Solutions Of Nonlinear Equations Research Articles

Witte’s conditions for uniqueness of solutions to a class of Fractal-Fractional ordinary differential equations

In this paper, Witte's conditions for the uniqueness solution of nonlinear differential equations with integer and non-integer order derivatives are investigated. We present a detailed analysis of the uniqueness solutions of four classes of nonlinear differential equations with nonlocal operators. These classes include classical and fractional ordinary differential equations in fractal calculus. For each case, theorems and lemmas and their proofs are presented in detail.

Boundedness of solutions of nonlinear retarded differential equations by using new integral inequalities

Boundedness of solutions of nonlinear retarded differential equations by using new integral inequalities

Numerical solution of nonlinear equations of traffic flow density using spectral methods by filter

Numerical solution of nonlinear equations of traffic flow density using spectral methods by filter

DG-IMEX method for a two-moment model for radiation transport in the [formula omitted] limit

We consider neutral particle systems described by moments of a phase-space density and propose a realizability-preserving numerical method to evolve a spectral two-moment model for particles interacting with a background fluid moving with nonrelativistic velocities. The system of nonlinear moment equations, with special relativistic corrections to O(v/c), expresses a balance between phase-space advection and collisions and includes velocity-dependent terms that account for spatial advection, Doppler shift, and angular aberration. The model is conservative for the correct O(v/c) Eulerian-frame number density and is consistent, to O(v/c), with Eulerian-frame energy and momentum conservation. This model is closely related to the one promoted by Lowrie et al. [1] and similar to models currently used to study transport phenomena in large-scale simulations of astrophysical environments. The proposed numerical method is designed to preserve moment realizability, which guarantees that the moments correspond to a nonnegative phase-space density. The realizability-preserving scheme consists of the following key components: (i) a strong stability-preserving implicit-explicit (IMEX) time-integration method; (ii) a discontinuous Galerkin (DG) phase-space discretization with carefully constructed numerical fluxes; (iii) a realizability-preserving implicit collision update; and (iv) a realizability-enforcing limiter. In time integration, nonlinearity of the moment model necessitates solution of nonlinear equations, which we formulate as fixed-point problems and solve with tailored iterative solvers that preserve moment realizability with guaranteed global convergence. We also analyze the simultaneous Eulerian-frame number and energy conservation properties of the semi-discrete DG scheme and propose a “spectral redistribution” scheme that promotes Eulerian-frame energy conservation. Through numerical experiments, we demonstrate the accuracy and robustness of this DG-IMEX method and investigate its Eulerian-frame energy conservation properties.

Uncovering Hidden Patterns: Approximate Resurgent Resummation from Truncated Series

We analyze truncated series generated as divergent formal solutions of non-linear ordinary differential equations. Motivating the study is a specific non-linear, first-order differential equation, which is the basis of the resurgent formulation of renormalized perturbation theory in quantum field theory. We use the Borel–Padé approximant and classical analysis to determine the analytic structure of the solution using the first few terms of its asymptotic series. Afterward, we build an approximant, consistent with the resurgent properties of the equation. The procedure gives an approximate expression for the Borel–Ecalle resummation of the solution useful for practical applications. Connections with other physical applications are also discussed.

Design and Applicability of Two-Step Fractional Newton–Raphson Method

Developing two-step fractional numerical methods for finding the solution of nonlinear equations is the main objective of this research article. In addition, we present a detailed study of convergence analysis for the methods that have been proposed. By comparing numerically, we can see that the proposed methods significantly improve convergence rate and accuracy. Additionally, we demonstrate how our main results can be applied to basins of attraction.

Nonlinear Elliptic Equations with Variable Exponents Anisotropic Sobolev Weights and Natural Growth Terms

Abstract The purpose of our paper is to prove the existence of the distributional solutions for anisotropic nonlinear elliptic equations with variable exponents, which contain lower order terms dependent on the gradient of the solution and on the solution itself. The terms are weighted, and the main results rely on the possibility of comparing the weights with each other, where the right-hand side is a sum of the natural growth term and the datum f ∈ L 1(Ω). Furthermore the weight function θ(·) is in W̊ 1,p→(·) (Ω), with θ(·) > 0 and connected with the coefficient b(·) ∈ L 1(Ω) of the lower order term.

