Order Parabolic Partial Differential Equations Research Articles

Stochastic differential equations with critically irregular drift coefficients

This paper is concerned with stochastic differential equations (SDEs for short) with irregular coefficients. By utilising a functional analytic approximation approach, we establish the existence and uniqueness of strong solutions to a class of SDEs with critically irregular drift coefficients in a new critical Lebesgue space, where the element may be only weakly integrable in time. To be more precise, let b:[0,T]×Rd→Rd be Borel measurable, where T>0 is arbitrarily fixed and d⩾1. We consider the following SDEXt=x+∫0tb(s,Xs)ds+Wt,t∈[0,T],x∈Rd, where {Wt}t∈[0,T] is a d-dimensional standard Wiener process. For p,q∈[1,+∞), we denote by C[q]([0,T];Lp(Rd)) the space of all Borel measurable functions f such that t1qf(t)∈C([0,T];Lp(Rd)). If b=b1+b2 such that |b1(T−⋅)|∈C[q]([0,T];Lp(Rd)) with 2/q+d/p=1 and ‖b1(T−⋅)‖C[q]([0,T];Lp(Rd)) is sufficiently small, and that b2 is bounded and Borel measurable, then we show that there exists a weak solution to the above equation, and if in addition limt↓0⁡‖t1qb(T−t)‖Lp(Rd)=0, the pathwise uniqueness holds. Furthermore, we obtain the strong Feller property of the semi-group and the existence of density associated with the above SDE. Besides, we extend the classical results concerning partial differential equations (PDEs) of parabolic type with Lq(0,T;Lp(Rd)) coefficients to the case of parabolic PDEs with L[q]∞(0,T;Lp(Rd)) coefficients, and derive the Lipschitz regularity for solutions of second order parabolic PDEs (see Theorem 3.1). Our results extend Krylov-Röckner and Krylov's profound results of SDEs with singular time dependent drift coefficients [20,23] to the critical case of SDEs with critically irregular drift coefficients in a new critical Lebesgue space.

Artificial neural network solver for time-dependent Fokker–Planck equations

Stochastic differential equations (SDEs) play a crucial role in various applications for modeling systems that have either random perturbations or chaotic dynamics at faster time scales. The time evolution of the probability distribution of a stochastic differential equation is described by the Fokker–Planck equation, which is a second order parabolic partial differential equation (PDE). Previous work combined artificial neural networks and Monte Carlo data to solve stationary Fokker–Planck equations. This paper extends this approach to time dependent Fokker–Planck equations. The main focus is on the investigation of algorithms for training a neural network that has multi-scale loss functions. Additionally, a new approach for collocation point sampling is proposed. A few 1D and 2D numerical examples are demonstrated.

NTIM solution of the fractional order parabolic partial differential equations

Abstract In this article, natural transform iterative method has been used to find the approximate solution of fractional order parabolic partial differential equations of multi-dimensions together with initial and boundary conditions. The method is applicable without any discretization or linearization. Three problems have been taken as test examples and the results are summarized through plots and tables to show the efficiency and reliability of the method. By practice of a few iterations, we observe that the approximate solution of the parabolic equations converges to the exact solution. The fractional derivatives are considered in the Caputo’s sense.

A multiscale domain decomposition approach for parabolic equations using expanded mixed method

We study the multiscale mortar expanded mixed method for second order parabolic partial differential equations. This technique involves non-overlapping domain decomposition which restrict a large problem into smaller pieces. Each block in the domain decomposition is discretized by expanded mixed method. The flux continuity is ensured by constructing a finite element space on the interface between subdomains. A pressure variable is introduced on the interfaces which play the role of Dirichlet boundary condition for each subdomain internal boundary. We demonstrate the unique solvability of discrete problem. We considered both continuous time and discrete time formulation and established a priori error estimates for local subdomain approximations. We derived the optimal order convergence for the appropriate choice of mortar space. An error estimate for mortar pressure variable is also provided. The numerical experiments are performed to show the accuracy and effectiveness of method.

