Class Of Partial Integro-differential Equations Research Articles

A space-time spectral order sinc-collocation method for the fourth-order nonlocal heat model arising in viscoelasticity

The purpose of this paper is to investigate a space-time Sinc-collocation method for solving the fourth-order nonlocal heat model arising in viscoelasticity, which is a class of partial integro-differential equations (PIDEs) whose solution typically exhibits the weak singularities at the initial time. Most of existing approximate methods for solving such kind PIDEs are unbalanced, since most work have used a low order scheme for integrating the temporal variable and a high order numerical framework for discretization of space variables. The current paper contributes to a space-time spectral-order Sinc-collocation method which is balanced in both time and space variables. In order to deal with the initial singularity of solution in the time direction, the Sinc-collocation method with single exponential (SE) transformation is used for the first time, which is an effective method to deal with the singularity of the equation. Also, we fully analyse the error bounds of the method which show the exponential convergence rate in space and time. Meanwhile, we consider several test problems for examining the suggested scheme. The experiment results highlight a good precision of the new Sinc-colloction method and the correctness of the theoretical prediction for this kind nonlocal heat problems.

Multidimensional Linear and Nonlinear Partial Integro-Differential Equation in Bessel Potential Spaces with Applications in Option Pricing

The purpose of this paper is to analyze solutions of a non-local nonlinear partial integro-differential equation (PIDE) in multidimensional spaces. Such class of PIDE often arises in financial modeling. We employ the theory of abstract semilinear parabolic equations in order to prove existence and uniqueness of solutions in the scale of Bessel potential spaces. We consider a wide class of Lévy measures satisfying suitable growth conditions near the origin and infinity. The novelty of the paper is the generalization of already known results in the one space dimension to the multidimensional case. We consider Black–Scholes models for option pricing on underlying assets following a Lévy stochastic process with jumps. As an application to option pricing in the one-dimensional space, we consider a general shift function arising from a nonlinear option pricing model taking into account a large trader stock-trading strategy. We prove existence and uniqueness of a solution to the nonlinear PIDE in which the shift function may depend on a prescribed large investor stock-trading strategy function.

A Modified Semi-Analytic Iterative Method for Solving a Class of Partial Integro-Differential Equations with Conformable Fractional Order Derivative

In this paper, we introduce a modified semi analytic iterative method for solving conformable fractional Partial Integro-Differential equations (CFPIDEs). The methodology is tested by some illustrative examples which are given to demonstrate its accuracy, applicability and efficiency.

Fractional calculus with power law: The cradle of our ancestors⋆

The application of differential operators based on convolution settings has become the centre of investigation of researchers from all the corners of the globe. Several discussions have been raised and many suggestions have been done. The aim of this paper is to answer some outstanding questions that have been asked and also correct some claims that have been made for many years within the field. We have, in addition to this, suggested some fractional operators that could be used in the future for modelling and theoretical investigations. We have constructed a new class of partial integro-differential equations with fractional operators based on the new generalized Mittag-Leffler function. We presented a detailed discussion underpinning the conditions for which the new equation is well-posed. We suggested to solve this equation, a new numerical methodology that will be used to handle such equation numerically.

Existence results for some nonlocal partial integrodifferential equations without compactness or equicontinuity

In this work, we provide a new approach to address the solvability of a class of partial integrodifferential equations with nonlocal conditions and two linear parts, the first one being the generator of a strongly continuous semigroup, while the second one is of integral type. We introduce new ideas to handle the measure of noncompactness of a class of integral operators. The methods developed in this paper are of independent interest and enable us to evolve a general existence theory for a large class integrodifferential equations and to give positive answers to some questions raised in earlier works. In our considerations the semigroup generated by the first linear part is not necessarily compact or equicontinuous, which is an extra interesting feature. To demonstrate the usefulness of our results we illustrate three examples.

