2,840 Results Found

The various studies of partial differential equations (PDEs) are hot topics of mathematical research. Among them, solving PDEs is a very important and difficult task. Since many partial differential equations do not have analytical solutions, numeric...

Partial differential equations are an integral part of modern scientific development. Hyperbolic partial differential equations are encountered in many fields and have many applications—both linear and nonlinear types, with some being semilinea...

The Dirac factorization method (DFM) is the key feature of the present investigation. It is addressed to the relevant use in diverse fields of research, regarding, e.g., the handling of pseudo-operators arising in quantum mechanics and fractional cal...

This paper mainly considers the parameter estimation problem for several types of differential equations controlled by linear operators, which may be partial differential, integro-differential and fractional order operators. Under the idea of data-dr...

In this paper, a technique generally known as meshless method is presented for solving fractional partial differential equations (FPDEs). Some physical linear and nonlinear experiments such as time-fractional convective-diffusion equation, timefracti...

Fourier spectral methods for solving some linear stochastic space-fractional partial differential equations perturbed by space-time white noises in the one-dimensional case are introduced and analysed. The space-fractional derivative is defined by us...

Partial differential equations are crucial in scientific computing, and this paper will consider some of the problems of partial differential equation singularities. The Composite Mesh Method (CGM) is a new and improved numerical method for solving p...

In this paper a new method called the least squares differential quadrature method (LSDQM) is introduced as a straightforward and efficient method to compute analytical approximate polynomial solutions for nonlinear partial differential equations wit...

This paper discusses the use of Equivariant Neural Networks (ENN) for solving Partial Differential Equations by exploiting their underlying symmetry groups. We first show that Group-Convolutionnal Neural Networks can be used to generalize Physics-Inf...

In this paper, we present an application of Wu’s method (differential characteristic set (dchar-set) algorithm) for computing the symmetry of (partial) differential equations (PDEs) that provides a direct and systematic procedure to obtain the...

In this paper, the solvability of nonlinear fractional partial differential equations (FPDEs) with mixed partial derivatives is considered. The invariant subspace method is generalized and is then used to derive exact solutions to the nonlinear FPDEs...

This paper mainly consists of two parts: (i) We study the uniqueness, existence, and stability of a new fractional nonlinear partial integro-differential equation in ${\mathbb{R}}^n$ with three-point conditions and variable coefficients in a Banach space using inv...

In this paper, we study Hyers–Ulam and generalized Hyers–Ulam–Rassias stability of a system of hyperbolic partial differential equations using Gronwall’s lemma and Perov’s theorem.

A multivariable technique has been incorporated for guesstimating solutions of Nonlinear Partial Differential Equations (NPDE) using bases set of B-Polynomials (B-polys). To approximate the anticipated solution of the NPD equation, a linear product o...

This review paper gives an overview of the method of multipliers for partial differential equations (PDEs). This method has made possible a lot of solutions to PDEs that are of interest in many areas such as applied mathematics, mathematical physics,...

The aim of this paper is to investigate generalized Ulam–Hyers stability and generalized Ulam–Hyers–Rassias stability for a system of partial differential equations of first order. More precisely, we consider a system of two nonline...

In this review paper, we discuss the relation between recent advances in the theory of partial differential equations and their applications to quantum field theory on curved spacetimes. In particular, we focus on hyperbolic propagators and the role...

In scientific and engineering calculation, the effective solution of partial differential equations (PDEs) has great significance. This paper presents an innovative method based on the combination of a U-Net neural network with Fourier neural operato...

An overview comparing ordinary and partial differential equations of fractional order, which arose as a new result of the author’s research and mathematical description of the differential constitutive relations of known and new models of fract...

In this paper, the double fuzzy Sumudu transform (DFST) method was used to find the solution to partial Volterra fuzzy integro-differential equations (PVFIDE) with convolution kernel under Hukuhara differentiability. Fundamental results of the double...

This study presents the Yang Transform Adomian Decomposition Method (YTADM), a semi-analytical framework for solving one-dimensional linear and nonlinear delay fractional partial differential equations involving the Caputo fractional derivative. The...

This paper introduces a new analytical technique (NAT) for solving a system of nonlinear fractional partial differential equations (NFPDEs) in full general set. Moreover, the convergence and error analysis of the proposed technique is shown. The appr...

In the present article, fractional-order partial differential equations with proportional delay, including generalized Burger equations with proportional delay are solved by using Natural transform decomposition method. Natural transform decompositio...

In this article, a new analytical scheme of the ARA transform is introduced to solve systems of fractional partial differential equations. The principle of the proposed technique is based on combining the ARA transform with the residual power series...

The present study aims to integrate the Adomian decomposition method with the Sumudu transform for solving delay fractional partial differential equations. Integrating these two methods enhances accuracy and computational efficiency, providing an eff...

The Kortweg–de Vries equations play an important role to model different physical phenomena in nature. In this research article, we have investigated the analytical solution to system of nonlinear fractional Kortweg–de Vries, partial diff...

Partial differential equations are known to be increasingly important in today’s research, and their solutions are paramount for tackling numerous real-life applications. This article extended the analytical scheme of the homotopy analysis meth...