Domain decomposition and expanded mixed method for parabolic partial differential equations

We developed and analyzed the multiblock mortar expanded mixed method for second order parabolic partial differential equations. This is a domain decomposition method in which the computational domain is expressed as the union of non-overlapping subdomains separated by interfaces. An auxiliary variable is introduced on the interface which represents the pressure and serves as Dirichlet boundary condition for local subdomain problems. The interface variable also plays the part of Lagrange multiplier to enforce flux matching condition on the interfaces. We explored the expanded mixed method to discretize each subdomain. We propose the semi-discrete formulation and address the solvability of the discrete problem. The optimal order convergence is provided for the continuous time case. We also investigate the fully discrete formulation and derived corresponding error estimates. The numerical experiments are conducted to demonstrate the theory developed in the paper.

Solution of Parabolic Partial Differential Equations Via Non-Polynomial Cubic Spline Technique

The discovery of parabolic partial differential equation (PDE) has made a profound impact on the scientific, engineering and technological community. A vast amount of research has been conducted to find the solution of parabolic PDEs. In this research, we introduced a novel technique to find the numerical solution of the fourth order PDEs. The novel technique is based upon the polynomial cubic spline method (PCSM) used along with Adomian decomposition method (ADM). The constraint of the alternative variables was decomposed by ADM to achieve successive approximation. Additionally, a numerical test problem of parabolic PDEs was solved by the proposed technique to check its viability.
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Uniform null controllability of a fourth-order parabolic equation with a transport term

In this paper we prove a uniform controllability result for a fourth order parabolic partial differential equation which includes a transport term, when the coefficients of higher order terms vanish. We prove the null controllability of the system with a single boundary control, and also it is obtained that the cost of the control decreases to zero, under the hypothesis of the control time being large enough. Moreover, we prove that, if the control time is small enough, the cost of controllability increases to infinity.

A Finite Element Method for a Fourth Order Surface Equation With Application to the Onset of Cell Blebbing

A variational problem for a fourth order parabolic surface partial differential equation is discussed. It contains nonlinear lower order terms, on which we only make abstract assumptions, and which need to be defined for specified problems. We derive a semi-discrete scheme based on the surface finite element method, show a-priori error estimates, and use the analytical results to prove well-posedness. Furthermore, we present a computational framework where specific problems can be conveniently implemented and, later on, altered with relative ease. It uses a domain specific language implemented in Python. The high level program control can also be done within the Python scripting environment. The computationally expensive step of evolving the solution over time is carried out by binding to an efficient C++ software back-end. The study is motivated by cell blebbing, which can be instrumental for cell migration. Starting with a force balance for the cell membrane, we derive a continuum model for some mechanical and geometrical aspects of the onset of blebbing in a form that fits into the abstract framework. It is flexible in that it allows for amending force contributions related to membrane tension or the presence of linker molecules between membrane and cell cortex. Cell membrane geometries given in terms of a parametrisation or obtained from image data can be accounted for by the software. The use of a domain specific language to describe the model makes is straightforward to add additional effects such as reaction-diffusion equations modelling some biochemistry on the cell membrane. Some numerical simulations illustrate the approach.

Highly accurate compact difference scheme for fourth order parabolic equation with Dirichlet and Neumann boundary conditions: Application to good Boussinesq equation

In this work, a three-level implicit compact difference scheme for the generalised form of fourth order parabolic partial differential equation is developed. The discretization is derived by approximating the lower order derivative terms using the governing differential equation with the imbedding technique and is fourth order accurate in space and second order accurate in time. The current approach is advantageous since the boundary conditions are completely satisfied and no further approximations are required to be carried out at the boundaries. The ability of the proposed scheme in handling linear singular problems is examined. The value of first order space derivative is computed alongwith the solution so it does not have to be estimated using the calculated value of the solution. The method successfully works for the highly nonlinear good Boussinesq equation for which more accurate solutions are obtained for the single and the double-soliton solutions in comparison with the existing numerical methods.

