Solutions Of Parabolic Equations Research Articles

Approximation Solution of the Fractional Parabolic Partial Differential Equation by the Half-Sweep and Preconditioned Relaxation

In this study, the numerical solution of a space-fractional parabolic partial differential equation was considered. The investigation of the solution was made by focusing on the space-fractional diffusion equation (SFDE) problem. Note that the symmetry of an efficient approximation to the SFDE based on a numerical method is related to the compatibility of a discretization scheme and a linear system solver. The application of the one-dimensional, linear, unconditionally stable, and implicit finite difference approximation to SFDE was studied. The general differential equation of the SFDE was discretized using the space-fractional derivative of Caputo with a half-sweep finite difference scheme. The implicit approximation to the SFDE was formulated, and the formation of a linear system with a coefficient matrix, which was large and sparse, is shown. The construction of a general preconditioned system of equation is also presented. This study’s contribution is the introduction of a half-sweep preconditioned successive over relaxation (HSPSOR) method for the solution of the SFDE-based system of equation. This work extended the use of the HSPSOR as an efficient numerical method for the time-fractional diffusion equation, which has been presented in the 5th North American International Conference on industrial engineering and operations management in Detroit, Michigan, USA, 10–14 August 2020. The current work proposed several SFDE examples to validate the performance of the HSPSOR iterative method in solving the fractional diffusion equation. The outcome of the numerical investigation illustrated the competence of the HSPSOR to solve the SFDE and proved that the HSPSOR is superior to the standard approximation, which is the full-sweep preconditioned SOR (FSPSOR), in terms of computational complexity.

The classical periodic solutions for nonlinear parabolic integro-differential equations

The classical periodic solutions for nonlinear parabolic integro-differential equations

Extinction in finite time of solutions to fractional parabolic porous medium equations with strong absorption

In this article we study the solutions of a general fractional parabolic porous medium equation with a non-Lipschitz absorption term. We obtain the existence of weak solutions, \(L^p\)-estimates, and decay estimates. Also, we show that weak solutions must vanish after a finite time, even for large initial data. For more information see https://ejde.math.txstate.edu/Volumes/2021/29/abstr.html

Random attractors via pathwise mild solutions for stochastic parabolic evolution equations

We investigate the longtime behavior of stochastic partial differential equations (SPDEs) with differential operators that depend on time and the underlying probability space. In particular, we consider stochastic parabolic evolution problems in Banach spaces with additive noise and prove the existence of random exponential attractors. These are compact random sets of finite fractal dimension that contain the global random attractor and are attracting at an exponential rate. In order to apply the framework of random dynamical systems, we use the concept of pathwise mild solutions.

Qualitative analysis of solutions for a Kirchhoff-type parabolic equation with multiple nonlinearities

In this work, the local and global existence of weak solutions by using the Faedo-Galerkin method, the finite time blow up of the weak solution with positive initial energy and the general decay of the solution are discussed. Finally, we consider the exponential growth of the solution with sufficient conditions. This work generalizes and improves earlier results in the literature, see [L.X. Truong and N. Van Y, On a class of nonlinear heat equations with viscoelastic term, Comput. Math. Appl., 2016] and [L.X. Truong and N. Van Y, Exponential growth with ${L}^{p}$-norm of solutions for nonlinear heat equations with viscoelastic term, Appl. Math. Comput., 2016].

Maximal regularity of multistep fully discrete finite element methods for parabolic equations

AbstractThis article extends the semidiscrete maximal $L^p$-regularity results in Li (2019, Analyticity, maximal regularity and maximum-norm stability of semi-discrete finite element solutions of parabolic equations in nonconvex polyhedra. Math. Comp., 88, 1--44) to multistep fully discrete finite element methods for parabolic equations with more general diffusion coefficients in $W^{1,d+\beta }$, where $d$ is the dimension of space and $\beta>0$. The maximal angles of $R$-boundedness are characterized for the analytic semigroup $e^{zA_h}$ and the resolvent operator $z(z-A_h)^{-1}$, respectively, associated to an elliptic finite element operator $A_h$. Maximal $L^p$-regularity, an optimal $\ell ^p(L^q)$ error estimate and an $\ell ^p(W^{1,q})$ estimate are established for fully discrete finite element methods with multistep backward differentiation formulae.

Complicated asymptotic behavior of solutions for the fourth-order parabolic equation with absorption

In this paper, we consider the relationships between the solutions of a fourth-order parabolic equation and the solutions for this equation with absorption. We use the relationships to study the asymptotic behavior of solutions and find that the complicated asymptotic behavior can happen in the solutions of this equation with absorption.

