Pseudoparabolic Partial Differential Equation Research Articles

New closed form solutions of some nonlinear pseudo-parabolic models via a new extended direct algebraic method

Abstract Our investigation delves into a specific category of nonlinear pseudo-parabolic partial differential equations (PDEs) that emerges from physical models. This set of equations includes the one-dimensional (1D) Oskolkov equation, the Benjamin-Bona-Mahony-Peregrine-Burgers (BBMPB) equation, the generalized hyperelastic rod wave (HERW) equation, and the Oskolkov Benjamin Bona Mahony Burgers (OBBMB) equation. We employ the new extended direct algebraic (NEDA) method to tackle these equations. The NEDA method serves as a powerful tool for our analysis, enabling us to obtain solutions grounded in various mathematical functions, such as hyperbolic, trigonometric, rational, exponential, and polynomial functions. As we delve into the physical implications of these solutions, we uncover complex structures with well-known characteristics. These include entities like dark, bright, singular, combined dark-bright solitons, dark-singular-combined solitons, solitary wave solutions, and others.

P-Biharmonic Pseudo-Parabolic Equation with Logarithmic Non linearity

This paper deals with the existence of solutions of a p-biharmonic pseudo parabolic partial differential equation with logarithmic nonlinearity in a bounded domain. We prove the global existence of the weak solutions using the Faedo-Galerkin method and applying the concavity approach, that the solutions blow up at a finite time. Further, we provide a maximal limit for the blow-up time.

Two approximation methods for fractional order Pseudo-Parabolic differential equations

In this study, fractional order pseudo-parabolic partial differential equation defined by Caputo derivative is investigated with initial-boundary conditions. Modified double Laplace decomposition method is used to find the exact solution of this equation. Explicit finite difference is constructed for this partial differential equation. Stability estimates are proved for these difference schemes. Error analysis table is obtained by compared the exact and approximate solutions. Figures showing the physical properties of the exact and approximate solutions are presented. From the error tables and figures, this applied method is an good and effective method for this equation.

Two Numerical Methods for Solving the Schrödinger Parabolic and Pseudoparabolic Partial Differential Equations

In this work, the initial-boundary value problems for one-dimensional linear time-dependent Schrödinger parabolic and pseudoparabolic partial differential equations are studied. The modified double Laplace decomposition method is applied to get the semianalytic solutions and the explicit finite difference method to get the approximate solutions of the problems. The von Neumann stability analysis of the presented problems is also investigated.

Lie symmetry analysis of a pseudoparabolic PDE: Power law in diffusion coefficient with constant viscosity

We perform symmetry analysis of a partial differential equation (PDE) of pseudoparabolic form which models solvent uptake in polymeric solids; a special case of power law for diffusion coefficient and constant viscosity is considered. Optimal systems of one-dimensional subalgebras for various model parameters are employed for symmetry reductions and construction of invariant solutions. • A phenomenon of case II diffusion in glassy polymers is considered. • A model equation is an implicit pseudoparabolic partial differential equation which incorporates the diffusion coefficient and viscosity of the polymer. • The model equation is studied from Lie symmetry analysis view point for a special case of power law in diffusion coefficient and constant viscosity. • Some exact solutions are obtained and presented graphically.

Double Laplace Decomposition Method and Finite Difference Method of Time-fractional Schrödinger Pseudoparabolic Partial Differential Equation with Caputo Derivative

In this paper, an initial-boundary value problem for a one-dimensional linear time-dependent fractional Schrödinger pseudoparabolic partial differential equation with Caputo derivative of order α ∈ 0,1 is being considered. Two strong numerical methods are employed to acquire the solutions to the problem. The first method used is the double Laplace decomposition method where closed-form solutions are obtained for any α ∈ 0,1 . As the second method, the implicit finite difference scheme is applied to obtain the approximate solutions. To clarify the performance of these two methods, numerical results are presented. The stability of the problem is also investigated.

Differential games in a Banach space on a fixed chain

The paper is devoted to obtaining the sufficient conditions for existence of epsilon-equilibrium in the sense of piecewise program strategies in antagonistic games associated with nonlinear non-autonomous controlled differential equation in a Banach space and cost functional of a general enough form. The concept of piecewise program strategies in such a game is defined on the base of a concept of Volterra set chain for a right-hand operator of the corresponding integral equation controlled by the opponent players and according to a given partition of the time segment. As example we consider the game associated with a nonlinear pseudoparabolic partial differential equation governing the evolution of electric field in a semiconductor.

On a final value problem for a nonlinear fractional pseudo-parabolic equation

<p style='text-indent:20px;'>In this paper, we investigate a final boundary value problem for a class of fractional with parameter <inline-formula><tex-math id="M1">$ \beta $</tex-math></inline-formula> pseudo-parabolic partial differential equations with nonlinear reaction term. For <inline-formula><tex-math id="M2">$ 0<\beta < 1, $</tex-math></inline-formula> the solution is regularity-loss, we establish the well-posedness of solutions. In the case that <inline-formula><tex-math id="M3">$ \beta >1 $</tex-math></inline-formula>, it has a feature of regularity-gain. Then, the instability of a mild solution is proved. We introduce two methods to regularize the problem. With the help of the modified Lavrentiev regularization method and Fourier truncated regularization method, we propose the regularized solutions in the cases of globally or locally Lipschitzian source term. Moreover, the error estimates is established.</p>

Fractional Order Pseudoparabolic Partial Differential Equation: Ulam–Hyers Stability

Using Gronwall inequality we will investigate the Ulam-Hyers and generalized Ulam–Hyers–Rassias stabilities for the solution of a fractional order pseudoparabolic partial differential equation.