Simulation of quantum states in solid-state ferromagnetic nanostructures

Purpose of the article. Identification and analysis of mathematical expressions for the energy spectrum of charge carriers in nano-sized magnetic films of nickel and Ni-Cu (substrate), as well as NiO-Ni-Cu (substrate) structures.Methods. In this work, based on basic quantum mechanical concepts and taking into account the boundary conditions for coupled quantum wells, expressions for the energy spectrum of electrons are obtained. The ferrimagnetic properties of nickel are taken into account through the effective mass value. The solution of nonlinear equations that determine discrete energy levels was achieved by numerical methods using the Mathcad mathematical package.Results. Transcendental expressions for the energies of free charge carriers in quantum wells of ferromagnetic nickel films are obtained, and the influence of the position of nickel on the copper substrate is shown. It has been shown that the use of a copper substrate leads to an increase in the density of energy levels in the nickel nanofilm. The influence of the oxide layer on single-electron states in nanofilms of nickel and its oxide is considered within the framework of the Anderson model. Based on the numerical solution of the transcendental equations obtained in the work, the influence of the ratio of the thicknesses of a ferromagnetic metal and its oxide on the energy levels of electrons localized in the oxide layer is shown.Conclusions. The formulas presented in the work for the energy spectrum take into account the energy relief of a complex quantum well, the dimensions of the film, surface oxide, and the significant effective mass in the region of the ferromagnetic film. It has been shown that an increase in the effective mass in magnetic heterostructures leads to an increase in the electron density of states. It was found that the density of electronic states in the region of surface nickel oxide is practically independent of the thickness of the nickel film. The results and conclusions of the work can be used for theoretical prediction of the physical properties of magnetic nanostructures, in particular spintronic elements.

Application of Chelyshkov polynomials in solving stochastic model with fractional Brownian motion

Abstract This paper introduces a computational spectral collocation approach utilizing orthonormal Chelyshkov polynomials to estimate solutions for both linear and nonlinear stochastic differential equations containing fractional Brownian motion characterized by the Hurst parameter H. The method employs Newton–Cotes nodes to transform these integral equations into manageable linear and nonlinear algebraic forms. The reliability of the proposed method is established through theorems on error estimation and convergence analysis. Moreover, some examples are given to demonstrate the viability, credibility, coherence, and dependability of the approach. Also, a comparative evaluation is conducted to contrast the results obtained from the suggested approach with those produced by the spectral collocation method utilizing orthonormal Bernoulli polynomials. Furthermore, a 95 % confidence interval is constructed for the error mean, revealing that the approximations maintain high accuracy.

A novel approach for the numerical solution of nonlinear Fredholm integral equations using Hosoya polynomial method

Abstract In this paper, we study the graph theoretical polynomial known as the Hosoya polynomial obtained from one of the standard classes of graphs called path. Using this polynomial applied for the numerical solution of the nonlinear Fredholm integral equation, which reduces in the algebraic system of equation with collocation points, then solving this system using Newton’s iterative with the help of MATLAB, we obtain the required approximate solution. The desired results in terms of a set of continuous polynomials over a closed interval [0, 1]. Illustrative applications show the efficiency, accuracy and validity of the proposed technique.

Modification of Adomian decomposition technique in multiplicative calculus and application for nonlinear equations

Multiplicative calculus is a mathematical system that offers an alternative to traditional calculus. Instead of using addition and subtraction to measure change, as in traditional calculus, it uses multiplication and division. The framework of nonlinear equations is an incredibly powerful tool that has proven invaluable in advancing our understanding of various phenomena across a wide range of applied sciences. This framework has enabled researchers to gain deeper insights into a vast array of scientific problems. The physical interpretation of iterative methods for nonlinear equations using multiplicative calculus offers a unique perspective on solving such equations and opens up potential applications across various scientific disciplines. Multiplicative calculus naturally aligns with processes characterized by exponential growth or decay. In many physical, biological, and economic systems, quantities change in a manner proportional to their current state. Multiplicative calculus models these processes more accurately than traditional additive approaches. For example, population growth, radioactive decay, and compound interest are all better described multiplicatively. The primary objective of this work is to modify and implement the Adomian decomposition method within the multiplicative calculus framework and to develop an effective class of multiplicative numerical algorithms for obtaining the best approximation of the solution of nonlinear equations. We build up the convergence criteria of the multiplicative iterative methods. To demonstrate the application and effectiveness of these new recurrence relations, we consider some numerical examples. Comparison of the multiplicative iterative methods with the similar ordinary existing methods is presented. Graphical comparison is also provided by plotting log of residuals. The purpose in constructing new algorithms is to show the implementation and effectiveness of multiplicative calculus.

