Order Partial Differential Equations Research Articles

Homotopy Analysis Method using Jumarie’s Approach for Multi-Dimensional Nonlinear Schrödinger Equations

In this paper, we suggest a fractional functional for the Homotopy Analysis Method (HAM) to solve the nonlinear fractional order partial differential equations with fractional order initial conditions by using the modified Riemann–Liouville fractional derivative proposed by G. Jumarie. Graphs have been plotted for different values of α , which clearly reflect the reliability of proposed scheme.

Fictitious finite integration method for solving high order partial differential equations

In this paper, we combine the finite integration method (FIM) with the technique of fictitious points to solve high order partial differential equations (HOPDEs). Two type of fictitious finite integration method (FFIM) are proposed and analyzed. The first type FFIM (FFIM-I) constructs the first order integration matrices by shifted Chebyshev polynomials so that the analytical high order integration matrices can be recursively obtained. Compared with the classical FIM, whose high order integration matrices are approximated directly through low order integration matrices, this FFIM-I leads to higher accuracy. Based on FFIM-I, the second type FFIM (FFIM-II) is proposed in which the fictitious points are located freely either inside or outside the problem domain. The FFIM-II can further improve the performance of FFIM-I in solving HOPDEs. The advantage of this fictitious finite integration method will be demonstrated by making comparisons among FIM, FFIM-I and FFIM-II through several 2D and 3D numerical experiments.

Hyers-Ulam-Rassias Stability of Third Order Partial Differential Equation

Hyers-Ulam-Rassias Stability of Third Order Partial Differential Equation

On the hyperbolic type differential equation with time involution

In the present paper, the initial value problem for the hyperbolic type involutory in t second order linear partial differential equation is studied. The initial value problem for the fourth order partial differential equations equivalent to this problem is obtained. The stability estimates for the solution and its first and second order derivatives of this problem are established.

Retraction

Retraction: Gnitchogna R, Atangana A. New two step Laplace Adam‐Bashforth method for integer a noninteger order partial differential equations. Numer. Methods Partial Differ. Eq. 2017; 34:1739–1758. https://doi.org/10.1002/num.22216.The above article, published online on 13 October 2017 in Wiley Online Library (wileyonlinelibrary.com), has been retracted by agreement between the journal Editor in Chief, Clayton Webster, and Wiley Periodicals, LLC. The article was re‐evaluated by the journal's editorial team and additional independent reviewers and a retraction has been agreed due to fundamental flaws in the paper which make the findings unreliable. The authors disagreed with the decision to retract the paper.

Hyperbolicity and nonconservativity of a hydrodynamic model of swarming rigid bodies

We study a nonlinear system of first order partial differential equations describing the macroscopic behavior of an ensemble of interacting self-propelled rigid bodies. Such system may be relevant for the modelling of bird flocks, fish schools or fleets of drones. We show that the system is hyperbolic and can be approximated by a conservative system through relaxation. We also derive viscous corrections to the model from the hydrodynamic limit of a kinetic model. This analysis prepares the future development of numerical approximations of this system.

The Fibonacci wavelets approach for the fractional Rosenau–Hyman equations

In this study, a novel approach, called the Fibonacci wavelet collocation technique (FWCT), is presented for the numerical solution of nonlinear fractional order partial differential equations (FPDEs). The chosen numerical strategy is based on the collocation method with the Fibonacci wavelet and its functional integration matrix. The newly developed numerical approach for the nonlinear time-fractional Rosenau–Hyman equation is the focus of this paper. The corresponding fractional order derivative is taken in the Caputo sense. This effective numerical approach converts the nonlinear problem into a set of algebraic-type nonlinear equations. This nonlinear system of equations is analyzed using the Newton–Raphson method to find the unknown coefficients. We provide the different types of error analysis of the present technique. The exact solution and the results obtained by the other literary methods are compared to the results achieved. The numerical outcomes demonstrate that the current analysis offers a more accurate approximation than the prior approaches. The result derived with the aid of this technique is in good agreement and indicates that the approach is easy to implement and accurate. These results reveal that the proposed method is computationally attractive, efficient, effective, reliable, and robust for solving various physical models in science and engineering. We made all calculations and plotted the graphs using Mathematica software.

Solutions of Some Wave Mechanics Models

We consider one-dimensional second order partial differential equations describing waves in inhomogeneous and nonlinear media. Contact transformations and Euler differential substitution are used to construct general solutions. General and partial solutions of some nonstationary continuum mechanics models are found.

Second order evolutionary problems driven by mixed quasi-variational–hemivariational inequalities

In this paper we introduce a differential quasi-variational inequality which consists of a second order partial differential equation involving history-dependent operators and a mixed quasi-variational–hemivariational inequality in Banach spaces. At first, by using the KKM theorem and monotonicity arguments, we show that the solution set of the mixed quasi-variational–hemivariational inequality is nonempty, bounded, closed and convex. Then, we establish the measurability and upper semicontinuity of the solution set with respect to the time variable and state variable. Finally, based on the theory of strongly continuous cosine operators and a fixed point theorem for condensing set-valued operators, we build the existence of mild solutions for the differential quasi-variational–hemivariational inequality.

