Solutions Of Nonlinear Equations Research Articles

Consistent travelling wave characteristic of space–time fractional modified Benjamin–Bona–Mahony and the space–time fractional Duffing models

Study on solitary wave phenomenon are closely related on the dynamics of the plasma and optical fiber system, which carry on broad range of wave propagation. The space–time fractional modified Benjamin–Bona–Mahony equation and Duffing model are important modeling equations in acoustic gravity waves, cold plasma waves, quantum plasma in mechanics, elastic media in nonlinear optics, and the damping of material waves. This study has effectively developed analytical wave solutions to the aforementioned models, which may have significant consequences for characterizing the nonlinear dynamical behavior related to the phenomenon. Conformable derivatives are used to narrate the fractional derivatives. The expanded tanh-function method is used to look into such kinds of resolutions. An ansatz for analytical traveling wave solutions of certain nonlinear evolution equations was originally a power sequence in tanh. The discovered explanations are useful, reliable, and applicable to chaotic vibrations, problems of optimal control, bifurcations to global and local, also resonances, as well as fusion and fission phenomena in solitons, scalar electrodynamics, the relation of relativistic energy–momentum, electromagnetic interactions, theory of one-particle quantum relativistic, and cold plasm. The solutions are drafted in 3D, contour, listpoint, and 2D patterns, and include multiple solitons, bell shape, kink type, single soliton, compaction solitary wave, and additional sorts of solutions. With the aid of Maple and MATHEMATICA, these solutions were verified and discovered that they were correct. The mentioned method applied for solving NLFPDEs has been designed to be practical, straightforward, rapid, and easy to use.

Generalization of Homotopy Analysis Method for q-Fractional Non-linear Differential Equations

This paper presents a generalization of the Homotopy analysis method (HAM) for finding the solutions of nonlinear q-fractional differential equations (q-FDEs). This method shows that the series solution in the case of generalized HAM is more likely to converge than that on HAM. In order that it is applicable to solve immensely non-linear problems and also address a few issues, such as the impact of varying the auxiliary parameter, auxiliary function, and auxiliary linear operator on the order of convergence of the method. The generalized HAM method is more accurate than the HAM.

The general Bernstein function: Application to χ‐fractional differential equations

In this paper, we present the general Bernstein functions for the first time. The properties of generalized Bernstein basis functions are given and demonstrated. The classical Bernstein polynomial bases are merely a subset of the general Bernstein functions. Based on the new Bernstein base functions and the collocation method, we present a numerical method for solving linear and nonlinear ‐fractional differential equations ( ‐FDEs) with variable coefficients. The fractional derivative used in this work is the ‐Caputo fractional derivative sense ( ‐CFD). Combining the Bernstein functions basis and the collocation methods yields the approximation solution of nonlinear differential equations. These base functions can be used to solve many problems in applied mathematics, including calculus of variations, differential equations, optimal control, and integral equations. Furthermore, the convergence of the method is rigorously justified and supported by numerical experiments.

Monotone Positive Solutions for Nonlinear Fractional Differential Equations with a Disturbance Parameter on the Infinite Interval

This paper is concerned with the existence and multiplicity of monotone positive solutions for a class of nonlinear fractional differential equation with a disturbance parameter in the integral boundary conditions on the infinite interval. By using Guo–Krasnosel’skii fixed-point theorem and the analytic technique, we divide the range of parameter for the existence of at least two, one and no positive solutions for the problem. In the end, an example is given to illustrate our main results.

Persistence and asymptotic analysis of solutions of nonlinear wave equations

Persistence and asymptotic analysis of solutions of nonlinear wave equations

Existence of the solution of nonlinear fractional differential equations via new best proximity point results

AbstractIn this paper, we obtain some best proximity point results by introducing the concepts of proximal p-contractions of the first type and proximal p -contractions of the second type on partial metric spaces. Thus, some famous results in the literature such as the main result of Altun et al. (Acta Math Hung 162:393–402, 2020) and Basha (J Approx Theory 163(11):1772–1781, 2011) have been extended. Also, we provide some examples where our results are applicable and the results in Haghi et al. (Topol Appl 160:450–454, 2013) are not. Hence, our results are a real generalization of some results in metric spaces and partial metric spaces. Finally, we obtain sufficient conditions for the existence of the solution of nonlinear fractional differential equations via our results.

