Fourth-order Parabolic Partial Differential Equation Research Articles

Approximate Solution to Fractional Order Models Using a New Fractional Analytical Scheme

In the present work, a new fractional analytical scheme (NFAS) is developed to obtain the approximate results of fourth-order parabolic fractional partial differential equations (FPDEs). The fractional derivatives are considered in the Caputo sense. In this scheme, we show that a Taylor series destructs the recurrence relation and minimizes the heavy computational work. This approach presents the results in the sense of convergent series. In addition, we provide the convergence theorem that shows the authenticity of this scheme. The proposed strategy is very simple and straightforward for obtaining the series solution of the fractional models. We take some differential problems of fractional orders to present the robustness and effectiveness of this developed scheme. The significance of NFAS is also shown by graphical and tabular expressions.

Space-time spectral methods for a fourth-order parabolic optimal control problem in three control constraint cases

<p style='text-indent:20px;'>In this paper, we are concerned with the space-time spectral discretization of an optimal control problem governed by a fourth-order parabolic partial differential equations (PDEs) in three control constraint cases. The dual Petrov-Galerkin spectral method in time and the spectral method in space are adopted to discrete the continuous system. By means of the obtained optimality condition for the continuous system and that of its spectral discrete system, we establish a priori error estimate for the spectral approximation in details. Four numerical examples are, subsequently, executed to confirm the theoretical results. The experiment results show the high efficiency and a good precision of the space-time spectral method for this kind of problems.</p>

A new strategy for the approximate solution of fourth-order parabolic partial differential equations with fractional derivative

PurposeThis study aims to purpose the idea of a new hybrid approach to examine the approximate solution of the fourth-order partial differential equations (PDEs) with time fractional derivative that governs the behaviour of a vibrating beam. The authors have also demonstrated the physical representations of the problem in different fractional order.Design/methodology/approachMohand transform is a new technique that the authors use to reduce the order of fractional problems, and then the homotopy perturbation method can be used to handle the further series solution in the form of convergence. The formulation of Mohand transform and the homotopy perturbation method is known as Mohand homotopy perturbation transform (MHPT). The fractional order in this paper is considered in the Caputo sense.FindingsThe results are formulated in the shape of iterative series and predict the solution close to the exact solution. This successive iteration demonstrates the authenticity and reliability of this scheme.Research limitations/implicationsThis paper presents the significance of MHPT such that, firstly, Mohand transform is coupled with homotopy perturbation method and, secondly, the fractional order a is used to show the physical behaviour of the graphical solution.Practical implicationsThis study presents the consistency and authenticity of the graphical solution with the exact solutions.Social implicationsThis study demonstrates that Mohand transform is capable to handle the fractional order problem without any constraints and assumptions.Originality/valueA new integral transform has been introduced without any restriction of variables that produces the results in a series form and confirms the validity of the proposed algorithm by graphical illustrations.

A Numerical Approach for the Analytical Solution of the Fourth-Order Parabolic Partial Differential Equations

In this study, we propose a new iterative scheme (NIS) to investigate the approximate solution of the fourth-order parabolic partial differential equations (PDEs) that arises in transverse vibration problems. We introduce the Mohand transform as a new operator that is very easy to implement coupled with the homotopy perturbation method. This NIS is capable of reducing the linearization, perturbation, and restrictive assumptions that ruin the nature of the numerical problems. Some numerical examples are demonstrated to legitimate the accuracy and authenticity of this NIS. The computational results are obtained in the shape of a series that converges only after a few iterations. The comparison of the graphical representations shows that NIS is a very simple but also an effective approach for other numerical problems involving complex variables.

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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A Comparative Study of Cubic B-spline-Based Quasi-interpolation and Differential Quadrature Methods for Solving Fourth-Order Parabolic PDEs

In this work, we present two approaches for simulation of fourth-order parabolic partial differential equations. In the first method, cubic B-spline quasi-interpolation is used to approximate the spatial derivative of the dependent variable and forward difference to approximate the time derivative. In the second method, we have used modified cubic B-spline functions-based differential quadrature method (DQM) for space discretization to get a system of ODEs and then this system is solved by SSP-RK43 method to get the results at knots. The numerical results demonstrate the accuracy of the proposed method. The stability analysis of the methods has also been discussed. It is observed that quasi-interpolation-based method is unconditionally stable, whereas for DQM, the stability has to be checked for a large number of space points. Moreover, for the small number of grid points, DQM gives better results, while for a large number of grid points, quasi-interpolation-based method is better.