Radiating stars and Riccati equations in higher dimensions

The objective of this study is to investigate spherically symmetric radiating stars undergoing gravitational collapse, in higher dimensional general relativity, inclusive of acceleration, expansion, shear, an electromagnetic field and a cosmological constant. Methods that can be used to obtain exact solutions to the boundary condition with/without a linear equation state are studied. Two distinct approaches are investigated. In the first approach, the boundary condition is expressed as a Riccati equation in terms of one of the dependent variables, and restrictions are placed to obtain new exact solutions. In the second approach, transformations that map the boundary condition into a new Riccati equation are investigated. The resulting new transformed equation is solved, by placing restrictions on the coefficients, to obtain new exact models. Special properties of the transformation are shown when appropriate restrictions on the parameters of the transformation are placed. This allows the order of the boundary condition to be reduced from a second order partial differential equation into a first order partial differential equation. The versatility of the transformation on other equations is exhibited when new solutions to the system of equations consisting of both the boundary condition and equation of state are obtained. When the dimension is set to four, some known solutions are recovered. It is shown that horizons can be identified by using a special case of the transformation. Our results elucidates the importance of the use of transformations that map the coordinates of differential equations into new and different coordinate systems.

Local Adaptiveness of Mixed Higher Order Partial Differential Equations and Its Application in Image Denoising

Background: Image denoising methods based on partial differential equations have attracted much attention due to their "infinite" local adaptation capabilities, high flexibility, and strong mathematical theoretical support. Methods: This paper proposes a mixed higher order partial differential equation denoising model for the step effect caused by the second-order denoising model and the edge blur caused by the fourth-order denoising model. The model combines the second-order and fourth-order terms based on the relationship between the variational energy minimization and the partial differential equations. The fourth-order term is used to remove noise in the uniform area of the image to avoid the step effect, and the second-order term is used at the edge to avoid boundary blur. Results: Theoretical analysis and numerical experiment results show that the proposed model has weak solutions and can effectively avoid the step effect and maintain the edge. Conclusion: The image denoising results of the model are better than those of other improved denoising models in subjective effect, and objective evaluation indicators, such as SNR, PSNR, and MSSIM.

Infrared-visible image fusion method based on sparse and prior joint saliency detection and LatLRR-FPDE

In this paper, a new infrared and visible image fusion method based on saliency detection and LatLRR-FPDE is proposed to ensure the saliency of the fused target and improve the clarity of the texture detail information of the latent low-rank representation (LatLRR) fusion method. In the feature representation stage, source images are decomposed into low-rank and sparse parts based on LatLRR. Then Fourth order Partial Differential Equation (FPDE) is used to extract high-frequency detail information from the low-rank part. In the fusion decision stage, according to the characteristic difference of the decomposed sub-bands, a multi-scale fusion scheme is carried out using rules such as EM (expectation-maximization), average (AVG), and the maximum absolute value. To guarantee the integrity of thermal radiation features in the target area in fusion results, an infrared image saliency detection algorithm is proposed, which utilizes the fusion information obtained from the sparse and prior saliency feature. Specifically, the algorithm uses convex hull center prior and contrast prior to obtain the background-suppressed saliency map. Meanwhile, to make full use of the high-dimensional saliency features in the image decomposition stage, the prior saliency map and the sparse saliency map are fused to improve the accuracy of saliency detection without the demand to extract redundant feature, which largely increases time-efficiency of the model. Finally, infrared feature compensation is performed on the target region of the fused image based on the saliency mapping strategy to make the final fused image having a better infrared target feature representation. The experiments are based on registered public datasets and compared with various popular image fusion algorithms. The subjective and objective results show that the proposed algorithm can effectively keep the integrity of salient infrared features and has superior visual quality.

Several families of new exact solutions for second order partial differential equations with variable coefficients

Several families of new exact solutions for a general second order linear partial differential equation with variable coefficients are derived in this paper. All the possible polynomial and polynomial-like solutions of this equation are derived. It is shown that there exist exactly two sets of such families of exact solutions. These solutions are extended to construct different families of exact solutions in terms of hypergeometric functions, which include polynomial solutions as particular cases. A total of eight families of exact solutions are derived using a novel method of balancing powers of the variables simultaneously. Several well known linear partial differential equations in applied mathematics and mechanics are special cases of the general equation considered in this paper and all the polynomial and polynomial-like solutions of these partial differential equations are also explicitly derived as special cases.