The expected discounted penalty function: from infinite time to finite time

ABSTRACTIn this paper we study the finite-time expected discounted penalty function (EDPF) and its decomposition in the classical risk model perturbed by diffusion. We first give the solution to a class of second-order partial integro-differential equations (PIDEs) with certain boundary conditions. We then show that the finite-time EDPFs as well as their decompositions satisfy this specific class of PIDEs so that their explicit expressions are obtained. Furthermore, we demonstrate that the finite-time EDPF may be expressed in terms of its ordinary counterpart (infinite-time) under the same risk model. Especially, the finite-time ruin probability due to oscillations and the finite-time ruin probability caused by a claim may also be expressed in terms of the corresponding quantities under the infinite-time horizon. Numerical examples are given when claims follow an exponential distribution.

Computational algorithm for solving fredholm time-fractional partial integrodifferential equations of dirichlet functions type with error estimates

Numerical modeling of partial integrodifferential equations of fractional order shows interesting properties in various aspects of science, which means increased attention to fractional calculus. This paper is concerned with a feasible and accurate technique for obtaining numerical solutions for a class of partial integrodifferential equations of fractional order in Hilbert space within appropriate kernel functions. The algorithm relies on the reproducing kernel Hilbert space method that provides the solutions in rapidly convergent series representations for the reproducing kernel based upon the Fourier coefficients of orthogonalization process. The Caputo fractional derivatives are introduced to address these issues. Moreover, the error estimate of the generated solutions is established as well as the convergence of the iterative method is investigated under some theoretical assumptions. The superiority and applicability of the present technique is illustrated by handling linear and nonlinear numerical examples. The outcomes obtained are compared with exact solutions and existing methods to confirm the effectiveness of the reproducing kernel method.

Cubic B-splines collocation method for a class of partial integro-differential equation

In this paper, a numerical method is proposed to estimate the solution of initial-boundary value problems for a class of partial integro-differential equations. This is based on the cubic B-splines method for spatial derivatives while the backward Euler method is used to discretize the temporal derivatives. Detailed discrete schemes are investigated. Next, we proved the convergence and the stability of the proposed method. The method is applied to some test examples and the numerical results have been compared with the exact solutions. The obtained results show the computational efficiency of the method. It can be concluded that computational efficiency of the method is effective for the initial-boundary value problems.

Applications of fractional calculus in solving Abel-type integral equations: Surface–volume reaction problem

In this paper we consider a class of partial integro-differential equations of fractional order, motivated by an equation which arises as a result of modeling surface–volume reactions in optical biosensors. We solve these equations by employing techniques from fractional calculus; several examples are discussed. Furthermore, for the first time, we encounter an order of the fractional derivative other than 12 in an applied problem. Hence, in this paper we explore the applicability of fractional calculus in real-world applications, further strengthening the true nature of fractional calculus.

A Convergent Difference Scheme for a Class of Partial Integro-Differential Equations Modeling Pricing under Uncertainty

In this paper we present a finite difference scheme to approximate viscosity solutions of a class of partial integro-differential equations describing pricing under model uncertainty. We establish ...

Quasi wavelet based numerical method for a class of partial integro-differential equation

In this paper we study the numerical solution of initial-boundary problem for a class of partial integro-differential equations. The quasi wavelet method is proposed to handle the spatial derivatives while the forward Euler method is used to discretize the temporal derivatives. Detailed discrete schemes are given and some numerical experiments are included to demonstrate the effectiveness of the discrete technique. The comparisons of the present numerical results with the exact analytical solutions show that the quasi wavelet based numerical method has distinctive local property and can achieve accurate results.

Asymptotic Periodicity for a Class of Partial Integrodifferential Equations

We study the existence of S-asymptotically ω-periodic solutions for a class of abstract partial integro-differential equations and for a class of abstract partial integrodifferential equations with delay. Applications to integral equations arising in the study of heat conduction in materials with memory are shown.