An Efficient Analytical Technique for Time-Fractional Parabolic Partial Differential Equations

In this work, we examine the time-fractional fourth-order parabolic partial differential equations with the aid of the optimal homotopy asymptotic method (OHAM). The 2nd order approximate results obtained by using the suggested scheme are compared with the exact solution. It has been noted that the results achieved via OHAM have a large convergence rate for the problems. The solutions are graphically analyzed, and the relative errors are presented in tabular form.

Modified Laplace variational iteration method for solving fourth-order parabolic partial differential equation with variable coefficients

In the present study, a new amendment in Laplace variational iteration method for the solution of fourth-order parabolic partial differential equations with variable coefficients is revealed i.e. modified Laplace variational iteration method (ML-VIM). The proposed modification is made by coupling of two methods: one of them is variational iteration method (VIM) and the other one is Laplace transformation (LT). Our modification has an important beauty that one has no need to compute Lagrange multiplier via integration nor by taking convolution theorem and has a simple way to use in the implementation of our proposed scheme. Moreover, homotopy perturbation method (HPM) with He’s polynomials is employed for the computation of nonlinear terms. The main improvement of our proposed scheme is the reduction of the problem to a simple one, which also embraces for solving a nonlinear term. Some illustrated examples are interpreted which unveil the robustness and accuracy of the newly developed scheme.

Tension spline technique for the solution of fourth-order parabolic partial differential equation

Tension spline technique for the solution of fourth-order parabolic partial differential equation

Computer-Assisted Proof of Heteroclinic Connections in the One-Dimensional Ohta--Kawasaki Model

We present a computer-assisted proof of heteroclinic connections in the one-dimensional Ohta--Kawasaki model of diblock copolymers. The model is a fourth-order parabolic partial differential equation subject to homogeneous Neumann boundary conditions, which contains as a special case the celebrated Cahn--Hilliard equation. While the attractor structure of the latter model is completely understood for one-dimensional domains, the diblock copolymer extension exhibits considerably richer long-term dynamical behavior, which includes a high level of multistability. In this paper, we establish the existence of certain heteroclinic connections between the homogeneous equilibrium state and local and global energy minimizers. The proof of the above statement is conceptually simple and combines several techniques from some of the authors' and Zgliczyński's works. Central for the verification is the rigorous propagation of a piece of the unstable manifold of the homogeneous state with respect to time. This propagation has to lead to small interval bounds, while at the same time entering the basin of attraction of the stable fixed point. For interesting parameter values the global attractor exhibits a complicated equilibrium structure, and the dynamical equation is rather stiff. This leads to a time-consuming numerical propagation of error bounds, with many integration steps. This problem is addressed using an efficient algorithm for the rigorous integration of partial differential equations forward in time. The method is able to handle large integration times within a reasonable computational time frame, and this makes it possible to establish heteroclinic connections at various nontrivial parameter values.

Application of the Adomian Decomposition Method (ADM) for Solving the Singular Fourth-Order Parabolic Partial Differential Equation

In this paper, the ADM method is used to construct the solution of the singular fourth-order partial differential equation.

A class of quasi-variable mesh methods based on off-step discretization for the numerical solution of fourth-order quasi-linear parabolic partial differential equations

Numerical schemes based on off-step discretization are developed to solve two classes of fourth-order time-dependent partial differential equations subjected to appropriate initial and boundary conditions. The difference methods reported here are second-order accurate in time and second-order accurate in space and, for a nonuniform grid, second-order accurate in time and third-order accurate in space. In case of a uniform grid, the second scheme is of order two in time and four in space. The presented methods split the original problem to a coupled system of two second-order equations and involve only three spatial grid points of a compact stencil without discretizing the boundary conditions. The linear stability of the presented methods has been examined, and it is shown that the proposed two-level finite difference method is unconditionally stable for a linear model problem. The new developed methods are directly applicable to fourth-order parabolic partial differential equations with singular coefficients, which is the main highlight of our work. The methods are successfully tested on singular problems. The proposed method is applied to find numerical solutions of the Euler-Bernoulli beam equation and complex fourth-order nonlinear equations like the good Boussinesq equation. Comparison of the obtained results with those for some earlier known methods show the superiority of the present approach.

