Nonlinear Parabolic Integro-differential Equations Research Articles

A finite volume approximations for one nonlinear and nonlocal integrodifferential equations

In this paper, the error analysis of the Petrov–Galerkin finite volume element method (FVEM) is investigated for a nonlinear parabolic integro-differential equation that arises in the mathematical modeling of the penetration of a magnetic field into a substance, accounting for temperature-dependent changes in electrical conductivity. Starting from Maxwell’s equations, we derive a one-dimensional model problem, which forms the basis of our analysis. Our main goal is to develop a general framework for obtaining finite volume element approximations and to study the error analysis. For simplicity, we consider only the lowest-order (linear and L-splines) finite volume elements. The novel contribution lies in the application of FVEM to this problem, leading to the establishment of an unconditionally stable numerical scheme and the derivation of optimal error estimates in the L∞(L2(Ω)) and L2(H01(Ω)) norms for both semi-discrete and linearized backward Euler fully-discrete schemes, using a generalized projection method that carefully manages the nonlinear terms. Lastly, numerical experiments are provided to support the theoretical conclusions.

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.

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.

Numerical Method for a Filtration Model Involving a Nonlinear Partial Integro-Differential Equation

In this paper, we propose an efficient numerical method for solving an initial boundary value problem for a coupled system of equations consisting of a nonlinear parabolic partial integro-differential equation and an elliptic equation with a nonlinear term. This problem has an important applied significance in petroleum engineering and finds application in modeling two-phase nonequilibrium fluid flows in a porous medium with a generalized nonequilibrium law. The construction of the numerical method is based on employing the finite element method in the spatial direction and the finite difference approximation to the time derivative. Newton’s method and the second-order approximation formula are applied for the treatment of nonlinear terms. The stability and convergence of the discrete scheme as well as the convergence of the iterative process is rigorously proven. Numerical tests are conducted to confirm the theoretical analysis. The constructed method is applied to study the two-phase nonequilibrium flow of an incompressible fluid in a porous medium. In addition, we present two examples of models allowing for prediction of the behavior of a fluid flow in a porous medium that are reduced to solving the nonlinear integro-differential equations studied in the paper.

The classical periodic solutions for nonlinear parabolic integro-differential equations

The classical periodic solutions for nonlinear parabolic integro-differential equations

A Mathematical Model of Tornado

We propose a mathematical model of tornado in the framework of continuum mechanics. According to this model, the swirling upward air motion in a tornado can be explained even without taking into account the Coriolis force or the vertical convection of warm air. Our results show that for a tornado to appear two factors are necessary: a powerful rotational motion of air in the upper atmosphere and the growth of air pressure from the center to the periphery of an expected region of tornado. The characteristics of air flow in tornado can be found by solving a boundary value problem for a nonlinear parabolic integro-differential equation for an unknown complex-valued temperature. We formulate this problem, propose an algorithm for its numerical solution, and examine its stationary solutions by the stabilization method.

Numerical blow-up analysis of linearly implicit Euler method for nonlinear parabolic integro-differential equations

The linearly implicit Euler method for nonlinear parabolic integro-differential equations (NPIDEs) on bounded domains is consideblue by approximating the local effects implicitly and nonlocal effects explicitly. Based on Nakagawa’s criteria, a suitable adaptive time-stepping strategy is introduced by the discrete energy instead of the infinite norm. With the help of lower discrete energies, the finite blow-up behaviors are replicated for any positive solution. Numerical simulations are carried out to examine the effectiveness of our blowup analysis, which also motivate furthermore to prove that a global numerical solution exists for certain NPIDEs with a weakly singular kernel.

A two-grid finite element method for nonlinear parabolic integro-differential equations

ABSTRACTIn this paper, we present a two-grid finite element method (FEM) for a two-dimensional nonlinear parabolic integro-differential equation. We solve a fully nonlinear system on a coarse grid space with a grid size H and derive a rough approximation of the exact solution, and then solve the corresponding linearized problem on a fine grid space with a grid size h. The optimal error estimates in -norm are obtained for spatially the semidiscrete two-grid FEM. Finally, numerical examples are given to testify the efficiency of the method.

On the null controllability of parabolic equations with nonlinear memory

ABSTRACT In this paper, we study the global null controllability of nonlinear parabolic integro-differential equations arising in reactor dynamics with internal distributed control. Unlike the known results for nonlinear parabolic equations, controllability is analyzed in the frame of strong solutions. The result implies the nonlinear growth of the system with memory which is null controllable can be higher than that of classical parabolic equations. This phenomenon is caused by the memory effects.

A front-fixing finite element method for pricing American options under regime-switching jump-diffusion models

A front-fixing finite element method is applied to solve the partial integro-differential equations (PIDEs) arising in pricing American options under Markov-modulated jump-diffusion models with the free boundaries feature. For this purpose, we used a front-fixing method to transform the pricing problem into a nonlinear parabolic integro-differential equation on a fixed domain. Then the variational form of the resulting problem is solved by a finite element method. Under some appropriate assumptions, we establish the stability of the method and illustrate some numerical results to examine the rate of convergence of the proposed method for the pricing problem and compare its accuracy to some recent works on pricing American options under regime-switching jump-diffusion models.