Thermal examination for double diffusive MHD Jeffrey fluid flow through the space of disc and cone apparatus subject to impact of multiple rotations

Non-Fourier’s and non-Fick’s models enhance predictions of transient heat transfer and mass transport, especially in situations where local equilibrium assumptions break down. In such fluids, the deformation rate and shear stress display a natural parabolic structure. In this study, we explore the flow of a Jeffrey fluid with dual diffusions, incorporating both non-Fourier's and non-Fick's assumptions, through the gap of disk-cone devices. Four scenarios have been investigated for fluid flow and heat transformation in this work comprising of (i) both disc and cone are rotating in opposite directions (ii) both disc and cone are rotating in same direction, (iii) cone is rotating while disc remains stationary, (iv) disc is rotating while cone remains stationary. After deriving the basic equations, a set of appropriate variables is used to convert them into dimension from form. RK-4 (Runge-Kutta 4th order) technique has been employed for the solution of nonlinear equations. As the magnetic and Maxwell factors experience a notable increase, whether the disk and cone move in tandem, in opposite directions, or involve a static cone with a gyrating disk, or a static disk with a gyrating cone, the transverse velocity panels consistently exhibit retardation in all scenarios. Additionally, a noteworthy observation is made when evaluating changes in both the Nusselt number and the Sherwood number, with significantly more pronounced variations detected at the surface of the disk compared to that of the cone. There has been a 35% increase observed in the thermal flow rate with variations in the thermophoresis factor.

Extension of Haar wavelet technique for numerical solution of three-dimensional linear and nonlinear telegraph equations

In this manuscript, a hybrid numerical technique is presented for solving three-dimensional hyperbolic telegraph equations. The proposed technique is based on the Haar wavelet collocation method and the finite difference method. In this technique, the space derivatives are estimated by truncated Haar wavelet series, while the time derivatives are approximated using finite difference method. The method is applied to some linear and non-linear three-dimensional hyperbolic telegraph equations. In the linear case, the resulting algebraic system is solved by the Gauss elimination technique. Whereas in the nonlinear case, the quasi-Newton linearization technique is utilized to remove the nonlinearity of the problem. The maximum absolute errors, root mean square errors and computational rate of convergence are calculated for a different number of collocation points at different time steps of the final time t=1. The stability analysis for the proposed method is also discussed. Overall, the recently devised numerical method boasts straightforward implementation, rapid convergence, and enhanced accuracy in addressing both linear and nonlinear problems.

Existence of solutions for nonlinear biharmonic Choquard equations on weighted lattice graphs

In this paper, let G=(ZN,E,μ,ω) be a weighted lattice graph, where μ is a uniformly bounded measure and ω is a positive symmetric weight. We are concerned with the following biharmonic Choquard equation with a nonlocal nonlinearity on ZN, for any α∈(0,N),Δ2u−Δu+hu=(∑y∈ZN,y≠xF(u(y))d(x,y)N−α)f(u), where h denotes a function on ZN, f:R→R is a continuous function, F(u)=∫0uf(s)ds represents the primitive function of f, and d(x,y) stands for the distance between x and y. In particular, under some suitable conditions on h, we prove that if the nonlinearity f satisfies distinct growth conditions, then there exists a mountain-pass solution and a ground state solution respectively. As far as we know, there are no existing results for the biharmonic Choquard equation with the nonlinearity f on weighted lattice graphs.

Usage of the Fuzzy Adomian Decomposition Method for Solving Some Fuzzy Fractional Partial Differential Equations

In this study, we examine the numerical solutions of nonlinear fuzzy fractional partial differential equations under the Caputo derivative utilizing the technique of fuzzy Adomian decomposition. This technique is used as an alternative method for obtaining approximate fuzzy solutions to various types of fractional differential equations and also investigated some new existence and uniqueness results of fuzzy solutions. Some examples are given to support the effectiveness of the proposed technique. We present the numerical results in graphical form for different values of fractional order and uncertainty γ∈0,1.

Solution of nonlinear stiff differential equations for a three-phase no-load transformer using a Runge–Kutta implicit method

Solution of nonlinear stiff differential equations for a three-phase no-load transformer using a Runge–Kutta implicit method

Convergence and data dependence results of the nonlinear Volterra integral equation by the Picard's three step iteration

<abstract> <p>Picard's three step iteration algorithm was one of the iteration algorithms that was recently shown to be faster than some other iterative algorithms in the existing literature. The purpose of this paper was to study using this iteration algorithm for the solution of nonlinear Volterra integral equations. It was investigated that the sequences obtained from this iteration algorithm converged to the solution of nonlinear Volterra integral equations. Moreover, data dependence was obtained for nonlinear Volterra integral equations. An example was given that confirmed the applicability of the newly proven theorems.</p> </abstract>

The efficacy of a new fixed point method in nonlinear equation solutions

The key objective of this research is to establish a new approach to the classical fixed point method for solving nonlinear equations by using the new theorem of a-fixed point established in [1] and the conformable fractional derivative defined in [6]. As an application, we establish a new a-fixed point algorithm as a numerical result.

