Exact solutions of nonlinear differential equations are of great importance to the theory and practice of complex systems. The main point of this review article is to discuss a specific methodology for obtaining such exact solutions. The methodology...
In this article, we will apply some of the algebraic methods to find great moving solutions to some nonlinear physical and engineering questions, such as a nonlinear (1 + 1) Ito integral differential equation and (1 + 1) nonlinear Schrödinger eq...
Some types of the generalized nonlinear Schrödinger equation of the second, fourth and sixth order are considered. The Cauchy problem for equations in the general case cannot be solved by the inverse scattering transform. The main objective of t...
We advance two nonlinear wave equations related to the nonextensive thermostatistical formalism based upon the power-law nonadditive
We present a short review of the methodology and applications of the Simple Equations Method (SEsM) for obtaining exact solutions to nonlinear differential equations. The applications part of the review is focused on the simple equations used, with e...
In recent years, symmetry in abstract partial differential equations has found wide application in the field of nonlinear integrable equations. The symmetries of the corresponding transformation groups for such equations make it possible to significa...
We discuss the Simple Equations Method (SEsM) for obtaining exact solutions of a class of nonlinear differential equations containing polynomial nonlinearities. We present an amended version of the methodology, which is based on the use of composite...
A modification to an existing iterative method for computing zeros with unknown multiplicities of nonlinear equations or a system of nonlinear equations is presented. We introduce preconditioners to nonlinear equations or a system of nonlinear equati...
This paper deals with several nonlinear partial differential equations (PDEs) of mathematical physics such as the concatenation model (perturbed concatenation model) from nonlinear fiber optics, the plane hydrodynamic jet theory, the Kadomtsev–...
The nonlinear perturbed Schrödinger equations (NPSEs) with nonlinear terms as Kerr law, power law, quadratic-cubic law, and dual-power law nonlinearity media play an important role in optical fibers. In this article we implement the rational sol...
In the paper, two new analytic methods using the decomposition and linearization technique on nonlinear differential/integral equations are developed, namely, the decomposition–linearization–sequential method (DLSM) and the linearized hom...
The optimal control problems of degenerate parabolic equations have many applications in economics, physics, climatology, and so on. Motivated by the applications, we consider the optimal control problems of a class of nonlinear degenerate parabolic...
An overview on the study of nonlinear evolution equations of soliton type is provided. In addition, 5th-order nonlinear evolution equations are shown to be connected to the Caudrey–Dodd–Gibbon–Sawada–Kotera (CDGSK) equation vi...
This research investigates the dynamics of higher-order nonlinear difference equations, specifically concentrating on seventh-order instances. Analytical solutions are obtained for particular equations, a formidable task owing to the absence of expli...
We consider novel discrete derivative nonlinear Schrödinger equations (ddNLSs). Taking the continuum derivative nonlinear Schrödinger equation (dNLS), we use for the discretisation of the derivative the forward, backward, and central differ...
The purpose of this article is to present some new nonlinear retarded integral inequalities which can be utilized to study the existence, stability, boundedness, uniqueness, and asymptotic behavior of solutions of nonlinear retarded integro-different...
In the present research work, we construct and examine the self-similarity of optical solitons by employing the Riccati Modified Extended Simple Equation Method (RMESEM) within the framework of non-integrable Coupled Nonlinear Helmholtz Equations (CN...
In this paper, an efficient decomposition method is constructed and used for solving system of nonlinear equations. The method based on the decomposition technique of Noor [M.A.Noor, K.I.Noor, Some iterative schemes for nonlinear equations, Appl. Mat...
This article discusses several forms of Ulam stability of nonlinear fractional delay differential equations. Our investigation is based on a generalised Gronwall’s inequality and Picard operator theory. Implementations are provided to demonstra...
In this study, a matrix method based on Legendre collocation points on interval [-1,1] is proposed for the approximate solution of the some first order nonlinear ordinary differential equations with the mixed conditions in terms of Legendre polynomia...
In this study, a collocation method based on Bernstein polynomials is developed for solution of the nonlinear ordinary differential equations with variable coefficients, under the mixed conditions. These equations are expressed as linear ordinary dif...
In this paper, we introduce a new three-step Newton method for solving a system of nonlinear equations. This new method based on Gauss quadrature rule has sixth order of convergence (with
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
We present a novel application of transcendental equations for nonlinear distance optimization in hyperbolic space. Through asymptotic approximations using Fourier and Taylor series expansions, we obtain approximations for the transcendental equation...