Strong convergence of a stochastic Rosenbrock-type scheme for the finite element discretization of semilinear SPDEs driven by multiplicative and additive noise

This paper aims to investigate the numerical approximation of a general second order parabolic stochastic partial differential equation(SPDE) driven by multiplicative and additive noise. Our main interest is on such SPDEs where the nonlinear part is stronger than the linear part, usually called stochastic dominated transport equations. Most standard numerical schemes lose their good stability properties on such equations, including the current linear implicit Euler method. We discretize the SPDE in space by the finite element method and propose a novel scheme called stochastic Rosenbrock-type scheme for temporal discretization. Our scheme is based on the local linearization of the semi-discrete problem obtained after space discretization and is more appropriate for such equations. We provide a strong convergence of the new fully discrete scheme toward the exact solution for multiplicative and additive noise and obtain optimal rates of convergence. Numerical experiments to sustain our theoretical results are provided.

Multiblock Mortar Mixed Approach for Second Order Parabolic Problems

In this paper, the multiblock mortar mixed approximation of second order parabolic partial differential equations is considered. In this method, the simulation domain is decomposed into the non-overlapping subdomains (blocks), and a physically-meaningful boundary condition is set on the mortar interface between the blocks. The governing equations hold locally on each subdomain region. The local problems on blocks are coupled by introducing a special approximation space on the interfaces of neighboring subdomains. Each block is locally covered by an independent grid and the standard mixed finite element method is applied to solve the local problem. The unique solvability of the discrete problem is shown, and optimal order convergence rates are established for the approximate velocity and pressure on the subdomain. Furthermore, an error estimate for the interface pressure in mortar space is presented. The numerical experiments are presented to validate the efficiency of the method.

Two-level implicit high order method based on half-step discretization for 1D unsteady biharmonic problems of first kind

In this work, compact difference scheme based on half-step discretization is proposed to solve fourth order time dependent partial differential equations subject to Dirichlet and Neumann boundary conditions. The difference method reported here is second order accurate in time and fourth order accurate in space. The scheme employs only three grid points at each time-level and the given boundary conditions are exactly satisfied with no further approximations at the boundaries. The linear stability of the presented method has been examined and it is shown that the proposed two-level finite difference method is unconditionally stable for a model linear problem. The developed method is directly applicable to fourth order parabolic partial differential equations with singular coefficients which is the main highlight of our work. The method is successfully tested on singular problem. The proposed method is applied to find the numerical solution of the classical nonlinear Kuramoto–Sivashinsky equation and the extended Fisher–Kolmogorov equation. Comparison of the obtained results with those of some earlier known methods demonstrate the superiority of the present approach.

Strong convergence of the linear implicit Euler method for the finite element discretization of semilinear SPDEs driven by multiplicative or additive noise

This paper aims to investigate the numerical approximation of a general second order parabolic stochastic partial differential equation(SPDE) driven by multiplicative or additive noise. The linear operator is not necessary self-adjoint, so more useful in concrete applications. The SPDE is discretized in space by the finite element method and in time by the linear implicit Euler method. The corresponding scheme is more stable and efficient to solve stochastic advection-dominated reactive transport in porous media. This extends the current results in the literature to not necessary self-adjoint operator. As a challenge, key part of the proof does not rely anymore on the spectral decomposition of the linear operator. The results reveal how the convergence orders depend on the regularity of the noise and the initial data. In particular for multiplicative trace class noise, we achieve optimal convergence order O(h2+Δt1/2) and for additive trace class noise, we achieve optimal convergence order in space and sup-optimal convergence order in time of the form O(h2+Δt1−ϵ), for an arbitrarily small ϵ > 0. Numerical experiments to sustain our theoretical results are provided.

Stability of fractional order parabolic partial differential equations using discretised backstepping method

This paper focuses on the application of backstepping control scheme for fractional order partial differential equations (FPDEs) of order with . Therefore to obtain highly accurate approximations for this derivative is of great importance. Here the discretised approach for the space variable is used to transform the FPDEs into a system of differential equations. These approximations arise mainly from the Caputo definition and the Grünwald-Letnikov definition. A Lyapunov function is defined at each stage and the negativity of an overall Lyapunov function is ensured by proper selection of the control law. Illustrative example is given to demonstrate the effectiveness of the proposed control scheme.