The main goal of this paper is to propose a new decomposition method for finding solutions to nonlinear partial fuzzy differential equations (NPFDE) through the fuzzy Sawi decomposition method (FSDM). This method is a combination of the fuzzy Sawi tr...

In this paper, we discuss the geometric structure, i.e., Dirac structure, underlying port-Hamiltonian systems. The paper has a tutorial character, and thus it contains questions/exercises. We start with the general definition of a Dirac structure and...

In this work, a unified lattice Boltzmann model is proposed for the fourth order partial differential equation with time-dependent variable coefficients, which has the form $u_t+\alpha \left(t\right){\left(p_1\left(u\right)\right)}_x+\beta \left(t\right){\left(p_2\left(u\right)\right)}_{xx}+\gamma \left(t\right){\left(p_3\left(u\right)\right)}_{xxx}+\eta \left(t\right){\left(p_4\left(u\right)\right)}_{}$...

There has been increasing research activity in recent years concerning the properties and the applications of nonlinear partial differential equations that are closely related to nonstandard entropic functionals, such as the Tsallis and Renyi entropi...

In this article, we developed the idea of q-time scale calculus in quantum geometry. It includes the q-time scale integral operators and${∆}_q$-differentials. It analyzes the fundamental principles which follow the calculus of q-time scales co...

The main difficulty posed by the parameter inversion of partial differential equations lies in the presence of numerous local minima in the cost function. Inversion fails to converge to the global minimum point unless the initial estimate is close to...

The main goal of this research is to present a new approach to double transforms called the double Laplace–ARA transform (DL-ARAT). This new double transform is a novel combination of Laplace and ARA transforms. We present the basic properties...

This article is to investigate the existence of entire solutions of several quadratic trinomial difference equations $f{(z+c)}^2+2\alpha f\left(z\right)f(z+c)+f{\left(z\right)}^2=e^{g\left(z\right)},$ and the partial differential difference equations $f{(z+c)}^2+2\alpha f(z+c)\frac{\partial f\left(z\right)}{\partial z_1}+{\left(\frac{\partial f\left(z\right)}{\partial z_1}\right)}^2=e^{g\left(z\right)},$$f{(z+c)}^2+$...

Robot manipulators have played an enormous role in the industry during the twenty-first century. Due to the advances in materials science, lightweight manipulators have emerged with low energy consumption and positive economic aspect regardless of th...

The Laplace residual power series method was introduced as an effective technique for finding exact and approximate series solutions to various kinds of differential equations. In this context, we utilize the Laplace residual power series method to g...

In this article, we apply the double ARA–Sumudu transformation (DARA-ST) to the nonlocal fractional Caputo derivative operator. We achieve interesting results and implement them to solve certain classes of fractional partial differential equati...

This paper introduces a unified analytical framework for solving a broad class of nonlinear partial differential equations by mapping them onto a fundamental second-order nonlinear template equation. While various direct methods such as the $G^{\prime }/$...

The purpose of this paper is to leverage the advantages of physics-informed neural network (PINN) and convolutional neural network (CNN) by using Legendre multiwavelets (LMWs) as basis functions to approximate partial differential equations (PDEs). W...

In this paper, we consider the parameter identification problem of partial differential equations with constraints. A nonlinear multigrid method is introduced to the process of parameter inversion. By keeping the objective functions on coarse grids c...

This paper proposes moving mesh strategies for the moving mesh methods when solving the nonlinear time dependent partial differential equations (PDEs). Firstly we analyse Huang’s moving mesh PDEs (MMPDEs) and observe that, after Euler discretion they...

This article is concerned with the description of the entire solutions of several Fermat type partial differential-difference equations (PDDEs) ${\left[\mu f\left(z\right)+\lambda f_{z_1}\left(z\right)\right]}^2+{[\alpha f(z+c)-\beta f\left(z\right)]}^2=1,$ and ${\left[\mu f\left(z\right)+{\lambda }_1{f}_{z_1}\left(z\right)+{\lambda }_2{f}_{z_2}\left(z\right)\right]}^{}$...

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 wi...

This paper presents a Pell–Lucas collocation method for time-fractional partial integro-differential equations (TFPIDEs) with initial and boundary conditions, where the fractional derivative is taken in the Caputo sense. The unknown solution is...

This paper establishes a method for solving partial differential equations using a multi-step physics-informed deep operator neural network. The network is trained by embedding physics-informed constraints. Different from traditional neural networks...

This article aims to provide a comprehensive review of the latest advancements in numerical methods and practical implementations in the field of fractional stochastic partial differential equations (FSPDEs). This type of equation integrates fraction...

We avoid as as much as possible the order reduction of Rosenbrock methods when they are applied to nonlinear partial differential equations by means of a similar technique to the one used previously by us for the linear case. For this we use a suitab...

The purpose of this paper is to investigate a class of initial value problems of fuzzy fractional coupled partial differential equations with Caputo $gH$-type derivatives. Firstly, using Banach fixed point theorem and the mathematical inductive method,...

In this study, local fuzzy fractional partial differential equations (LFFPDEs) are considered using a hybrid local fuzzy fractional approach. Fractal model behavior can be represented using fuzzy partial differential equations (PDEs) with local fract...