Computer-assisted equilibrium validation for the diblock copolymer model

The diblock copolymer model is a fourth-order parabolic partial differential equation which models phase separation with fine structure. The equation is a gradient flow with respect to an extension of the standard van der Waals free energy functional which involves nonlocal interactions. Thus, the long-term dynamical behavior of the diblock copolymer model is described by its finite-dimensional attractor. However, even on one-dimensional domains, the full structure of the underlying equilibrium set is not fully understood. In this paper, we develop a rigorous computational approach for establishing the existence of equilibrium solutions of the diblock copolymer model. We consider both individual solutions, as well as pieces of solution branches in a parameter-dependent situation. The results are presented for the case of one-dimensional domains, and can easily be implemented using standard interval arithmetic packages.

Application Of Reduced Differential Transformation Method For Solving Fourth-order Parabolic Partial Differential Equations

The purpose of this paper is to obtain the approximate solution of fourth-order parabolic partial differential equations by the reduced differential transform method (RDTM).This method provides the solution in the form of a convergent series with easily calculable terms. Comparing RDTM with some other methods in the literature shows present approach is very simple, effective, powerful and can be easily applied to other linear or nonlinear PDEs in science and engineering.

A coupling method based on new MFE and FE for fourth-order parabolic equation

In this article, a coupling method of new mixed finite element (MFE) and finite element (FE) is proposed and analyzed for fourth-order parabolic partial differential equation. First, the fourth-order parabolic equation is split into the coupled system of second-order equations. Then, an equation is solved by finite element method, the other equation is approximated by the new mixed finite element method, whose flux belongs to the square integrable space replacing the classical H(div;Ω) space. The stability for fully discrete scheme is derived, and both semi-discrete and fully discrete error estimates are obtained. Moreover, the optimal a priori error estimates in L 2 and H 1-norm for both the scalar unknown u and the diffusion term γ and a priori error estimate in (L 2)2-norm for its flux σ are derived. Finally, some numerical results are provided to validate our theoretical analysis.

Well-posedness for a class of fourth order diffusions for image processing

A number of image denoising models based on higher order parabolic partial differential equations (PDEs) have been proposed in an effort to overcome some of the problems attendant to second order methods such as the famous Perona–Malik model. However, there is little analysis of these equations to be found in the literature. In this paper, methods of maximal regularity are used to prove the existence of unique local solutions to a class of fourth order PDEs for noise removal. The proof is laid out explicitly for two newly proposed fourth order models, and an outline is given for how to apply the techniques to other proposed models.

Analytical Aspect of Fourth-Order Parabolic Partial Differential Equations with Variable Coefficients

In this work, the homotopy analysis method (HAM) is applied to solve the fourth-order parabolic partial differential equations. This equation practically arises in the transverse vibration problems. The proposed iterative scheme finds the solution without any discritization, linearization, or restrictive assumptions. Some applications are given to verify the reliability and efficiency of the method. The convergence control parameter h in the HAM solutions has provided a convenient way of controlling the convergence region of series solutions. It is also shown that the solutions that are obtained by Adomian decomposition method (ADM) and variational iteration method (VIM) are special cases of the solutions obtained by HAM.

He's homotopy perturbation method for fourth-order parabolic equations

In this paper, He's homotopy perturbation method is applied to fourth-order parabolic partial differential equations with variable coefficients to obtain the analytic solution. The method is tested on six examples, which reveal its effectiveness and simplicity.

Nonnegative Solutions for a Long-Wave Unstable Thin Film Equation with Convection

We consider a nonlinear fourth-order degenerate parabolic partial differential equation that arises in modeling the dynamics of an incompressible thin liquid film on the outer surface of a rotating horizontal cylinder in the presence of gravity. The parameters involved determine a rich variety of qualitatively different flows. Depending on the initial data and the parameter values, we prove the existence of nonnegative periodic weak solutions. In addition, we prove that these solutions and their gradients cannot grow any faster than linearly in time; there cannot be a finite-time blowup. Finally, we present numerical simulations of solutions.

The Solution of the Variable Coefficients Fourth-Order Parabolic Partial Differential Equations by the Homotopy Perturbation Method

Abstract In this work, the homotopy perturbation method proposed by Ji-Huan He [1] is applied to solve both linear and nonlinear boundary value problems for fourth-order partial differential equations. The numerical results obtained with minimum amount of computation are compared with the exact solution to show the efficiency of the method. The results show that the homotopy perturbation method is of high accuracy and efficient for solving the fourth-order parabolic partial differential equation with variable coefficients. The results show also that the introduced method is a powerful tool for solving the fourth-order parabolic partial differential equations.