Meshfree numerical approach for some time-space dependent order partial differential equations in porous media

<abstract><p>In this article, the meshfree radial basis function method based on the Gaussian function is proposed for some time-space dependent fractional order partial differential equation (PDE) models. These PDE models have significant applications in chemical engineering and physical science. Some main advantages of the proposed method are that it is easy to implement, and the output response is quick and highly accurate, especially in the higher dimension. In this method, the time-dependent derivative terms are treated by Caputo fractional derivative while space-dependent derivative terms are treated by Riesz, Riemann-Liouville, and Grünwald-Letnikov derivatives. The proposed method is tested on some numerical examples and the accuracy is analyzed by $ \|L\|_\infty $.</p></abstract>

On the well-posedness of a tensor-based second order PDE with bilateral term for image super-resolution

<p style='text-indent:20px;'>In this paper, we introduce a second-order equation based on a diffusion tensor combined with a bilateral total variation (BTV) term that takes the benefit from the diffusion model of Perona-Malik in the homogeneous regions, the Weickert model near sharp edge and the BTV term in reducing blur. Compared to the other partial differential equations (PDE) used in the SR context, the proposed super-resolution (SR) PDE can efficiently preserve image components such as corners and flat regions with less apparition of blur in homogeneous regions. Moreover, since the SR approaches are always ill-posed, a mathematical study of the existence and uniqueness of the solution is also checked in a Sobolev space. From the numerical experiments we can observe that the proposed PDE can efficiently improve the quality of the high-resolution (HR) image. Surprisingly, the apparition of blur in the restored image is less compared to the other methods. In addition to visual evaluation, two-based metrics are computed to confirm the improvements offered by the proposed PDE.</p>

求解一类时间分数阶偏微分方程的高阶格式

求解一类时间分数阶偏微分方程的高阶格式

New approximate solutions to time fractional order partial differential equations optimal auxilary function method

In this article, approximate solutions of some PDE of fractional order are investi?gated with the help of a new semi-analytical method called the optimal auxiliary function method. The proposed method was tested upon the time-fractional Fisher equation, the time-fractional Fornberg-Whitham equation, and the time-fractional Inviscid Burger equation. The beauty of this method is that there is no need for discretization and assumptions of small or large parameters and provides an ap?proximate solution after only one iteration. The numerical results obtained by the proposed method compared with the other existing methods used in the literature. From the numerical and graphical results, it is clear that the proposed method gives a better solution than existing methods. The MATHEMATICA software pack?age has been used for the huge computational work.

Stability of Partial Differential Equations by Mahgoub Transform Method

The stability theory is an important research area in the qualitative analysis of partial differential equations. The Hyers-Ulam stability for a partial differential equation has a very close exact solution to the approximate solution of the differential equation and the error is very small which can be estimated. This study examines Hyers-Ulam and Hyers-Ulam Rassias stability of second order partial differential equations. We present a new method for research of the Hyers-Ulam stability of partial differential equations with the help of the Mahgoub transform. The Mahgoub transform method is practical as a fundamental tool to demonstrate the original result on this study. Finally, we give an example to illustrate main results. Our findings make a contribution to the topic and complete those in the relevant literature.

On two‐dimensional weakly singular fractional partial integro‐differential equations and dual Bernstein polynomials

AbstractThis article presents an efficient numerical scheme based on some properties of dual Bernstein operational matrices for a class of linear and nonlinear two‐dimensional weakly singular fractional partial integro‐differential equations (WSFPID). A novel property of the suggested scheme is the conversion of the WSFPID into an algebraic system of equations. The corresponding linear and nonlinear system of equations are solved by well‐known Newton–Raphson scheme. Convergence and an upper error bound, of the scheme are also analyzed. Finally, by providing several examples and reviewing the numerical results and comparing them with other available methods, we show that the mentioned method has acceptable accuracy and efficiency. The main important applications of the proposed scheme is that it can be applied on linear as well as nonlinear problems and can be applied on higher order partial differential equations too. Also, with a small number of bases, we can find a reasonable approximate solution.

Finite line method for solving high-order partial differential equations in science and engineering

In this paper, a completely new numerical method, named Finite Line Method (FLM), is proposed for solving general linear and non-linear high-order partial differential equations (PDEs) in science as well as engineering problems in heat conduction and mechanics. In this method, the computational domain is discretized into a number of collocation points as in the free element method (Gao et al., 2019; Liu et al., 2019), and at each collocation point a set of straight or curved lines crossing the point is formed, which is represented by a number of nodes distributed over lines. The shape functions for each line are constructed using the Lagrange interpolation formulation and the first and high order spatial partial derivatives of the shape functions with respect to global coordinates are derived through an ingenious technique. The derived partial derivatives can be directly substituted into the governing differential equations and related boundary conditions to set up the discretized system of equations.FLM is a type of collocation method, not needing any integration to establish the solution scheme. And since the Lagrange interpolation formulation is used to construct the shape functions, arbitrarily high order lines can be easily formulated for solving any types of linear and non-linear high order PDEs and engineering problems. A number of numerical examples for PDEs, heat conduction and solid mechanics problems are given to demonstrate the correctness and stability of the proposed method.