Exploring Soliton Solutions for Fractional Nonlinear Evolution Equations: A Focus on Regularized Long Wave and Shallow Water Wave Models with Beta Derivative

Exploring Soliton Solutions for Fractional Nonlinear Evolution Equations: A Focus on Regularized Long Wave and Shallow Water Wave Models with Beta Derivative

Results on solutions of several product type nonlinear partial differential equations in [formula omitted]

This paper is devoted to exploring the solutions of the product type nonlinear partial differential equations (PDEs) with three complex variables. By making use of Nevanlinna theory with several complex variables and Hadamard factorization theory of meromorphic functions, and combining with the properties of the full rank determinants and algebraic cofactor, we prove that equation∏i=13(ai1uz1+ai2uz2+ai3uz3)=1 in C3 has no any finite order transcendental entire solution under the condition that the rank of the following matrixA=(a11a12a13a21a22a23a31a32a33) is full rank, i.e. R(A)=3, where aij(i,j=1,2,3) are constants in C. Our results are some improvements and generalizations of the previous results given by Saleeby, Li, Lü and Xu. Meantime, we list some examples to explain that the condition in our theorem can not be removed.

Travelling wave solutions of nonlinear conformable Bogoyavlenskii equations via two powerful analytical approaches

Travelling wave solutions of nonlinear conformable Bogoyavlenskii equations via two powerful analytical approaches

Analysis of Solutions for Nonlinear ψ-Caputo Fractional Differential Equations with Fractional Derivative Boundary Conditions in Banach Algebra

This article explores the solutions of nonlinear implicit ψ-Caputo fractional-order ordinary differential equations (NLIFDEs) with two-point fractional derivatives boundary conditions in Banach algebra. The research aims to establish the existence and uniqueness of solutions for this complex class of differential equations. Utilizing Banach’s and Krasnoselskii’s fixed point theorems, the study conducts a rigorous analysis of the solutions, ensuring their existence and uniqueness. This comprehensive investigation contributes to enhancing the understanding of the behavior of solutions of nonlinear fractional differentials within a challenging mathematical framework.

Identification of Hamiltonian systems using neural networks and first integrals approaches

This research introduces a class of non-parametric identifiers based on differential neural networks represented by Hamiltonian dynamics. The structure of the identifier corresponds to the form of a canonical Hamiltonian system that uses the evolution of generalized coordinates and momentums. The learning laws of the identifier come from applying the first integrals approach, which justifies the design of an exact identifier considering the time invariance of the Hamiltonian, with a finite number of activation functions in the identifier structure. The first integrals approach derives several learning laws for the proposed class of identifiers. The learning laws design uses the estimated derivative of the generalized momentum assessed via a super-twisting differentiator with multiple inputs and outputs. All proposed laws require the solution of differential continuous-time Riccati equations and nonlinear differential equations for the learning laws, which depend on the identification error and state constraints. The developed identifier was evaluated compared to an identifier that did not consider the Hamiltonian constraint using first integrals. This comparison included a numerical evaluation of the identifier considering its application to a classical Hamiltonian system associated with the Kepler dynamics representing satellite orbital evolution. This evaluation confirmed that the identification results were improved with the proposed learning laws regarding the class of Hamiltonian structures, and the quality indicators based on the mean square error were several times lower.

Exploring solutions to the fractional Boiti–Leon–Manna–Pempinelli equation characterizing wave dynamics in incompressible fluids

In this study, the (3 + 1)-dimensional space-time fractional Boiti–Leon–Manna–Pempinelli (BLMP) equation is investigated utilizing the Kudryashov method (KM) and the modified Kudryashov method (MKM). These two efficient methods are implemented to acquire exact closed-form solutions to the considered fractional BLMP equation, and novel solitary wave solutions are constructed involving hyperbolic functions, and exponential rational functions with arbitrary constant parameters. With the aid of a unique wave transformation, the nonlinear fractional differential equation is transformed into an ordinary differential equation. In this unique wave transformation, the conformable fractional derivative is considered to which the chain rule is applied. The obtained solutions are indeed beneficial for analyzing the dynamic behavior of the fractional BLMP equation in describing fluid propagation. A few three-dimensional representations of the attained exact analytic solutions are offered by accounting for the proper selection of the appropriate parameters with the help of mathematical software. As a result, novel soliton solutions, such as the anti-kink type and singular kink type waves, are acquired from the attained solutions. This approach is relatively straightforward and efficient, making it possible to utilize it to obtain exact solutions for higher-order nonlinear partial differential equations.

