators. Further, a complete integro-differential equation and its conjugate equation have been investigated. It is shown that in every cases of characteristic equation roots the homogeneous integro-differential equation can have a nontrivial solutions. Non-model equation is investigated by the regularization method. Regularization of non-model equation is based on selecting a model part of equation. On the basis of the analysis of a model part of equation the solution of non-model equation reduced to the solution of a second kind Volterra integral equations with super singular kernel. It is important to emphasize that in contrast to the usual theory of Volterra integral equations, the studied homogeneous integral equation has nontrivial solutions. It is easy to see that the presence of a non-model part in the equation does not affect to the general structure of the obtained solutions. From here investigation of the model equations for given class of the integro-differential equations becomes important.

The Approximation Solutions for Higher Dimensional Integro-Differential Equations

This work deals with approximation solutions to a type of integro-differential equations in several complex variables. It concerns the Cauchy formula on higher dimensional domains. In our study, we make use of multiple power series expansions and an iterative computation method to solve a kind of integro-differential equation. We introduce a symmetrized topology product area which is called a bicylinder. We expand functions and derivatives of them to power series. Moreover we obtain unknown functions by comparing coefficients of the series on both sides of equations. We express the approximation solutions by a regular product of matrixes.

Existence and Uniqueness of Mild Solution for Mixed type of Integro-Differential Equation

Existence and Uniqueness of Mild Solution for Mixed type of Integro-Differential Equation

On multivariate modifications of Cramer–Lundberg risk model with constant intensities

ABSTRACTThe paper considers very general multivariate modifications of Cramer–Lundberg risk model. The claims can be of different types and can arrive in groups. The groups arrival processes have constant intensities. The counting groups processes are dependent multivariate compound Poisson processes of Type I. We allow empty groups and show that in that case we can find stochastically equivalent Cramer–Lundberg model with non-empty groups. The investigated model generalizes the risk model with common shocks, the Poisson risk process of order k, the Poisson negative binomial, the Polya-Aeppli, the Polya-Aeppli of order k among others. All of them with one or more types of policies. The numerical characteristics, Cramer–Lundberg approximations, and probabilities of ruin are derived. During the paper, we show that the theory of these risk models intrinsically relates to the special types of integro differential equations. The probability solutions to such differential equations provide new insights, typically overseen from the standard point of view.

A Local Galerkin Integral Equation Method for Solving Integro-differential Equations Arising in Oscillating Magnetic Fields

The current work investigates a computational method to solve a class of integro-differential equations which describes the charged particle motion for certain configurations of oscillating magnetic fields. To handle the method, we first convert these types of integro-differential equations to two-dimensional Volterra integral equations of the second kind. Afterward, the solution of the mentioned Volterra integral equations can be estimated using the Galerkin method based on the moving least squares (MLS) approach constructed on scattered points. The MLS methodology is an effective technique for approximating an unknown function that involves a locally weighted least squares polynomial fitting. Since the scheme does not need any background meshes, it can be identified as a meshless method. The scheme is simple and effective to solve integro-differential equations and its algorithm can be easily implemented. Finally, numerical examples are included to show the validity and efficiency of the new technique.

Numerical solution of integro-differential equations of high order by wavelet basis, its algorithm and convergence analysis

This paper presents, for the first time, numerical solutions for this particular type of integro-differential equations. According to equations which will be introduced, suitable wavelet Galerkin method is provided using wavelet basis in the space C α ( R ) ⋂ L 2 ( R ) , α > 0 , that C α ( R ) is the Hölder space of exponent α . This approach has two advantages. First, the wavelets basis are arbitrary. It means that any differentiable wavelets basis can be used. Second, the desired orders for this equation are the reasons for involving a wide variety of these types of equations. The Algorithm and convergence analysis of this scheme are described. Numerical examples, plots and tablets of errors confirm the applicability and the validity of the proposed method.

Numerical Algorithms for Solving One Type of Singular Integro-Differential Equation Containing Derivatives of the Time Delay States

This study presents numerical algorithms for solving a class of equations that partly consists of derivatives of the unknown state at previous certain times, as well as an integro-differential term containing a weakly singular kernel. These equations are types of integro-differential equation of the second kind and were originally obtained from an aeroelasticity problem. One of the main contributions of this study is to propose numerical algorithms that do not involve transforming the original equation into the corresponding Volterra equation, but still enable the numerical solution of the original equation to be determined. The feasibility of the proposed numerical algorithm is demonstrated by applying examples in measuring the maximum errors with exact solutions at every computed nodes and calculating the corresponding numerical rates of convergence thereafter.

An operator-based approach to the analysis of ruin-related quantities in jump diffusion risk models

Recent developments in ruin theory have seen the growing popularity of jump diffusion processes in modeling an insurer’s assets and liabilities. Despite the variations of technique, the analysis of ruin-related quantities mostly relies on solutions to certain differential equations. In this paper, we propose in the context of Lévy-type jump diffusion risk models a solution method to a general class of ruin-related quantities. Then we present a novel operator-based approach to solving a particular type of integro-differential equations. Explicit expressions for resolvent densities for jump diffusion processes killed on exit below zero are obtained as by-products of this work.

On symmetry group properties and general similarity forms of the Benney equations in the Lagrangian variables

In the present study, we investigate the symmetry groups of Benney equations in Lagrangian variables in the form of the system of the nonlinear integro-differential equations. We obtain the Lie point symmetries by using the invariance criterion for a specific type of integro-differential equation and find some reduced forms that have fewer independent variables by using the symmetry groups.

Contact stresses in an infinite plate containing a rigid circular inclusion under thermal loads

The problem of an infinite plate containing a rigid circular inclusion under thermal loads is solved in this paper. The thermal loads considered here include the temperature gradient applied at infinity and a uniform temperature change. One of the major difficult parts in solving the present problem is that separation may occur between an insert or inclusion and the surrounding matrix under a non-uniform expansion of the matrix due to a temperature change. Unlike the corresponding problem with perfect contact along the interface between the inclusion and the surrounding matrix, there is no exact solution available for the current problem with incomplete contact. Based on the complex potential theory and the method of analytical continuation, a Prandtl type of integro-differential equation corresponding to the incomplete contact problem is derived. By expressing the normal stress function in terms of series form, the system of simultaneous equations is then established and solved numerically. Numerical results of the normal compressive stress and the circumferential stress for different thermal loading conditions are discussed in detail and shown in graphic form.

A numerical solution technique for a class of integro-differential equations in elastodynamic crack propagation problems

A strategy for the numerical solution of a type of integro-differential equation arising from the analysis of dynamic crack propagation problems is presented. Specifically, the problem of interest is that of a mode III crack dynamically propagating in a homogeneous linear elastic medium while the crack tip consists of a nonlinear rate dependent cohesive (or failure) zone. The mode of propagation is general, that is, not restricted to steady state or other special regimes. Furthermore, the presented solution technique is in no way restricted to mode III problems.

Singular Integro-differential Equations and Product Spaces

For certain types of integro-differential equations with weakly singular kernels we investigate well-posedness in a state space with product space structure. This state space is obtained as the extrapolation space corresponding to the solution semigroup of the problem on a weighted L2-space, where well-posedness has been established recently.

On an equivalence theorem for integro-differential equations occurring in field theories

A direct proof is given for the equivalence of two types of integro-differential equations treated formerly byKrolikowski andRzewuski. The calculation shows a close connection between the equivalence theorem and the well-known initial value problem.