Pseudo-parabolic Problem Research Articles

Recovering initial population density of fractional pseudo-parabolic problem associated with a nonlinear reaction

Recovering initial population density of fractional pseudo-parabolic problem associated with a nonlinear reaction

An inverse problem for pseudoparabolic equation: existence, uniqueness, stability, and numerical analysis

In this paper, we study an inverse problem for a linear third-order pseudoparabolic equation. The investigated inverse problem consists of determining a space-dependent coefficient of the right-hand side of a pseudoparabolic equation. As an additional information a final overdetermination condition is considered. Under the suitable conditions on the data of the problem, the unique solvability of the considered inverse problem is established. The stability of solutions is also proved. The established results are also true for inverse problems for parabolic equations, which could be obtained as a regularization of the studied pseudoparabolic equation. In addition, the pseudoparabolic problem is discretized using the cubic B-spline functions and recast as a nonlinear least-squares minimization of the Tikhonov regularization function. Numerically, this is effectively solved using the MATLAB subroutine lsqnonlin. Both exact and noisy data are inverted. Numerical results for three benchmark test examples are presented and discussed. Moreover, the von Neumann stability analysis is also discussed.

An inverse source problem for a pseudoparabolic equation with memory

<abstract><p>This paper is devoted to investigating the well-posedness, as well as performing the numerical analysis, of an inverse source problem for linear pseudoparabolic equations with a memory term. The investigated inverse problem involves determining a right-hand side that depends on the spatial variable under the given observation at a final time along with the solution function. Under suitable assumptions on the problem data, the existence, uniqueness and stability of a strong generalized solution of the studied inverse problem are obtained. In addition, the pseudoparabolic problem is discretized using extended cubic B-spline functions and recast as a nonlinear least-squares minimization of the Tikhonov regularization function. Numerically, this problem is effectively solved using the MATLAB subroutine <italic>lsqnonlin</italic>. Both exact and noisy data are inverted. Numerical results for a benchmark test example are presented and discussed. Moreover, the von Neumann stability analysis is also discussed.</p></abstract>

A computational method for the singularly perturbed delay pseudo-parabolic differential equations on adaptive mesh

This paper aims to establish the finite difference scheme on Bakhvalov-type mesh for the singularly perturbed pseudo-parabolic problems with time-delay. Using the energy estimates, the bounds of the solution are investigated. An error analysis is derived in the discrete energy norm and the almost second order convergence is obtained. The reliability of the suggested method is displayed by means of a numerical example.

Grow-up of weak solutions in a [formula omitted]-Laplacian pseudo-parabolic problem

This paper deals with a p-Laplacian pseudo-parabolic equation with a homogeneous Dirichlet boundary condition in bounded domains. By using potential well method and the logarithmic Sobolev inequality, we study the grow-up phenomena and the bounds of growth rate, which are classified by the energy functional and the Nehari functional for the initial data.

An inverse boundary value problem for a two-dimensional pseudo-parabolic equation of third order

In the present work, we consider an inverse boundary value problem for a two-dimensional pseudo-parabolic equation of the third-order. Using analytical and operator-theoretic methods, as well as the Fourier method, the existence and uniqueness of the classical solution of this problem is proved. This inverse problem is formulated as an auxiliary inverse problem which, in turn, is reduced to the operator equation in a specified Banach space using the method of spectral analysis. In addition, the two-dimensional pseudo-parabolic problem is discretized using the FDM and reshaped as nonlinear least-squares optimization of the Tikhonov regularization function. This is numerically solved by means of the MATLAB subroutine lsqnonlin tool. Both analytical and perturbed data are inverted. Numerical outcomes for benchmark test example is reported and discussed.

