Third-order Partial Differential Equation Research Articles

A Note on Fractional Third-Order Partial Differential Equations and the Generalized Laplace Transform Decomposition Method

This paper establishes a unique approach known as the multi-generalized Laplace transform decomposition method (MGLTDM) to solve linear and nonlinear dispersive KdV-type equations. This method combines the multi-generalized Laplace transform (MGLT) with the decomposition method (DM), and offers a strong procedure for handling complicated equations. To verify the applicability and validity of this method, some ideal problems of dispersive KDV-type equations are discussed and the outcoming approximate solutions are stated in sequential form. The results show that the MGLTDM is a dependable and powerful technique to deal with physical problems in diverse implementations.

Self-Similar Solutions of a Multidimensional Degenerate Partial Differential Equation of the Third Order

When studying the boundary value problems’ solvability for some partial differential equations encountered in applied mathematics, we frequently need to create systems of partial differential equations and explicitly construct linearly independent solutions explicitly for these systems. Hypergeometric functions frequently serve as solutions that satisfy these systems. In this study, we develop self-similar solutions for a third-order multidimensional degenerate partial differential equation. These solutions are represented using a generalized confluent Kampé de Fériet hypergeometric function of the third order.

Absolutely stable difference scheme for the delay partial differential equation with involution and Robin boundary condition

This paper examines the initial value problem for a third-order delay partial differential equation with involution and Robin boundary condition. We construct a first-order accurate difference scheme to obtain the numerical solution for this equation. Illustrative numerical results are provided.

Unveiling advection–dominated interactions: Efficacy of neural networks in natural systems modelling

Studying the interactions between advection and dispersion in natural systems, especially in cases where advection predominates, are important because it is necessary to accurately model physical phenomena in domains like hydrology, atmospheric science, environmental engineering, etc. Conventional analytical and numerical methods often have drawbacks, such as high computational costs and challenges in accurately modeling complex behaviors. Improved simulation and a deeper understanding of these interactions are made possible by neural networks’ capacity to learn from big datasets and simulate complex relationships. To demonstrate that deep neural networks effectively capture scenarios within physical processes where advection plays a dominant role, the effectiveness of applying deep neural networks to a third-order dispersive partial differential equation incorporating advection and dispersion terms has been investigated in this study.

Solvability of an Inverse Problem for an Elliptic-Type Equation

In this study, we consider an inverse problem of determining an unknown source function in the right-hand side of an elliptic equation which is ill-posed in the Hadamard sense. To investigate the solvability of the problem, we reduce it to a Dirichlet problem for a third-order partial differential equation with homogeneous boundary condition. Since the problem is linear, the proof of the uniqueness theorem is based on the Fredholm Alternative Theorem. We prove the existence of the solution to the problem by using the Galerkin method.

Point- and contact-symmetry pseudogroups of dispersionless Nizhnik equation

Applying an original megaideal-based version of the algebraic method, we compute the point-symmetry pseudogroup of the dispersionless (potential symmetric) Nizhnik equation. This is the first example of this kind in the literature, where there is no need to use the direct method for completing the computation. The analogous studies are also carried out for the corresponding nonlinear Lax representation and the dispersionless counterpart of the symmetric Nizhnik system. We also first apply the megaideal-based version of the algebraic method to find the contact-symmetry (pseudo)group of a partial differential equation. It is shown that the contact-symmetry pseudogroup of the dispersionless Nizhnik equation coincides with the first prolongation of its point-symmetry pseudogroup. We check whether the subalgebras of the maximal Lie invariance algebra of the dispersionless Nizhnik equation that naturally arise in the course of the above computations define the diffeomorphisms stabilizing this algebra or its first prolongation. In addition, we construct all the third-order partial differential equations in three independent variables that admit the same Lie invariance algebra. We also find a set of geometric properties of the dispersionless Nizhnik equation that exhaustively defines it.

