Parabolic Integro-differential Equations Research Articles

Development of integrated radial basis function Kriging interpolation for linear and nonlinear parabolic integro-differential equations

In this study, we explore linear and nonlinear parabolic integro-differential equations in one and two dimensions. We employ a semi-implicit scheme to discretize the temporal variable and discretize the spatial variable using an integrated radial basis function based on the moving Kriging interpolation (MKI) method. Unlike the global integrated radial basis function (IRBF) method, our proposed approach is particularly well-suited for addressing large-scale problems. Moreover, issues such as the selection of shape parameters leading to matrices with high condition numbers can be mitigated by devising an algorithm to determine optimal shape parameters, resulting in lower condition numbers. To assess the accuracy of the proposed method, we utilize two criteria, namely, L2 and H1. We compare the computed results obtained from the IRBF-MKI method with those from two grid and finite element methods. To demonstrate the accuracy and efficiency of our approach, we consider irregular areas and scattered Halton points in the two-dimensional case.

HDG Method for Nonlinear Parabolic Integro-Differential Equations

Abstract The hybridizable discontinuous Galerkin (HDG) method has been applied to a nonlinear parabolic integro-differential equation. The nonlinear functions are considered to be Lipschitz continuous to analyze uniform in time a priori bounds. An extended type Ritz–Volterra projection is introduced and used along with the HDG projection as an intermediate projection to achieve optimal order convergence of O ⁢ ( h k + 1 ) O(h^{k+1}) when polynomials of degree k ≥ 0 k\geq 0 are used to approximate both the solution and the flux variables. By relaxing the assumptions in the nonlinear variable, super-convergence is achieved by element-by-element post-processing. Using the backward Euler method in temporal direction and quadrature rule to discretize the integral term, a fully discrete scheme is derived along with its error estimates. Finally, with the help of numerical examples in two-dimensional domains, computational results are obtained, which verify our results.

A New A Posteriori Error Estimates for Optimal Control Problems Governed by Parabolic Integro-Differential Equations

A New A Posteriori Error Estimates for Optimal Control Problems Governed by Parabolic Integro-Differential Equations

BDF2 ADI orthogonal spline collocation method for the fractional integro-differential equations of parabolic type in three dimensions

In this paper, we are concerned with constructing a fast and an efficient alternating direction implicit (ADI) scheme for the fractional parabolic integro-differential equations (FPIDE) with a weakly singular kernel in three dimensions (3D). Our constructed scheme is based on a second-order backward differentiation formula (BDF2) for temporal discretization, orthogonal spline collocation (OSC) method for spatial discretization and a second-order fractional quadrature rule proposed by Lubich for the Riemann-Liouville fractional integral. The stability and convergence of the constructed numerical scheme are derived. Finally, some numerical examples are given to illustrate the accuracy and validity of the BDF2 ADI OSC method. Based on the obtained results, the numerical results are in line with the theoretical ones.

Uniqueness of the kernel determination problem in an integro-differential parabolic equation with variable coefficient

Abstract. We investigate the inverse problem of determining the time and space dependent kernel of the integral term in the n-dimensional integro-differential equation of heat conduction from the known solution of the Cauchy problem for this equation. First, the original problem is replaced by the equivalent problem where an additional condition contains the unknown kernel without integral. We study the question of the uniqueness of the determining of this kernel. Next, assuming that there are two solutions k1(x, t) and k2(x, t) of the stated problem, it is formed an equation for the difference of this solution. Further research is being conducted for the difference k1(x, t) - k2(x, t) of solutions of the problem and using the techniques of integral equations estimates.

Uniqueness of the Kernel Determination Problem in a Integro-Differential Parabolic Equation with Variable Coefficients

Uniqueness of the Kernel Determination Problem in a Integro-Differential Parabolic Equation with Variable Coefficients

A Conforming Virtual Element Method for Parabolic Integro-Differential Equations

Abstract This article develops and analyses a conforming virtual element scheme for the spatial discretization of parabolic integro-differential equations combined with backward Euler’s scheme for temporal discretization. With the help of Ritz–Voltera and L 2 L^{2} projection operators, optimal a priori error estimates are established. Moreover, several numerical experiments are presented to confirm the computational efficiency of the proposed scheme and validate the theoretical findings.

An existence result for two-dimensional parabolic integro-differential equations involving CEV model

Abstract In this paper, we present an existence result of weak solutions for some parabolic equations involving the so-called CEV model with jumps.

Numerical Analysis of Direct and Inverse Problems for a Fractional Parabolic Integro-Differential Equation

A mathematical model consisting of weakly coupled time fractional one parabolic PDE and one ODE equations describing dynamical processes in porous media is our physical motivation. As is often performed, by solving analytically the ODE equation, such a system is reduced to an integro-parabolic equation. We focus on the numerical reconstruction of a diffusion coefficient at finite number space-points measurements. The well-posedness of the direct problem is investigated and energy estimates of their solutions are derived. The second order in time and space finite difference approximation of the direct problem is analyzed. The approach of Lagrangian multiplier adjoint equations is utilized to compute the Fréchet derivative of the least-square cost functional. A numerical solution based on the conjugate gradient method (CGM) of the inverse problem is studied. A number of computational examples are discussed.

Perturbations of Time Optimal Control Problems for Parabolic Integro-Differential Equations

Perturbations of Time Optimal Control Problems for Parabolic Integro-Differential Equations

Regularity for a special case of two-phase Hele-Shaw flow via parabolic integro-differential equations

We establish that the C1,γ regularity theory for translation invariant fractional order parabolic integro-differential equations (via Krylov-Safonov estimates) gives an improvement of regularity mechanism for solutions to a special case of a two-phase free boundary flow related to Hele-Shaw. The special case is due to both a graph assumption on the free boundary of the flow and an assumption that the free boundary is C1,Dini in space. The free boundary then must immediately become C1,γ for a universal γ depending upon the Dini modulus of the gradient of the graph. These results also apply to one-phase problems of the same type.