Fixed point results for generalized almost contractions and application to a nonlinear matrix equation

<abstract><p>The goal of this paper was to improve some known results of fixed points by using $ w $-distances and properties of locally symmetric $ \mathcal{H} $-transitivity of binary relations. Also, we gave the application of the obtained results for finding the solution of nonlinear matrix equations. Finally, we gave a numerical example to demonstrate the applicability of our results.</p></abstract>

Green's functions and existence of solutions of nonlinear fractional implicit difference equations with Dirichlet boundary conditions

This article is devoted to deduce the expression of the Green's function related to a general constant coefficients fractional difference equation coupled to Dirichlet conditions. In this case, due to the points where some of the fractional operators are applied, we are in presence of an implicit fractional difference equation. So, due to such a property, it is more complicated to calculate and manage the expression of the Green's function than in the explicit case studied in a previous work of the authors. Contrary to the explicit case, where it is shown that the Green's function is constructed as finite sums, the Green's function constructed here is an infinite series. This fact makes necessary to impose more restrictive assumptions on the parameters that appear in the equation. The expression of the Green's function will be deduced from the Laplace transform on the time scales of the integers. We point out that, despite the implicit character of the considered equation, we can have an explicit expression of the solution by means of the expression of the Green's function. These two facts are not incompatible. Even more, this method allows us to have an explicit expression of the solution of an implicit problem. Finally, we prove two existence results for nonlinear problems, via suitable fixed point theorems.

On the Initial Value Problem for Fuzzy Nonlinear Fractional Differential Equations

In this paper, we study the existence result of solutions for fuzzy nonlinear fractional differential equations involving Caputo differentiability of an arbitrary order 0 < q < 1. As application, an example is included to show the applicability of our result.

Breathers of Gerdjikov-Ivanov equation under background of elliptic function

As one specific type of local solutions of nonlinear evolution equation, the breathers have the characteristic of envelope oscillation structure. This kind of oscillation is periodic. According to the periodicity of the distribution and evolution directions, there are three kinds of breathers, namely, the Kuznetsov-Ma breather (KMB), the Akhmediev breather (AB), and the general breather (GB). In recent years, the propagation of envelope breathers under the periodic background has been observed in many nonlinear physical fields, including nonlinear optical fibers and hydrodynamics. It is believed that the breathers can arise due to the modulational instability of the periodic waves, and they demonstrate many rich physical properties and dynamic behaviors of interactions. Therefore, recently great attention has been paid to the breathers under the periodic background in nonlinear science. As an important integrable model, the Gerdjikov-Ivanov (GI) equation can be used to describe various nonlinear phenomena in many physical fields such as in the quantum field theory, weak nonlinear dispersive water wave, and nonlinear optics. It is very meaningful to solve various types of solutions of this model to describe the propagation of nonlinear waves. As far as we know, the breather solutions for the GI equation have not been given under the elliptic function background. In this study, firstly, elliptic function solutions of the GI equation are solved by the modified squared wave (MSW) function approach and the traveling wave transformation. Then, we obtain the basic solution of the Lax pair corresponding to the Jacobi elliptic function seed solution. Based on the elliptic function transformation formulas and the integral formulas, the potential function solution can be expressed in terms of the Weierstrass elliptic function. Secondly, by the once iterated Darboux transformation, three types of breather solutions under the elliptic function background are constructed including the GB, the KMB and the AB. In addition, we analyze the dynamic behaviors of these three kinds of breathers, and present their three-dimensional space-time structures. By the twice iterated Darboux transformation, under the dn-periodic background we exhibit three types of interactions between two breathers, i.e. a GB and a KMB, an AB and a KMB, and a GB and an AB. Finally, we also present three types of interactions between two breathers under the general periodic background.

On the Usage of Orthogonal Polynomials with Picard Iteration Method to Find Numerical Solutions of Neutrosophic Non-Linear Ordinary and Partial Differential Equations

This research aims to modify the Picard iteration method by hybridizing it with some orthogonal polynomials and then applying the hybrid method in solving neutrosophic nonlinear elementary value problems. This method is based on modifying the Picard iteration method by approximating the right-hand side of the neutrosophic differential equation of the studied problem either by Legendre polynomials or by Chebyshev polynomials of the first kind to obtain two different hybrids of the Picard iteration method. Also, we apply this modification to neutrosophic elementary value problems represented by neutrosophic nonlinear and right-handed nonlinear differential equations to demonstrate the reliability and efficiency of the proposed modified method. For this goal, we prove how effective this method is, we calculate the neutrosophic absolute error of approximate solutions resulting from the application of the proposed modification of the Picard iteration method and with the exact solution.

Parallel Solution of Nonlinear Projection Equations in a Multitask Learning Framework.

Nonlinear projection equations (NPEs) provide a unified framework for addressing various constrained nonlinear optimization and engineering problems. However, when it comes to solving multiple NPEs, traditional numerical integration methods are not efficient enough. This is because traditional methods solve each NPE iteratively and independently. In this article, we propose a novel approach based on multitask learning (MTL) for solving multiple NPEs. The solution procedure is outlined as follows. First, we model each NPE as a system of ordinary differential equations (ODEs) using neurodynamic optimization. Second, for each ODE system, we use a physics-informed neural network (PINN) as the solution. Third, we use a multibranch MTL framework, where each branch corresponds to a PINN model. This allows us to solve multiple NPEs in parallel by training a single neural network model. Experimental results show that our approach has superior computational performance, especially when the number of NPEs to be solved is large.

Normalized multi-bump solutions of nonlinear Kirchhoff equations

Normalized multi-bump solutions of nonlinear Kirchhoff equations