$ H^1 $ solutions for a modified Korteweg-de Vries-Burgers type equation

<p>This paper modeled the dynamics of microbubbles coated with viscoelastic shells using the modified Korteweg-de Vries-Burgers equation, a nonlinear third-order partial differential equation. This study focused on the well-posedness of the Cauchy problem associated with this equation.</p>

Stability analysis and numerical implementation of the third-order fractional partial differential equation based on the Caputo fractional derivative

Stability analysis and numerical implementation of the third-order fractional partial differential equation based on the Caputo fractional derivative

General solutions’ laws of linear partial differential equations II

This paper uses Z transformations to obtain the general solutions of a large number of second-order, third-order and fourth-order linear partial differential equations for the first time, including one-dimensional inhomogeneous wave equation. Using general solutions, we have obtained plenty of exact solutions of the typical definite solution problems, which proves the important value of general solutions. We propose the Z4 transformation for the first time and initially use it to solve a specific case. We successfully obtain the Fourier series solution using the series general solution of the one-dimensional homogeneous wave equation, which successfully solves a famous unresolved debate in the history of mathematics.

Solution of Time-Fractional Third-Order Partial Differential Equations of One and Higher Dimensions

Solution of Time-Fractional Third-Order Partial Differential Equations of One and Higher Dimensions

Approximate Solutions for Time-Fractional Fornberg–Whitham Equation with Variable Coefficients

In this research, three numerical methods, namely the variational iteration method, the Adomian decomposition method, and the homotopy analysis method are considered to achieve an approximate solution for a third-order time-fractional partial differential Equation (TFPDE). The equation is obtained from the classical (FW) equation by replacing the integer-order time derivative with the Caputo fractional derivative of order η=(0,1] with variable coefficients. We consider homogeneous boundary conditions to find the approximate solutions for the bounded space variable l<χ<L and l,L∈R. To confirm the effectiveness of the proposed methods of non-integer order η, the computation of two test problems was presented. A comparison is made between the obtained results of the (VIM), (ADM), and (HAM) through tables and graphs. The numerical results demonstrate the effectiveness of the three numerical methods.

Comparison of third-order fractional partial differential equation based on the fractional operators using the explicit finite difference method

In this research paper, the third-order fractional partial differential equation (FPDE) in the sense of the Caputo fractional derivative and the Atangana-Baleanu Caputo (ABC) fractional derivative is investigated for the first time. The importance of the proposed problem is more general than the third-order linear time-varying systems model. This fractional partial differential equation is converted to the abstract form of the ordinary differential equation at β = 1. A first-order finite difference scheme is created for the converted problem. The stability inequality of the (FDS) is demonstrated for this abstract form problem using Cardano's relation method in Hilbert space. The explicit finite difference method (EFDM) is used to obtain an approximate solution of the third-order fractional partial differential equation (FPDE) based on the Caputo fractional derivative and the (ABC) fractional derivative of order β∈(0,1]. Concerning these equations, the obtained approximate solutions using the proposed method are contrasted with the exact solution. Finally, the simulation for the third-order (FPDE) based on the fractional derivatives of Caputo and (ABC) for both the exact and approximate solutions is given.

Observer-Based Fuzzy Quantized Control for a Stochastic Third-Order Parabolic PDE System

The work addresses an observer-based fuzzy quantized control for stochastic third-order parabolic partial differential equations (PDEs) using discrete point measurements. For the first time, we contribute in introducing three types of quantizer—logarithmic quantizer, uniform quantizer, and hysteresis quantizer into the controller designs for the stochastic PDE system. The main advantage of the quantized control law lies in reducing the energy that the system spends. The stability analysis is nontrivial due to the complex stochastic PDE model. Different stability results are presented for each case. Sufficient LMI-based conditions are investigated to ensure the internal mean-square exponential stability and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{\infty }$ </tex-math></inline-formula> performance of the perturbed actuated system. Consistent simulation results that support the proposed theoretical statements are provided.

