1,170 Results Found

In this paper, a second-order operator splitting method combined with the barycentric Lagrange interpolation collocation method is proposed for the nonlinear Schrödinger equation. The equation is split into linear and nonlinear parts: the linear...

This paper introduces the optimal auxiliary function method (OAFM) for solving a nonlinear system of Belousov–Zhabotinsky equations. The system is characterized by its complex dynamics and is treated using the Caputo operator and concepts from...

Reduction operators, i.e., the operators of nonclassical (or conditional) symmetry of a class of variable coefficient nonlinear wave equations with power nonlinearities, are investigated within the framework of a singular reduction operator. A classi...

Recent decades have witnessed the emergence of interesting models of fractional partial differential equations. In the current work, a class of parabolic equations with regularized Hyper-Bessel derivative and the exponential source is investigated. M...

An algorithm for the symbolic computation of recursion operators for systems of nonlinear differential-difference equations (DDEs) is presented. Recursion operators allow one to generate an infinite sequence of generalized symmetries. The existence o...

In this paper, we consider a boundary value problem for a nonlinear partial differential equation of mixed type with Hilfer operator of fractional integro-differentiation in a positive rectangular domain and with spectral parameter in a negative rect...

This article proposed two novel techniques for solving the fractional-order Boussinesq equation. Several new approximate analytical solutions of the second- and fourth-order time-fractional Boussinesq equation are derived using the Laplace transform...

The fractional Schrödinger–Korteweg-de Vries (S-KdV) equation is an important mathematical model that incorporates the nonlinear dynamics of the KdV equation with the quantum mechanical effects described by the Schrödinger equation. M...

In this study, we aim to provide reliable methods for the initial value problem of the fractional modified Korteweg–de Vries (mKdV) equations. Fractional differential equations are essential for more precise simulation of numerous processes. Th...

This paper proposes a Chebyshev spectral collocation method for nonlinear Volterra–Hammerstein integral equations of the second kind with smooth kernels. We approximate the unknown solution using a finite Chebyshev expansion and place collocati...

We present recent advances in the analysis of nonlinear problems involving singular (degenerate) operators. The results are obtained within the framework of p-regularity theory, which has been successfully developed over the past four decades. We ill...

This paper presents the optimal auxiliary function method (OAFM) implementation to solve a nonlinear fractional system of the Jaulent–Miodek Equation with the Caputo operator. The OAFM is a vital method for solving different kinds of nonlinear...

Schrödinger’s operator relations combined with Einstein’s special relativistic energy-momentum equation produce the linear Klein–Gordon partial differential equation. Here, we extend both the operator relations and the energy-m...

In this paper, we design a new third order Newton-like method and establish its convergence theory for finding the approximate solutions of nonlinear operator equations in the setting of Banach spaces. First, we discuss the convergence analysis of ou...

In this paper, we study the Liouville-type behaviors of the generalized Greiner operators with nonlinear boundary value conditions. We use the technique based upon the method of moving planes.

Tautenhahn (2002) studied the Lavrentiev regularization method (LRM) to approximate a stable solution for the ill-posed nonlinear equation $\kappa \left(u\right)=v$, where $\kappa :D\left(\kappa \right)\subseteq X⟶X$ is a nonlinear monotone operator and X is a Hilbert spac...

In this research, we employ a dual-approach that combines the Laplace residual power series method and the novel iteration method in conjunction with the Caputo operator. Our primary objective is to address the solution of two distinct, yet intricate...

In this study, we present a spectral method for solving nonlinear Volterra integral equations with weakly singular kernels based on the Genocchi polynomials. Many other interesting results concerning nonlinear equations with discontinuous symmetric k...

This paper presents the application of an economic model predictive control (MPC) for the operational management of water distribution networks (WDNs) with periodic operation and nonlinear constraint relaxation. In addition to minimizing operational...

Considering the case of a continual bundle of controlled dynamic processes, the authors have studied the functional-geometric conditions of existence of non-stationary coefficients-operators from the differential realization of this bundle in the cla...

The analysis of differential equations using Lie symmetry has been proved a very robust tool. It is also a powerful technique for reducing the order and nonlinearity of differential equations. Lie symmetry of a differential equation allows a dynamic...

We investigate the Cauchy problem for a nonlinear fractional diffusion equation, which is modified using the time-fractional hyper-Bessel derivative. The source function is a gradient source of Hamilton–Jacobi type. The main objective of our cu...

This paper aims to introduce a real four-component integrable extension of the complex Kaup–Newell soliton hierarchy. Following a general idea for extending the standard Kaup–Newell spectral matrix, we propose a specific matrix eigenvalue...

In this paper, we focus on the existence of positive solutions for a singular tempered sub-diffusion fractional model involving a quasi-homogeneous nonlinear operator. By using the spectrum theory and computing the fixed point index, some new suffici...