On initial-boundary value problem for nonlinear integro-differential equations with variable exponents of nonlinearity

Some nonlinear parabolic integro-differential equations with variable exponents of the nonlinearity are considered. The initial-boundary value problem for these equations is investigated and the existence theorem for the problem is proved.

Finite difference scheme for one nonlinear parabolic integro-differential equation

Initial–boundary value problem with mixed boundary conditions for one nonlinear parabolic integro-differential equation is considered. The model is based on Maxwell system describing the process of the penetration of a electromagnetic field into a substance. Unique solvability and asymptotic behavior of solution are fixed. Main attention is paid to the convergence of the finite difference scheme. More wide cases of nonlinearity that already were studied are investigated.

A two-grid block-centered finite difference method for nonlinear non-Fickian flow model

In this paper, a two-grid block-centered finite difference scheme is introduced and analyzed to solve the nonlinear parabolic integro-differential equation arising in modeling non-Fickian flow in porous media. This method is considered where the nonlinear problem is solved only on a coarse grid of size H and a linear problem is solved on a fine grid of size h. Error estimates are established on non-uniform rectangular grid which show that the discrete L∞(L2) and L2(H1) errors are O(▵t+h2+H3). Finally, some numerical experiments are presented to show the efficiency of the two-grid method and verify that the convergence rates are in agreement with the theoretical analysis.

Long-time behavior of solution and semi-discrete scheme for one nonlinear parabolic integro-differential equation

Long-time behavior of solution and semi-discrete scheme for one nonlinear parabolic integro-differential equation are studied. Initial–boundary value problem with mixed boundary conditions are considered. Attention is paid to the investigation of more wide cases of nonlinearity than already were studied. Considered model is based on Maxwell’s system describing the process of the penetration of a magnetic field into a substance.

Numerical Blow-Up of Nonlinear Parabolic Integro-Differential Equations on Unbounded Domain

The efficient numerical methods of the nonlinear parabolic integro-differential PDEs on unbounded spatial domains whose solutions blow up in finite time are considered. Based on the unified approach proposed in Zhang et al. (Phys Rev E 78:026709, 2008), Zhang et al. (Phys Rev E 79:046711, 2009), the nonlinear absorbing boundary conditions for one-dimensional and two-dimensional nonlinear parabolic integro-differential PDEs are derived. Thus the original problem on the unbounded spatial domain is reduced to an initial-boundary-value (IBV) problem on a bounded computational domain. Secondly, a simple but efficient adaptive time-stepping scheme for the reduced IBV problem is achieved by using the fixed point method to make the finite difference approximation stable at each time level. At each time level, we also prove that the lower bound and upper bound of the blow-up time can be bounded by the numerical blow-up times of the forward and backward Euler schemes. Finally, the theoretical results are illustrated by a broad range of numerical examples, including a problem with a circle line blow-up.

BSDEs with jumps and path-dependent parabolic integro-differential equations

This paper deals with backward stochastic differential equations with jumps, whose data (the terminal condition and coefficient) are given functions of jump-diffusion process paths. The author introduces a type of nonlinear path-dependent parabolic integrodifferential equations, and then obtains a new type of nonlinear Feynman-Kac formula related to such BSDEs with jumps under some regularity conditions.

Finite element method for a nonlinear parabolic integro-differential equation in higher spatial dimensions

In this paper, we extend and analyze the Galerkin finite element method for the spatial discretization (Jangveladze et al., 2011) to the higher spatial dimensions and develop a numerical algorithm based on θ scheme for the time discretization for solving a parabolic integro-differential equation which arises in the magnetic field penetration process. A-priori bounds are derived for the exact solution. The semi discrete and fully discrete error estimates are derived in L∞(L2(Ω)) and L2(H1(Ω)) norms using energy arguments. Further, we present a numerical experiment which supports the theoretical results.

Regularity for fully nonlinear nonlocal parabolic equations with rough kernels

We prove space and time regularity for solutions of fully nonlinear parabolic integro-differential equations with rough kernels. We consider parabolic equations $$u_t = \mathrm{I}u$$ , where $$\mathrm{I}$$ is translation invariant and elliptic with respect to the class $$\mathcal L_0(\sigma )$$ of Caffarelli and Silvestre, $$\sigma \in (0,2)$$ being the order of $$\mathrm{I}$$ . We prove that if $$u$$ is a viscosity solution in $$B_1 \times (-1,0]$$ which is merely bounded in $$\mathbb {R}^n \times (-1,0]$$ , then $$u$$ is $$C^\beta $$ in space and $$C^{\beta /\sigma }$$ in time in $$\overline{B_{1/2}} \times [-1/2,0]$$ , for all $$\beta < \min \{\sigma , 1+\alpha \}$$ , where $$\alpha >0$$ . Our proof combines a Liouville type theorem—relaying on the nonlocal parabolic $$C^\alpha $$ estimate of Chang and Davila—and a blow up and compactness argument.