Order Integro-differential Equations Research Articles

A Numerical Algorithm and a Variational Iteration Technique for Solving Higher Order Fuzzy Integro-differential Equations

In this paper, we present an algorithmic form of the variational iteration method (VIM) to handle both linear and nonlinear higher order fuzzy integro-differential equations. Using parametric form of fuzzy numbers to convert higher order fuzzy integro-differential equation to a system of higher order integro-differential equations in crisp case. By using the variational integration method we find the approximate solution of this system and consequently we obtain an approximation for fuzzy solution of the higher order fuzzy integro-differential equations. The numerical results are examined.

Modification Adomian Decomposition Method for solving Seventh OrderIntegro-Differential Equations

In this paper, a method based on modified adomian decomposition method for solving Seventh order integro-differential equations (MADM). The distinctive feature of the method is that it can be used to find the analytic solution without transformation of boundary value problems. To test the efficiency of the method presented two examples are solved by proposed method. Keyword: Adomian decomposition method; boundary-value problems; integro-differential equation

A new algorithm for higher order integro-differential equations

A new modified Laplace decomposition is established for approximate solution of fifth order integro-differential equations. The approximate numerical results confirm the reliability and efficiency of the proposed new algorithm. The proposed mathematical technique is verified on linear and nonlinear problems.

Positive Solutions of Initial Value Problem for the Systems of Second Order Integro-Differential Equations in Banach Space

In this paper, by establishing a new comparison result, we investigate the existence of positive solutions for initial value problems of nonlinear systems of second order integro-differential equations in Banach space.We improve and generalize some results(see[5,6]),and the results is new even in finite dimensional spaces. Keywords—Systems of integro-differential equations, monotone iterative method, comparison result, cone.

Compact finite difference methods for high order integro-differential equations

High order integro-differential equations (IDEs), especially nonlinear, are usually difficult to solve even numerically. In this chapter, we give a highly accurate compact finite difference method to efficiently solve integro-differential equations, including high order and nonlinear problems. By numerical experiments, we show that compact finite difference method for integro-differential equations is fast and can obtain a high accuracy.

Existence results for boundary value problems of arbitrary order integrodifferential equations in Banach spaces

Abstract We study a boundary value problem of fractional integrodifferential equations involving Caputo's derivative of order α ∈ (n-1,n) in a Banach space. Existence and uniqueness results for the problem are established by means of the Hölder's inequality together with some standard fixed point theorems

Multiple positive solutions of boundary value problems for fractional order integro-differential equations in a Banach space

In this paper, we obtain the existence of multiple positive solutions of a boundary value problem for α-order nonlinear integro-differential equations in a Banach space by means of fixed point index theory of completely continuous operators. MSC:26A33, 34B15.

The Perturbed Sparre Andersen Model with Interest and a Threshold Dividend Strategy

In this paper, we consider a Sparre Andersen model perturbed by diffusion (in which the inter-claim times are generalized Erlang(n)-distributed) with a constant interest under a threshold dividend payment strategy. Under such a strategy, no dividends are paid if the insurer’s surplus is below a certain threshold level. When the surplus is above the threshold level, part of the premium income and all of the interest income are paid out as dividends. Integro-differential equations with certain boundary conditions for the moment generating functions and moment functions of the present value of all dividends until ruin are derived. We also derive the integro-differential equations with boundary conditions for the Gerber–Shiu functions. Explicit expressions are given in terms of some functions related to high order integro-differential equations when the inter-claim times are Erlang(2) and Erlang(1) distributed.

On a perturbed by diffusion compound Poisson risk model with delayed claims and multi-layer dividend strategy

In this paper we consider an extension to the compound Poisson risk process perturbed by diffusion in which two types of dependent claims, main claims and by-claims, are incorporated. Every by-claim is induced by the main claim and may be delayed for one time period with a certain probability. An integro-differential equation system for the Gerber–Shiu expected discounted penalty functions is derived and solved by proving that the Gerber–Shiu function satisfies some defective renewal equation. An exact representation for the solution of this equation is derived through an associated compound geometric distribution and an analytic expression for this quantity is given when both the claim and the by-claim amounts belong to the rational family of distributions. Further, the same risk model is considered in the presence of a multi-layer dividend strategy. A system of integro-differential equations for the expected discounted penalty functions depending on the current surplus level, with certain initial and boundary conditions, is obtained. To solve this, we derive a general solution to a certain second order integro-differential equation system. This solution is obtained by transforming this system to a Volterra-type system of integral equations of second kind, which is solved by using Laplace transforms provided an explicit expression for the Gerber–Shiu functions depending on the current surplus level. Finally, numerical results for the ruin probability are given to illustrate the applicability of our results.

On a nonlinear functional second order integrodifferential equation in Banach Spaces

In this paper, we study the existence, uniqueness and other properties of solutions of a nonlinear functional second order Volterra integrodifferential equation in a general Banach space. The techniques used in our analysis are the theory of the strongly continuous cosine family, Schauder fixed point theorem and Pachpatte’s integral inequality.