Global solutions and blow up solutions to a class of pseudo-parabolic equations with nonlocal term

In this paper, we investigate an initial boundary value problem to a class of pseudo-parabolic partial differential equations with Newtonian nonlocal term. First, the local existence and uniqueness of a weak solution is established. In virtue of the energy functional and the related Nehari manifold, we also describe the exponent decay behavior and the blow up phenomenon of weak solutions with different kinds of initial data. Our second conclusion states that some solutions starting in a potential well exist globally, whereas solutions with suitable initial data outside the potential well must blow up. Furthermore, the instability of a ground state equilibrium solution is studied.

Poro-Visco-Elastic Compaction in Sedimentary Basins

The porosity of a visco-elastic medium is shown to satisfy a nonlinear pseudoparabolic partial differential equation of the form $ u' + A(u) (\alpha(u) + \eta u' ) = G(t,u)$ in which $u'$ denotes t...

Computing numerical solutions of the pseudo-parabolic Buckley–Leverett equation with dynamic capillary pressure

We present numerical approaches for solving a pseudo-parabolic partial differential equation, which models incompressible two phase flow in porous media taking into account dynamic effects in the capillary pressure. First, we briefly discuss two numerical schemes based on the operator splitting technique. Our numerical experiments show that the standard splitting, widely used to solve parabolic problems, may fail when applied to pseudo-parabolic models. As an illustration, we give an example for this case. So we present an operator splitting scheme based on a dispersive-like character that obtains correct numerical solutions. Then, we discuss an unsplit efficient numerical modelling, locally conservative by construction. This framework is based on a fully coupled space–time mixed hybrid finite element/volume discretization approach in order to account for the delicate local nonlinear balance between the numerical approximations of the hyperbolic flux and the pseudo-parabolic term, but linked to a natural dispersive-like character of the full pseudo-parabolic equation. We compare our numerical results with approximate solutions constructed with methods recently introduced in the specialized literature, in order to establish that we are computing the expected qualitative behaviour of the solutions.

Ulam-Hyers-Rassias stability of pseudoparabolic partial differential equations

The aim of this paper is to give some types of Ulam stability for a pseudoparabolic partial differential equation. In this case we consider Ulam-Hyers stability and generalized Ulam-Hyers-Rassias stability. We investigate some new applications of the Gronwall lemmas to the Ulam stability of a nonlinear pseudoparabolic partial differential equations.

A modified F-expansion method for solving nonlinear pseudoparabolic BBMB equations

In this paper, the modified F-expansion method with the aid of symbolic computational system Mathematica is implemented to obtain many exact travelling wave solutions for a special form of generalized nonlinear pseudo-parabolic Benjamin-Bona-Mohany-Burgers (BBMB) equation. The obtained solutions are expressed by the hyperbolic functions and the trigonometric functions. This method presents a wider applicability for handling nonlinear pseudo-parabolic partial differential equations arising from the mathematical physics.

Групповая классификация одного класса полулинейных псевдопараболических уравнений

Group classification is implemented for a pseudoparabolic partial differential equation with two parameters. Equivalence transformations groups are found and used for classification of the equation parameters. Kernels of principal symmetries groups are found for the equations. Principal symmetries groups are found for specifications of parameters expanding the kernel of transformations groups. The obtained submodels are summarized in a table at the end of the paper.

The Buckley–Leverett Equation with Dynamic Capillary Pressure

The Buckley-Leverett equation for two phase flow in a porous medium is modified by including a dependence of capillary pressure on the rate of change of saturation. This model, due to Gray and Hassanizadeh, results in a nonlinear pseudo-parabolic partial differential equation. Phase plane analysis, including a separation function to measure the distance between invariant manifolds, is used to determine when the equation supports traveling waves corresponding to undercompressive shocks. The Riemann problem for the underlying conservation law is solved and the structures of the various solutions are confirmed with numerical simulations of the partial differential equation.

Continuous Dependence for the Pseudoparabolic Equation

We determine the continuous dependence of solution on the parameters in a Dirichlet-type initial-boundary value problem for the pseudoparabolic partial differential equation.

On a Pseudo parabolic System for a Viscoelastic Shallow Shell

AbstractWe deal with the system describing moderately large deflections of thin viscoelastic shallow shells. The nonlinear straindisplacement relations due to von Kármán‐Donnell theory is considered. The system consists of an pseudoparabolic partial differential equation for a deflection and an equation with the biharmonic left‐hand side and the pseudoparabolic right‐hand side for the Airy stress function. The existence and uniqueness of a solution is verified by transforming the problem into the sequence of stationary canonical equations of von Kármán elastic type. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Identification of a Pseudoparabolic System by Use of a Quasi-Newton Method

A quasi-Newton method (QNM) in Hilbert spaces for identifying parameters involved in distributed parameter systems is presented in this paper. Next, the linear convergence of a sequence generated by the algorithm QNM is also proved. Furthermore, the algorithm QNM is applied to an identification problem for a pseudoparabolic partial differential equation to illustrate the efficiency of the algorithm QNM.

On pseudoparabolic equations and their approximation by parabolic equations

Pseudoparabolic partial differential equations (of composite type and of third order) has been used for the approximation of solutions of the parabolic equation M0[u]=0, and vice versa. We refer e.g. to the work of Ewing2, who considered the improperly posed backward time problem for the parabolic equation and its approximation by the properly posed initial value problem for the pseudoparabolic equation. This motivates our recent study of the solution to the equation (1) and their approximation for small parameters p. Our approach is a generalization of the method of complex integral operators due to S. Bergman and I. N. Vekua.