MONOTONICITY AND ASYMPTOTIC PROPERTIES OF SOLUTIONS FOR PARABOLIC EQUATIONS VIA A GIVEN INITIAL VALUE CONDITION ON GRAPHS

In this paper, we establish several results involving the minimum and maximum principles and the comparison principles for elliptic equations and parabolic equations on finite graphs. The results are then used to prove the monotonicity and asymptotic properties of solutions for parabolic equations whose initial values are given by the equation [Formula: see text] with Dirichlet boundary conditions. Finally, an illustration with numerical experiments is provided to demonstrate our main results.

Least squares formulation for ill-posed inverse problems and applications

In this paper we propose a least squares formulation for ill-posed inverse problems. For example, ill-posed inverse problems in partial differential equations are those such that a solution of inverse problem exists for a smooth data but it does not depend continuously on data, or there is no solution for inverse problem, e.g. the Cauchy problem for elliptic equations and backward solution of parabolic equations. We develop the least squares formulation in which the sum of equations error over domain and data fitting criterion and Tikhonov regularization terms is minimized over entire solutions. In this way we can establish the existence and uniqueness of an inverse solution and establish the continuity of the inverse solution for noisy data in . The method can be applied to a general class of non-linear inverse problems and an operator theoretic stability analysis is developed. We describe how one can apply the method for various PDE inverse problems. Numerical tests using backward heat equation and Cauchy problem for elliptic equations are presented to demonstrate the applicability and performance of the proposed method.

Local null controllability of a model system for strong interaction between internal solitary waves

In this paper, we prove the local null controllability property for a nonlinear coupled system of two Korteweg–de Vries equations posed on a bounded interval and with a source term decaying exponentially on [Formula: see text]. The system was introduced by Gear and Grimshaw to model the interactions of two-dimensional, long, internal gravity waves propagation in a stratified fluid. We address the controllability problem by means of a control supported on an interior open subset of the domain and acting on one equation only. The proof consists mainly on proving the controllability of the linearized system, which is done by getting a Carleman estimate for the adjoint system. While doing the Carleman, we improve the techniques for dealing with the fact that the solutions of dispersive and parabolic equations with a source term in [Formula: see text] have a limited regularity. A local inversion theorem is applied to get the result for the nonlinear system.

Numerical solution for solving fractional parabolic partial differential equations

In this paper, A reliable numerical scheme is developed and reviewed in order to obtain approximate solution of time fractional parabolic partial differential equations. The introduced scheme is based on Legendre tau spectral approximation and the time fractional derivative is employed in the Caputo sense. TheL2convergence analysis of the numerical method is analyzed. Numerical results for different examples are examined to verify the accuracy of spectral method and justification the theoretical analysis, and to compare with other existing methods in the literatures

Extinction Phenomenon and Decay Estimate for a Quasilinear Parabolic Equation with a Nonlinear Source

By energy estimate approach and the method of upper and lower solutions, we give the conditions on the occurrence of the extinction and nonextinction behaviors of the solutions for a quasilinear parabolic equation with nonlinear source. Moreover, the decay estimates of the solutions are studied.

A gradient maximum principle of solutions for a quasilinear parabolic equation

This short note is concerned with a quasilinear diffusion equation under initial and Neumann boundary value condition. To be more precise, the authors establish a gradient maximum principle of classical solutions via the maximum principle and Hopf’s lemma. The result generalizes a recent work obtained by Kim (Proc Amer Math Soc 145:1203–1208, 2017).

Solving Parabolic and Hyperbolic Equations with Variable Coefficients Using Space-Time Localized Radial Basis Function Collocation Method

In this paper, we investigate the numerical approximation solution of parabolic and hyperbolic equations with variable coefficients and different boundary conditions using the space-time localized collocation method based on the radial basis function. The method is based on transforming the original d -dimensional problem in space into d + 1 -dimensional one in the space-time domain by combining the d -dimensional vector space variable and 1 -dimensional time variable in one d + 1 -dimensional variable vector. The advantages of such formulation are (i) time discretization as implicit, explicit, θ -method, method-of-line approach, and others are not applied; (ii) the time stability analysis is not discussed; and (iii) recomputation of the resulting matrix at each time level as done for other methods for solving partial differential equations (PDEs) with variable coefficients is avoided and the matrix is computed once. Two different formulations of the d -dimensional problem as a d + 1 -dimensional space-time one are discussed based on the type of PDEs considered. The localized radial basis function meshless method is applied to seek for the numerical solution. Different examples in two and three-dimensional space are solved to show the accuracy of such method. Different types of boundary conditions, Neumann and Dirichlet, are also considered for parabolic and hyperbolic equations to show the sensibility of the method in respect to boundary conditions. A comparison to the fourth-order Runge-Kutta method is also investigated.