On the null controllability of integer order integro-differential equations

Abstract This paper is addressed to the study of the null controllability for integer order integro-differential equations. Unlike the known results for partial differential equations, we need to consider the equation involving a $\beta -$power of the Laplace operator $(-\varDelta )^\beta $ and an integral term. The key point is to construct a suitable state space of the controlled system at the final time. We first discuss a class of hyperbolic integro-differential equation. We prove that the controlled system is null controllable by an Ingham-type estimate. Also, the controllability time is given. On the other hand, by reduction to absurdity, we deduce that the null controllability property fails for a class of parabolic integro-differential equation with $\beta \in \mathbb{N}^+$.

HDG method for linear parabolic integro-Differential equations

This paper develops the hybridizable discontinuous Galerkin (HDG) method for a linear parabolic integro-differential equation and analyzes uniform in time apriori error bounds. To handle the integral term, an extended Ritz-Volterra projection is introduced, which helps in achieving optimal order convergence of O(hk+1) for the semi-discrete problem when polynomials of degree k≥0 are used to approximate both the solution and the flux variables. Further, element-by-element post-processing is proposed, and it is established that it achieves convergence of the order O(hk+2) for k≥1. Using the backward Euler method in temporal direction and quadrature rule to discretize the integral term, a fully discrete scheme is derived along with its error estimates. Finally, with the help of numerical examples in two-dimensional domains, computational results are obtained, which verify our results.

ON DETERMINATION OF THE COEFFICIENT AND KERNEL IN AN INTEGRO -DIFFERENTIAL EQUATION OF PARABOLIC TYPE

The inverse problem of determination of x-dependent coefficient a(x) at u and the kernel k(t) functions in the one-dimensional integro–differential parabolic equation is investigated. The direct problem is the initial-boundary problem for this equation. Firstly, we studied the solvability of the direct problem, by used to the Fourier method and approximation series methods. As additional information for solving inverse problem, the solution of the direct problem by over determining condition is given. The problem is reduced to an equivalent closed system of Volterra-type integral equations with respect to unknown functions. Applying the method of contraction mappings to this system in the continuous class of functions, we prove the main result of the article, which is a local existence and uniqueness theorem of inverse problem solution.

Two-grid methods of finite element approximation for parabolic integro-differential optimal control problems

<abstract><p>In this paper, we present a two-grid scheme of fully discrete finite element approximation for optimal control problems governed by parabolic integro-differential equations. The state and co-state variables are approximated by a piecewise linear function and the control variable is discretized by a piecewise constant function. First, we derive the optimal a priori error estimates for all variables. Second, we prove the global superconvergence by using the recovery techniques. Third, we construct a two-grid algorithm and discuss its convergence. In the proposed two-grid scheme, the solution of the parabolic optimal control problem on a fine grid is reduced to the solution of the parabolic optimal control problem on a much coarser grid; additionally, the solution of a linear algebraic system on the fine grid and the resulting solution maintain an asymptotically optimal accuracy. Finally, we present a numerical example to verify the theoretical results.</p></abstract>

Deep learning schemes for parabolic nonlocal integro-differential equations

In this paper we consider the numerical approximation of nonlocal integro differential parabolic equations via neural networks. These equations appear in many recent applications, including finance, biology and others, and have been recently studied in great generality starting from the work of Caffarelli and Silvestre by Lius and Lius (Comm PDE 32(8):1245–1260, 2007). Based on the work by Hure, Pham and Warin by Hure et al. (Math Comp 89:1547–1579, 2020), we generalize their Euler scheme and consistency result for Backward Forward Stochastic Differential Equations to the nonlocal case. We rely on Lévy processes and a new neural network approximation of the nonlocal part to overcome the lack of a suitable good approximation of the nonlocal part of the solution.

Numerical approximate controllability for unidimensional parabolic integro-differential equations

This paper discusses numerical solutions for the control functions of parabolic integro-differential equations under the assumption of approximate controllability. It is proved theoretically that the numerical solution for the control function converges to the exact solution, and the validity of the theory is demonstrated from different perspectives through four numerical simulation examples.

Optimal error estimates of Galerkin method for a nonlinear parabolic integro-differential equation

In this paper, the error analysis of Galerkin finite element method (FEM) is investigated for a nonlinear parabolic integro-differential equation in two dimensions. By skillfully and rigorously manipulating the nonlinear term, optimal error estimates in L∞(L2(Ω)) and L∞(H1(Ω)) are obtained for a linearized backward Euler fully-discrete scheme, which improves the suboptimal approximation in L∞(L2(Ω)) in the previous literature. Finally, some numerical results are provided to verify the theoretical findings.

The two-phase Stefan problem with anomalous diffusion

The non-local in space two-phase Stefan problem (a prototype in phase change problems) can be formulated via a singular nonlinear parabolic integro-differential equation which admits a unique weak solution. This formulation makes Stefan problem to be part of the General Filtration Problems; a class which includes the Porous Medium Equation. In this work, we prove that the weak solutions to both Stefan and Porous Media problems are continuous.

Two-Grid Algorithm of $H^{1}$-Galerkin Mixed Finite Element Methods for Semilinear Parabolic Integro-Differential Equations

Two-Grid Algorithm of $H^{1}$-Galerkin Mixed Finite Element Methods for Semilinear Parabolic Integro-Differential Equations