On Some Nonlinear Integrodifferential Inequalities for Functions of Independent Variables

The object of this paper is to establish some nonlinear integrodifferential integral inequalities in independent variables. These new inequalities represent a generalization of the results obtained by Pachpatte in the case of a function with one and two variables. Our results can be used as tools in the qualitative theory of a certain class of partial integrodifferential equation.

Efficient solution of a class of partial integro-differential equation in reproducing kernel space

In [Y.l. Wang, T. Chaolu, Z. Chen, Using reproducing kernel for solving a class of singular weakly nonlinear boundary value problems, Int. J. Comput. Math. 87(2) (2010), pp. 367–380], we present three algorithms to solve a class of ordinary differential equations boundary value problems in reproducing kernel space. It is worth noting that our methods can get the solution of partial integro-differential equation. In this note, we use method 2 [M. Dehghan, Solution of a partial integro-differential equation arising from viscoelasticity, Int. J. Comput. Math. 83(1) (2006), pp. 123–129] to solve a class of partial integro-differential equation in reproducing kernel space. Numerical example shows our method is effective and has high accuracy.

Positivity of Green's functions for a class of partial integro-differential equations including viscoelasticity

Positivity of Green's functions is established for a class of linear partial integro-differential equations in d ≤ 3 space dimensions with Volterra kernels satisfying appropriate conditions. In particular it is assumed that the Volterra kernel is completely monotone. This class includes the equations of motion of scalar hereditary viscoelasticity along with the extreme cases of elasticity and Newtonian viscosity. The positivity of Green's functions is also established under a less restrictive assumption. Positivity results are extended to a class of systems of partial integro-differential equations with completely monotone matrix-valued Volterra kernels. Positivity for longitudinal and transverse modes of 3D isotropic viscoelasticity is also established. Weaker results are obtained for higher space dimensions.

ADI orthogonal spline collocation methods for parabolic partial integro-differential equations

Alternating direction implicit (ADI) orthogonal spline collocation schemes are formulated and analysed for a class of partial integro-differential equations of parabolic type. These techniques are based on the θ-method, for θ ∈ [1/2, 1], (where θ = 1 yields the backward Euler method and θ = 1/2 yields the Crank-Nicolson method) and the second-order backward differentiation formula (BDF) method. For each method, optimal estimates in various norms at each time step are derived and confirmed by results of numerical experiments.

On the analytical separation of variables solution for a class of partial integro-differential equations

The present work deals with the derivation of analytical solutions for a particular class of partial intregro-differential equations, by employing the separation of variables technique. The equation under consideration consists of parabolic and hyperbolic terms, so that the character of the equation is determined by their relative weight. The key feature for the analysis presented here is that the eigenfunctions of the problem, which can be found in a closed form, do not comprise an orthogonal set of functions. The convergence of the new series solution is getting faster as the parabolic term of the integro-differential equation dominates over the hyperbolic (integral) one.

A Class of Partial Integro-Differential Equations with Correlation-Convolution Integral II

A Class of Partial Integro-Differential Equations with Correlation-Convolution Integral II

A Class of Partial Integro-Differential Equations with Correlation-Convolution Integral I

By means of the iteration method with weighted norms existence and uniqueness theorems are proved for three classes of nonlinear integral equations and first order integro-differential equations in two variables. The quadratic non-linearity is given by the correlation-convolution integral. Existence and uniqueness of the solutions are shown in Lebesgue spaces with mixed norms in rectangle and strip.

The link between integral equations and higher order ODEs

We study a class of partial integro-differential equations defined on a spatially extended domain that arise in the modeling of neuronal networks. In previous studies the Fourier transform was used to derive associated fourth order ordinary differential equations when the solution of the time independent integral equation is a homoclinic orbit. This gives rise to the question of whether solutions of the ODE whose Fourier transform is not well defined are also solutions of the time independent integral equation. We address this question and show that any solution of the ODE that satisfies a fairly relaxed growth condition is also a solution of the integral equation. Applications to two specific examples are given.