We consider a class of nonlinear integro-differential equations that model degenerate nonlocal diffusion. We investigate whether the strong maximum principle is valid for this nonlocal equation. For degenerate parabolic PDEs, the strong maximum princ...
The paper is devoted to the problem of the local existence for a solution to a nonlinear wave equation, with the dissipation given by a nonlinear form with the presence of a nonlinear memory term. Moreover, the global nonexistence of a solution is es...
In this study, a set of accurate dispersive nonlinear wave equations is established, using the wave velocity and free surface elevation as variables. These equations improve upon previous equations in which the velocity potential is used as a variabl...
Differential equations are useful mathematical tools for solving complex problems. Differential equations include ordinary and partial differential equations. Nonlinear equations can express the nonlinear relationship between dependent and independen...
An explicit method for solving time fractional wave equations with various nonlinearity is proposed using techniques of Laplace transform and wavelet approximation of functions and their integrals. To construct this method, a generalized Coiflet with...
This paper explores the linearized stability of nonlinear delay differential equations (DDEs) with impulses. The classical results on the existence of periodic solutions are extended from ordinary differential equations (ODEs) to DDEs with impulses....
In this work, we propose a new fourth-order Jarratt-type method for solving systems of nonlinear equations. The local convergence order of the method is proven analytically. Finally, we validate our results via some numerical experiments including an...
In this paper, we investigate the iterative methods for the solution of different types of nonlinear matrix equations. More specifically, we consider iterative methods for the minimal nonnegative solution of a set of Riccati equations, a nonnegative...
The existence of solutions is used the premise of discussing other properties of dynamic systems. The goal of this paper is to investigate the fundamental properties of nonlinear stochastic differential equations via the Khasminskii test, including t...
Recent developments on the generalizations of two important equations of quantum physics, namely the Schroedinger and Klein–Gordon equations, are reviewed. These generalizations present nonlinear terms, characterized by exponents depending on an inde...
In this article, we solve the second type of nonlinear Volterra picture fuzzy integral equation (NVPFIE) using an accelerated form of the Adomian decomposition method (ADM). Based on
The aim of this paper is to find formulas for the solutions of the nonlinear system of difference equations related to symmetry
A new family of iterative methods for solving systems of nonlinear equations is introduced, extending and unifying several existing approaches within a general framework. In the single-variable case, Newton’s method was accelerated using Pad&ea...
In this article, a class of scalar nonlinear integro-differential equations of first order with fading memory is investigated. For the considered fading memory problem, we discuss the effects of the memory over all the values of the parameter in the...
Tautenhahn (2002) studied the Lavrentiev regularization method (LRM) to approximate a stable solution for the ill-posed nonlinear equation
This paper considers the existence and uniqueness of stochastic entropy solution for a nonlinear transport equation with a stochastic perturbation. The uniqueness is based on the doubling variable method. For the existence, we develop a new scheme of...
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...
This work explores the numerical translation of the weak or integral solution of nonlinear partial differential equations into a numerically efficient, time-evolving scheme. Specifically, we focus on partial differential equations separable into a qu...
Several previous results valid for one-dimensional nonlinear Fokker-Planck equations are generalized to N-dimensions. A general nonlinear N-dimensional Fokker-Planck equation is derived directly from a master equation, by considering nonlinearitiesin...
This research article investigates a tripled system of nonlinear fractional differential equations with n terms. The study explores this novel class of differential equations to establish existence and stability results. Utilizing Schaefer’s an...
Nonlocal nonlinear Schrödinger equations are among the important models of nonlocal integrable systems. This paper aims to present a general formula for arbitrary-order breather solutions to multi-component nonlocal nonlinear Schrödinger eq...
The singularity of Jacobian happens when we are looking for a root, with multiplicity greater than one, of a system of nonlinear equations. The purpose of this article is two-fold. Firstly, we will present a modification of an existing method that co...
The study gives a brief overview of existing modifications of the method of functional separation of variables for nonlinear PDEs. It proposes a more general approach to the construction of exact solutions to nonlinear equations of applied mathematic...
Approximate symmetries of a class of perturbed nonlinear wave equations are computed using two newly-developed methods. Invariant solutions associated with the approximate symmetries are constructed for both methods. Symmetries and solutions are comp...
In the paper, we provide sufficient conditions for the oscillatory and asymptotic behavior of a new type of third-order nonlinear dynamic equations with mixed nonlinear neutral terms. Our theorems not only improve and extend existing theorems in the...
The paper deals with the results of a study of a third-order nonlinear differential equation with moving singular points and critical poles. So far, this type of equation cannot be solved in quadratures. The development of the author’s approach...
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