The nonlinear Bagley–Torvik equation is of fundamental importance, as it captures a realistic and intricate interplay among memory effects, nonlinearity, and functional dependence—making it a powerful model for a wide range of natural and...

This paper investigates the solitary wave solutions for the perturbed nonlinear Schrödinger equation with six different nonlinearities with the essence of the generalized classical derivative, which is known as the beta derivative. The aforementioned...

In this article, we study the JF iterative algorithm to approximate the fixed points of a non-linear operator that satisfies condition (E) in uniformly convex Banach spaces. Further, some weak and strong convergence results are presented for the same...

In this paper, we investigate functional inverse operators associated with a class of fractional integro-differential equations. We further explore the existence, uniqueness, and stability of solutions to a new integro-differential equation featuring...

This paper applies a recently proposed numerical algorithm to discuss the deflection of viscoelastic micro-beams in the time domain with direct access. A nonlinear-fractional order model for viscoelastic micro-beams is constructed. Before solving the...

In this paper, we extend the invariant subspace method to a class of short pulse-type equations. Complete classification results with invariant subspaces from 2 to 5 dimensions are provided. The key step is to take subspaces of solutions of linear or...

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...

This paper delves into the asymptotic and oscillatory behavior of all classes of solutions of fourth-order nonlinear neutral delay differential equations in the noncanonical form with damping terms. This research aims to improve the relationships bet...

The primary aim of this manuscript is to establish unique fixed point results for a class of $\Psi$-contraction operators in complete G-metric spaces. By combining and extending various fixed point theorems in the context of $\Psi$-contraction operator...

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...

Two new methods for handling a system of nonlinear fractional differential equations are presented in this investigation. Based on the characteristics of fractional calculus, the Caputo fractional partial derivative provides an easy way to determine...

We study the topological degree for densely defined operators of class $\left(S_{+}\right)$ with compact multivalued perturbations. It is assumed that a perturbation can be represented as a sum of two terms: a multivalued mapping which is itself a composition of a c...

In this paper we present a two-step solver for nonlinear equations with a nondifferentiable operator. This method is based on two methods of order of convergence $1+\sqrt{2}$ . We study the local and a semilocal convergence using weaker conditions...

This paper establishes new fixed-point theorems in the framework of complete p-normed spaces, where $p\in (0,1]$. By extending the classical Banach, Schauder, and Krasnosel’skii fixed-point theorems, we derive several results for the sum of con...

This paper introduces a collocation algorithm for numerically solving the third-order Gilson–Pickering equation (GPE) and the classical Rosenau–Hyman equation (RHE). We employ newly developed shifted Pell polynomials as basis functions. N...

The aim of this paper is the approximation of nonlinear equations using iterative methods. We present a unified convergence analysis for some two-point type methods. This way we compare specializations of our method using not necessarily the same con...

This paper applies modified analytical methods to the fractional-order analysis of one and two-dimensional nonlinear systems of coupled Burgers and Hirota–Satsuma KdV equations. The Atangana–Baleanu fractional derivative operator and the...

Using both analytical and numerical techniques, we discuss wave solutions within the framework of an extended nonlinear Schrödinger equation with constant coefficients equipped with spatiotemporal dispersion, self-steepening effects, and a Raman...

This paper presents a novel and general analytical approach: the rational sine-Gordon expansion method and its applications to the nonlinear Gardner and (3+1)-dimensional mKdV-ZK equations including a conformable operator. Some trigonometric, periodi...

This paper presents a modeling framework for hysteretically nonlinear piezoelectric composite beams using functional differential equations (FDEs). While linear piezoelectric models are well established, they fail to capture the complex nonlinear beh...

The necessary and sufficient conditions of existence of the nonlinear operator equations’ branches of solutions in the neighbourhood of branching points are derived. The approach is based on the reduction of the nonlinear operator equations to...

This article solves the nonlinear fractional integral equation (NFrIE) using the Genocchi polynomial method (GPM). We have provided proof to demonstrate the existence of a unique solution to the second sort of NFrIE in Hilbert space. The proof of the...

This paper investigates the fractional calculus of distributions supported on ${\mathbb{R}}^{+}$ in the sense of L. Schwartz, based on distributional convolutions. We further study a generalized Bagley–Torvik equation involving an arbitrary number of fractiona...

This paper introduces a novel approach employing the fast cosine transform to tackle the 2-D and 3-D fractional nonlinear Schrödinger equation (fNLSE). The fractional Laplace operator under homogeneous Neumann boundary conditions is first define...

In this paper, we construct and analyze a class of high-order and dissipation-preserving schemes for the nonlinear space fractional generalized wave equations by the newly introduced scalar auxiliary variable (SAV) technique. The system is discretize...

Fractional differential equations play a significant role in various scientific and engineering disciplines, offering a more sophisticated framework for modeling complex behaviors and phenomena that involve multiple independent variables and non-inte...