Analysis of fractional solitary wave propagation with parametric effects and qualitative analysis of the modified Korteweg-de Vries-Kadomtsev-Petviashvili equation

This study explores the fractional form of modified Korteweg-de Vries-Kadomtsev-Petviashvili equation. This equation offers the physical description of how waves propagate and explains how nonlinearity and dispersion may lead to complex and fascinating wave phenomena arising in the diversity of fields like optical fibers, fluid dynamics, plasma waves, and shallow water waves. A variety of solutions in different shapes like bright, dark, singular, and combo solitary wave solutions have been extracted. Two recently developed integration tools known as generalized Arnous method and enhanced modified extended tanh-expansion method have been applied to secure the wave structures. Moreover, the physical significance of obtained solutions is meticulously analyzed by presenting a variety of graphs that illustrate the behaviour of the solutions for specific parameter values and a comprehensive investigation into the influence of the nonlinear parameter on the propagation of the solitary wave have been observed. Further, the governing equation is discussed for the qualitative analysis by the assistance of the Galilean transformation. Chaotic behavior is investigated by introducing a perturbed term in the dynamical system and presenting various analyses, including Poincare maps, time series, 2-dimensional 3-dimensional phase portraits. Moreover, chaotic attractor and sensitivity analysis are also observed. Our findings affirm the reliability of the applied techniques and suggest its potential application in future endeavours to uncover diverse and novel soliton solutions for other nonlinear evolution equations encountered in the realms of mathematical physics and engineering.

Solutions of Second-Order Nonlinear Implicit ψ-Conformable Fractional Integro-Differential Equations with Nonlocal Fractional Integral Boundary Conditions in Banach Algebra

In this paper, we introduce and thoroughly examine new generalized ψ-conformable fractional integral and derivative operators associated with the auxiliary function ψ(t). We rigorously analyze and confirm the essential properties of these operators, including their semigroup behavior, linearity, boundedness, and specific symmetry characteristics, particularly their invariance under time reversal. These operators not only encompass the well-established Riemann–Liouville and Hadamard operators but also extend their applicability. Our primary focus is on addressing complex fractional boundary value problems, specifically second-order nonlinear implicit ψ-conformable fractional integro-differential equations with nonlocal fractional integral boundary conditions within Banach algebra. We assess the effectiveness of these operators in solving such problems and investigate the existence, uniqueness, and Ulam–Hyers stability of their solutions. A numerical example is presented to demonstrate the theoretical advancements and practical implications of our approach. Through this work, we aim to contribute to the development of fractional calculus methodologies and their applications.

Investigation of Space-Time Dynamics of Akbota Equation using Sardar Sub-Equation and Khater Methods: Unveiling Bifurcation and Chaotic Structure

This paper focuses on obtaining exact solutions of nonlinear Akbota equation through the application of the modified Khater method and Sardar sub-equation method. Renowned as one of the latest and precise analytical schemes for nonlinear evolution equations, this method has proven its efficacy by generating diverse solutions for the model under consideration. The equation is crucial in the study of optical solitons, which are stable pulses of light that maintain their shape over long distances. The Akbota equation helps in understanding the behavior and stability of these solitons. The governing equation undergoes transformation into an ordinary differential equation through a well-suited wave transformation. This analytical simplification paves the way for the derivation of trigonometric, hyperbolic, and rational solutions through the proposed methods. To illuminate the physical behavior of the model, the study presents graphical plots of the selected solutions of Khater and Sardar sub-equation method. This visual representation, achieved by selecting appropriate values for arbitrary parameters, enhances the understanding of the system’s dynamics. All calculations in this study are meticulously conducted using the Mathematica and Maple software, ensuring accuracy and reliability in the analysis of the obtained solution. Furthermore we investigate the sensitivity analysis of the dynamical system.