Solution Blow-Up and Global Solvability of the Cauchy Problem for a Model Third-Order Partial Differential Equation

Solution Blow-Up and Global Solvability of the Cauchy Problem for a Model Third-Order Partial Differential Equation

Solution of the Mixed Problem for a Third-Order Partial Differential Equation

Solution of the Mixed Problem for a Third-Order Partial Differential Equation

An Efficient Technique of Fractional-Order Physical Models Involving ρ-Laplace Transform

In this article, the ρ-Laplace transform is paired with a new iterative method to create a new hybrid methodology known as the new iterative transform method (NITM). This method is applied to analyse fractional-order third-order dispersive partial differential equations. The suggested technique procedure is straightforward and appealing, and it may be used to solve non-linear fractional-order partial differential equations effectively. The Caputo operator is used to express the fractional derivatives. Four numerical problems involving fractional-order third-order dispersive partial differential equations are presented with their analytical solutions. The graphs determined that their findings are in excellent agreement with the precise answers to the targeted issues. The solution to the problems at various fractional orders is achieved and found to be correct while comparing the exact solutions at integer-order problems. Although both problems are the non-linear fractional system of partial differential equations, the present technique provides its solution sophisticatedly. Including both integer and fractional order issues, solution graphs are carefully drawn. The fact that the issues’ physical dynamics completely support the solutions at both fractional and integer orders is significant. Moreover, despite using very few terms of the series solution attained by the present technique, higher accuracy is observed. In light of the various and authentic features, it can be customized to solve different fractional-order non-linear systems in nature.

Applications of Solvable Lie Algebras to a Class of Third Order Equations

A family of third-order partial differential equations (PDEs) is analyzed. This family broadens out well-known PDEs such as the Korteweg-de Vries equation, the Gardner equation, and the Burgers equation, which model many real-world phenomena. Furthermore, several macroscopic models for semiconductors considering quantum effects—for example, models for the transmission of electrical lines and quantum hydrodynamic models—are governed by third-order PDEs of this family. For this family, all point symmetries have been derived. These symmetries are used to determine group-invariant solutions from three-dimensional solvable subgroups of the complete symmetry group, which allow us to reduce the given PDE to a first-order nonlinear ordinary differential equation (ODE). Finally, exact solutions are obtained by solving the first-order nonlinear ODEs or by taking into account the Type-II hidden symmetries that appear in the reduced second-order ODEs.

A Stable Difference Scheme for a Third-Order Partial Differential Equation

The nonlocal boundary-value problem for a third-order partial differential equation$$ \left\{\begin{array}{l}\frac{d^3u(t)}{dt^3}+A\frac{du(t)}{dt}=f(t),\kern1em 0<t<1,\\ {}u(0)=\gamma u\left(\lambda \right)+\varphi, \kern1em {u}^{\prime }(0)=\alpha {u}^{\prime}\left(\lambda \right)+\psi, \left|\gamma \right|<1,\\ {}{u}^{{\prime\prime} }(0)=\beta {u}^{{\prime\prime}}\left(\lambda \right)+\xi, \kern1em \left|1+\beta \alpha \right|>\left|\alpha +\beta \right|,0<\lambda \le 1\end{array}\right. $$in a Hilbert space H with a self-adjoint positive definite operator A is considered. A stable three-step difference scheme for the approximate solution of the problem is presented. The main theorem on stability of this difference scheme is established. As applications, the stability estimates for the solution of difference schemes of the approximate solution of three nonlocal boundary-value problems for third-order partial differential equations are obtained. Numerical results for one- and two-dimensional third-order partial differential equations are provided.

Analytical Investigation of Third-Order Time-Fractional Dispersive Partial Differential Equations Using Sumudu Transform Iterative Method

This paper investigates the approximate analytical solutions of third-order timefractional dispersive partial differential equations in one-and higher-dimensional spaces by employing a newly developed analytical method, the Sumudu transform iterative method. To express fractional derivatives, the Caputo operator is used. Furthermore, the results of this investigation are graphically represented, and the solution graphs reveal that the approximate solutions are closely connected to the exact solutions.

On a nonlinear boundary-value problem for a third-order partial differential equations

On a nonlinear boundary-value problem for a third-order partial differential equations