Analytical evaluation of transformation matrix between functional spaces and its application in obtaining exact solutions of thermo-mechanical stresses in disks

Thermo-mechanical stresses developed in a disk, inner boundary of which is subject to the nonstationary thermo-mechanical loadings due to variation in temperature and pressure inside the disk is considered in the presented paper. Problems of theoretical analysis are divided and described as follows: The first problem implies analytical solution of the heat conduction equation with nonhomogeneous boundary conditions of Robin type, and the second problem deals with obtaining analytical solution of dynamic equation of displacement for nonisothermal elastic body (Navier equation) with traction boundary conditions in the plane strain case of uncoupled thermo-elasticity. Finite Hankel transform is used to solve the problems. Two sets of orthogonal basis functions necessary for integral transform are obtained. The first one corresponds to the boundary conditions written for the heat conduction equation, and the second fits the boundary conditions for Navier equation. Integrals defining elements of transformation matrix between the two sets of basis functions are evaluated analytically. In general, convenient method of obtaining analytical solution of linear hyperbolic equation with nonhomogeneous term, which itself is solution of nonhomogeneous parabolic equation, is elaborated within the presented work. Using finite difference method, discrete counterparts of the described problems are constructed and, via the built-in ordinary differential equation (ODE) solvers in MATLAB, numerical solutions are obtained, which exhibit excellent accordance with analytical ones.

Radon Measure-valued Solutions for Nonlinear Strongly Degenerate Parabolic Equations with Measure Data

In this paper, we prove the existence of Radon measure-valued solutions for nonlinear degenerate parabolic equations with nonnegative bounded Radon measure data. Moreover, we show the uniqueness of the measure-valued solutions when the Radon measure as a forcing term is diffuse with respect to the parabolic capacity and the Radon measure as a initial value is diffuse with respect to the Newtonian capacity. We also deduce that the concentrated part of the solution with respect to the Newtonian capacity depends on time.

A Kolmogorov-type theorem for stochastic fields

We generalize the Kolmogorov continuity theorem and prove the continuity of a class of stochastic fields with the parameter. As an application, we derive the continuity of solutions for nonlocal stochastic parabolic equations driven by non-Gaussian Lévy noises.

Entropy Solutions for Nonlinear Parabolic Equations with Nonstandard Growth in Non-reflexive Orlicz Spaces

We prove in this paper an existence result of entropy solutions for nonlinear parabolic equations of the form: $$\begin{aligned} \displaystyle \frac{\partial u}{\partial t}-{\text {div}}\, a(x,t,u,\nabla u)-\mathrm{div}\Phi (x,t,u)= f \quad \text {in }{Q_T=\Omega \times (0,T)}, \end{aligned}$$ where the lower order term $$\Phi $$ satisfies only a natural growth condition prescribed by the N-function M defining the Orlicz spaces framework and the data f are an element of $$L^1(Q_T)$$ . We do not assume any restriction neither on M nor on its complementary $$\overline{M}$$ . No particular growth is considered on $$\Phi $$ .

Cubic B-spline collocation method on non-uniform mesh for solving non-linear parabolic partial differential equation

In this paper, an approximate solution of non-linear parabolic partial differential equation is obtained for a non-uniform mesh. The scheme for partial differential equation subject to Neumann boundary is based on cubic B-spline collocation method. Modified cubic B-splines are proposed over non-uniform mesh to deal with the Dirichlet boundary conditions. This scheme produces a system of first order ordinary differential equations. This system is solved by Crank Nicholson method. The stability is also discussed using Von Neumann stability analysis. The accuracy and efficiency of the scheme is shown by numerical experiments. We have compared the approximate solutions with that in the literature.

L^{infty } decay estimates of solutions of nonlinear parabolic equation

In this paper, we are interested in L^{infty } decay estimates of weak solutions for the doubly nonlinear parabolic equation and the degenerate evolution m-Laplacian equation not in the divergence form. By a modified Moser’s technique we obtain L^{infty } decay estimates of weak solutiona.