Degenerate irregular SDEs with jumps and application to integro-differential equations of Fokker-Planck type

We investigate stochastic differential equations with jumps and irregular coefficients, and obtain the existence and uniqueness of generalized stochastic flows. Moreover, we also prove the existence and uniqueness of $L^p$-solutions or measure-valued solutions for second order integro-differential equation of Fokker-Planck type.

Existence of Solutions for Integro-differential Equations of Mixed Type in Banach Spaces

The initial value problem of differential equation can be reduced to the equivalent integral equation.Recently,this method is used successfully to discuss the existence of the solution to initial value problem of nonlinear differential equation.Using the fixed point theorem of convexpower condensing operator,we study the existence of solutions to initial value problems for nonlinear second order integro-differential equations of mixed type in Banach spaces.

Semilinear fractional order integro-differential equations with infinite delay in Banach spaces

This paper concerns the existence of mild solutions for fractional order integro-differential equations with infinite delay. Our analysis is based on the technique of Kuratowski’s measure of noncompactness and Mönch’s fixed point theorem. An example to illustrate the applications of main results is given.

Spectral-Collocation Methods for Fractional Pantograph Delay-Integrodifferential Equations

We propose and analyze a spectral Jacobi-collocation approximation for fractional order integrodifferential equations of Volterra type with pantograph delay. The fractional derivative is described in the Caputo sense. We provide a rigorous error analysis for the collocation method, which shows that the error of approximate solution decays exponentially inL∞norm and weightedL2-norm. The numerical examples are given to illustrate the theoretical results.

Simple Recipe for Accurate Solution of Fractional Order Equations

Fractional order integrodifferential equations cannot be directly solved like ordinary differential equations. Numerical methods for such equations have additional algorithmic complexities. We present a particularly simple recipe for solving such equations using a Galerkin scheme developed in prior work. In particular, matrices needed for that method have here been precisely evaluated in closed form using special functions, and a small Matlab program is provided for the same. For equations where the highest order of the derivative is fractional, differential algebraic equations arise; however, it is demonstrated that there is a simple regularization scheme that works for these systems, such that accurate solutions can be easily obtained using standard solvers for stiff differential equations. Finally, the role of nonzero initial conditions is discussed in the context of the present approximation method.

Numerical Solution of Arbitrary-Order Ordinary Differential and Integro-Differential Equations with Separated Boundary Conditions Using Optimal Control Technique

In this paper, a new method for solving arbitrary order ordinary differential equations and integro-differential equations of Fredholm and Volterra kind is presented. In the proposed method, these equations with separated boundary conditions are converted to a parametric optimization problem subject to algebraic constraints. Finally, control and state variables will be approximated by a Chebychev series. In this method, a new idea has been used, which offers us the ability of applying the mentioned method for almost all kinds of ordinary differential and integro-differential equations with different types of boundary conditions. The accuracy and efficiency of the proposed numerical technique have been illustrated by solving some test problems.

Application of Homotopy Perturbation and Variational Iteration Methods for Fredholm Integrodifferential Equation of Fractional Order

This paper presents the application of homotopy perturbation and variational iteration methods as numerical methods for Fredholm integrodifferential equation of fractional order with initial‐boundary conditions. The fractional derivatives are described in Caputo sense. Some illustrative examples are presented.

Solutions of System of Volterra Integro Differential Equation of Second Kind by Using Piecewise Constant Functions

In this paper we may use piece wise constant functions for the special type of system of second kind integro differential equation of the first order. The main problem is reduced to linear system of algebraic equations. Some numerical examples are dedicated for showing efficiency and validity of the method.

Numerical Solution of Nonlinear Fredholm Integrodifferential Equations of Fractional Order by Using Hybrid Functions and the Collocation Method

A numerical method to solve nonlinear Fredholm integral equations of second kind is presented in this work. The method is based upon hybrid function approximate. The properties of hybrid of block‐pulse functions and Taylor series are presented and are utilized to reduce the computation of nonlinear Fredholm integral equations to a system of algebraic equations. Some numerical examples are selected to illustrate the effectiveness and simplicity of the method.

Existence and iteration of positive solutions for a three-point boundary value problem of second order integro-differential equation with p-Laplace operator

In the paper, we obtain the existence of positive solutions and establish a corresponding iterative scheme for the following three-point boundary value problem $$\left\{\begin{array}{l}(\phi_p(u'))'(t)+q(t)f\left(u(t),u'(t),Tu(t),Su(t)\right)=0,\quad0 < t< 1,\\u'(0)=\alpha u'(\eta),\quad u(1)=g(u'(1)),\end{array}\right.$$ where ϕ p (s)=|s| p−2 s,p>1,α∈[0,1),η∈(0,1), T and S are all linear operators, g(t) is continuous and nonincreasing on (−∞,0]. The main tools are monotone iterative technique and numerical simulation. We illustrate our results by one example, and give its numerical results by iterative scheme.