Unconditionally stable high accuracy compact difference schemes for multi-space dimensional vibration problems with simply supported boundary conditions

• We have derived 2-level implicit methods for 4th order vibration problems. • The methods are compact and unconditionally stable. • We do not require to discretize the boundary conditions. • We require only a tri-diagonal solver to compute the proposed numerical methods. • We have solved 09 practical examples arising from physics. The Euler–Bernoulli beam equation is a fourth order parabolic partial differential equation governing the transverse vibrations of a long and slender beam and is thus of interest in various engineering applications. In this study, we propose new two-level implicit difference formulas for the solution of vibration problem in one, two and three space dimensions subject to appropriate initial and boundary conditions. The proposed methods are fourth order accurate in space and second order accurate in time and are based upon a single compact stencil. The boundary conditions are incorporated in a natural way without any discretization or introduction of fictitious nodes. The derived methods are shown to be unconditionally stable for model linear problems. Some physical examples and their numerical results are given to illustrate the accuracy of the proposed methods. The test problems confirm that the computed solutions are not only in good agreement with the exact solutions but also competent with the solutions derived in earlier research studies.

Stochastic Reaction-diffusion Equations Driven by Jump Processes

We establish the existence of weak martingale solutions to a class of second order parabolic stochastic partial differential equations. The equations are driven by multiplicative jump type noise, with a non-Lipschitz multiplicative functional. The drift in the equations contains a dissipative nonlinearity of polynomial growth.

A class of two-level implicit unconditionally stable methods for a fourth order parabolic equation

In this paper, we discuss two-level implicit methods for the solution of a special type of fourth order parabolic partial differential equation of the form uxxxx-2uxxt+utt=f(x,t,u), 0<x<1, t>0 subject to appropriate initial and Dirichlet boundary conditions by converting the original problem to a coupled system of two second order parabolic equations. We use only three spatial grid points and it is not required to discretize the boundary conditions. The proposed Crank-Nicolson type scheme is second order accurate in both the temporal and spatial dimensions while the compact Crandall's type scheme is second order accurate in temporal and fourth order accurate in spatial dimension. The methods do not require any fictitious nodes outside the solution domain for handling the boundary conditions. For a fixed mesh ratio parameter (∆t/∆x2), the proposed Crandall's type method behaves like a fourth order method in space. Using matrix stability analysis, the proposed methods are shown to be unconditionally stable. The resulting implicit difference formulas gives block tri-diaginal matrix structure which is solved efficiently using block Gauss-Seidel method or block Newton method depending on linear or nonlinear behaviour of the equations. The methods compute the numerical value of u and time-dependent Laplacian uxx−ut, simultaneously. Numerical results are provided to demonstrate the accuracy and efficiency of the proposed methods.

On Some Lag Synchronization and Higher Order Parabolic Systems

Chaos synchronization is a topic of great interest, due to its observation in a huge variety of phenomena of different nature. We study synchronization of two chaotic oscillators in a Master- Slave configuration. The two dynamic systems are coupled via a directed feedback that randomly switches among a finite set of given constant function at a prescribed time rate. And we use Lyapunov stability theory. This paper discussed the using of lag synchronization approach, and provided the equilibrium solutions of a new class of higher order parabolic partial differential equations to be applicable for Lorenz chaotic system in order to minimize the dynamical error of large Lorenz chaotic system

Isomorphism theorems for some parabolic initial-boundary value problems in Hörmander spaces

Abstract In Hörmander inner product spaces, we investigate initial-boundary value problems for an arbitrary second order parabolic partial differential equation and the Dirichlet or a general first-order boundary conditions. We prove that the operators corresponding to these problems are isomorphisms between appropriate Hörmander spaces. The regularity of the functions which form these spaces is characterized by a pair of number parameters and a function parameter varying regularly at infinity in the sense of Karamata. Owing to this function parameter, the Hörmander spaces describe the regularity of functions more finely than the anisotropic Sobolev spaces.

Numerical solution of fourth order parabolic partial differential equation using parametric septic splines

Numerical solution of fourth order parabolic partial differential equation using parametric septic splines