Recovering a source term in the higher-order pseudo-parabolic equation via cubic spline functions

In this paper, we considered an inverse problem of recovering the space-dependent source coefficient in the third-order pseudo-parabolic equation from final over-determination condition. This inverse problem appears extensively in the modelling of various phenomena in physics such as the motion of non-Newtonian fluids, thermodynamic processes, filtration in a porous medium, etc. The unique solvability theorem for this inverse problem is supplied. However, since the governing equation is yet ill-posed (very slight errors in the final input may cause relatively significant errors in the output source term), we need to regularize the solution. Therefore, to get a stable solution, a regularized cost function is to be minimized for retrieval of the unknown force term. The third-order pseudo-parabolic problem is discretized using the Cubic B-spline (CB-spline) collocation technique and reshaped as non-linear least-squares optimization of the Tikhonov regularization function. Numerically, this is effectively solved using the lsqnonlin routine from the MATLAB toolbox. Both perturbed data and analytical solutions are inverted. Numerical outcomes are reported and discussed. The computational efficiency of the method is investigated by small values of CPU time. In addition, the von Neumann stability analysis for the proposed numerical approach has also been discussed.

Identifying an unknown heat source term in the third-order pseudo-parabolic equation from nonlocal integral observation

The aim of this paper is to find the time-dependent heat source term numerically in the third-order pseudo-parabolic equation with initial and boundary conditions supplemented by nonlocal integral observation. This is a very interesting and challenging nonlinear inverse source problem with important applications in various fields of mechanics and physics. Unique solvability theorem of this inverse problem is supplied. However, since the problem is still ill-posed (small errors in the integral input data cause large errors in the output force) the solution needs to be regularized. Therefore, to obtain a stable solution, a regularized objective function is minimized to retrieve the unknown heat source coefficient. The third-order pseudo-parabolic problem is discretized using the finite difference method (FDM) and recast as nonlinear least-squares minimization of the Tikhonov regularization function with simple bounds on the unknown force coefficient. Numerically, this is effectively solved using the MATLAB subroutine lsqnonlin. Both analytical and perturbed (noisy) data are inverted. Numerical results for three benchmark test examples are presented and discussed. In addition, the von Neumann stability analysis is also discussed.

Reconstructing an unknown potential term in the third-order pseudo-parabolic problem

Reconstructing an unknown potential term in the third-order pseudo-parabolic problem

Well-posedness of two pseudo-parabolic problems for electrical conduction in heterogeneous media

We prove a well-posedness result for two pseudo-parabolic problems, which can be seen as two models for the same electrical conduction phenomenon in heterogeneous media, neglecting the magnetic field. One of the problems is the concentration limit of the other one, when the thickness of the dielectric inclusions goes to zero. The concentrated problem involves a transmission condition through interfaces, which is mediated by a suitable Laplace-Beltrami type equation.

Regularized solution approximation of a fractional pseudo-parabolic problem with a nonlinear source term and random data

In this paper, we consider a multi-dimensional fractional pseudo-parabolic problem with nonlinear source in case the input data is measured on a discrete set of points instead of the whole domain. For any number of dimensions, the solution is not stable. This makes the problem we are interested in be ill-posed. Here, we construct regularized solutions for this problem in two cases of number of dimensions (denoted by m ) including m = 2 and m is arbitrary. In each case, we show the uniqueness of the regularized solution and give the error estimates. Finally, the convergence rate is also investigated numerically.

Numerical treatment for the solution of singularly perturbed pseudo-parabolic problem on an equidistributed grid

AbstractThe initial-boundary value problem for a pseudo-parabolic equation exhibiting initial layer is considered. For solving this problem numerically independence of the perturbation parameter, we propose a difference scheme which consists of the implicit-Euler method for the time derivative and a central difference method for the spatial derivative on uniform mesh. The time domain is discretized with a nonuniform grid generated by equidistributing a positive monitor function. The performance of the numerical scheme is tested which confirms the expected behavior of the method. The existing method is compared with other methods available in the recent literature.

P.120 Plasma lipidomics in patients with affective disorders

In this paper, we consider a multi-dimensional fractional pseudo-parabolic problem with nonlinear source in case the input data is measured on a discrete set of points instead of the whole domain. For any number of dimensions, the solution is not stable. This makes the problem we are interested in be ill-posed. Here, we construct regularized solutions for this problem in two cases of number of dimensions (denoted by m) including m=2 and m is arbitrary. In each case, we show the uniqueness of the regularized solution and give the error estimates. Finally, the convergence rate is also investigated numerically.

High order numerical schemes for solving fractional powers of elliptic operators

In many recent applications when new materials and technologies are developed it is important to describe and simulate new nonlinear and nonlocal diffusion transport processes. A general class of such models deals with nonlocal fractional power elliptic operators. In order to solve these problems numerically it is proposed (Vabishchevich, 2015) to consider equivalent local nonstationary initial value pseudo-parabolic problems. Previously such problems were solved by using the standard implicit backward and symmetrical Euler methods. In this paper we use the one-parameter family of three-level finite difference schemes for solving the initial value problem for the first order nonstationary pseudo-parabolic problem. The fourth-order approximation scheme is developed by selecting the optimal value of the weight parameter. The results of the theoretical analysis are supplemented by results of extensive computational experiments.

A degenerate pseudo-parabolic equation with memory

Abstract We prove the existence and uniqueness for a degenerate pseudo-parabolic problem with memory. This kind of problem arises in the study of the homogenization of some differential systems involving the Laplace-Beltrami operator and describes the effective behaviour of the electrical conduction in some composite materials.

High-order finite difference technique for delay pseudo-parabolic equations

One dimensional initial–boundary delay pseudo-parabolic problem is being considered. To solve this problem numerically, we construct higher order difference method for approximation to the considered problem and obtain the error estimate for its solution. Based on the method of energy estimate the fully discrete scheme is shown to be convergent of order four in space and of order two in time. Numerical example is presented.

Parallel algorithms for three-dimensional parabolic and pseudoparabolic problems with different boundary conditions

In this paper, three-dimensional parabolic and pseudo-parabolic equations with classical, periodic and nonlocal boundary conditions are approximated by the full approximation backward Euler method, locally one dimensional and Douglas ADI splitting schemes. The stability with respect to initial conditions is investigated. We note that the stability of the proposed numerical algorithms can be proved only if the matrix of discrete operator can be diagonalized and eigenvectors make a complete basis system.Parallel versions of all algorithms are constructed and scalability analysis is done. It is shown that discrete one-dimensional problems with periodic and nonlocal boundary conditions can be efficiently solved with similar modifications of the parallel Wang algorithm.

On Stability Analysis of Finite-Difference Schemes for Some Parabolic Problems with Nonlocal Boundary Conditions

In this article, one-dimensional parabolic and pseudo-parabolic equations with nonlocal boundary conditions are approximated by the implicit Euler finite-difference scheme. For a parabolic problem, the stability analysis is done in the weak H −1 type norm, which enables us to generalize results obtained in stronger norms. In the case of a pseudo-parabolic problem, the stability analysis is done in the discrete analog of the norm. It is shown that a solution of the proposed finite-discrete scheme satisfies stronger stability estimates than a discrete solution of the parabolic problem. Results of numerical experiments are presented.

ON CONSTRUCTION AND ANALYSIS OF FINITE DIFFERENCE SCHEMES FOR PSEUDOPARABOLIC PROBLEMS WITH NONLOCAL BOUNDARY CONDITIONS

In this paper the one- and two-dimensional pseudoparabolic equations with nonlocal boundary conditions are approximated by the Euler finite difference scheme. In the case of classical boundary conditions the stability of all schemes is investigated by the spectral method. Stability regions of finite difference schemes approximating pseudoparabolic problem are compared with the stability regions of the classical discrete parabolic problem. These results are generalized for problems with nonlocal boundary conditions if a matrix of the finite difference scheme can be diagonalized. For the two-dimensional problem an efficient algorithm is constructed, which is based on the combination of the FFT method and the factorization algorithm. General stability results, known for the three level finite difference schemes, are applied to investigate the stability of some explicit approximations of the two-dimensional pseudoparabolic problem with classical boundary conditions. A connection between the energy method stability conditions and the spectrum Hurwitz stability criterion is shown. The obtained results can be applied for pseudoparabolic problems with nonlocal boundary conditions, if a matrix of the finite difference scheme can be diagonalized.

On the solvability of a nonlinear pseudoparabolic problem

This paper is concerned with the study of an initial boundary value problem for a nonlinear second order pseudoparabolic equation arising in the unidirectional flow of a thermodynamic compatible third grade fluid. We establish some a priori bounds for the solution and prove its existence.