Solutions Of Nonlinear Equations Research Articles

The High-Order ADI Difference Method and Extrapolation Method for Solving the Two-Dimensional Nonlinear Parabolic Evolution Equations

In this paper, the numerical solution for two-dimensional nonlinear parabolic equations is studied using an alternating-direction implicit (ADI) Crank–Nicolson (CN) difference scheme. Firstly, we use the CN format in the time direction, and then use the CN format in the space direction before discretizing the second-order center difference quotient. In addition, we strictly prove that the proposed ADI difference scheme has unique solvability and is unconditionally stable and convergent. The extrapolation method is further applied to improve the numerical solution accuracy. Finally, two numerical examples are given to verify our theoretical results.

On the Solutions of Nonlinear Implicit ω-Caputo Fractional Order Ordinary Differential Equations with Two-Point Fractional Derivatives and Integral Boundary Conditions in Banach Algebra

This article delves into the analysis of nonlinear implicit ω-Caputo fractional-order ordinary differential equations (NLIFDEs) with two-point fractional derivatives and integral boundary conditions within the context of Banach algebra. The primary focus is on demonstrating the existence and uniqueness of solutions for these complex fractional differential equations by utilizing Banach's and Krasnoselskii's fixed point theorems. Furthermore, the study explores the stability of these solutions through the Ulam-Hyers and Ulam-Hyers-Rassias stability criteria, thereby assessing the robustness of the proposed model. To illustrate the versatility of the generalized model, several special cases are examined, showcasing its ability to encompass various classical models. The practical applicability of the theoretical findings is underscored through a numerical example, which demonstrates the feasibility and relevance of the proposed methodology. This thorough investigation advances the comprehension of nonlinear fractional differential equations with integral boundary conditions, highlighting the intricate relationship between fractional derivatives, nonlinearities, and integral terms. The results offer significant insights into the behavior and stability of solutions within this demanding mathematical framework.

Analysis of Cauchy Reaction-Diffusion Equations Involving Atangana-Baleanu Fractional Operator

This study investigates the Cauchy reaction-diffusion equation (CRDE) with the Atangana-Baleanu differential operator. The existence and uniqueness of solutions to fractional starting value issues are begun using the fixed-point theorem and contraction principle, respectively. The proposed study uses the natural variation iteration technique (NVIM) to get an approximate solution for nonlinear fractional reaction-diffusion equations. This study's approximate answers are compared to other solutions found using known methodologies, and the results are discussed. The devised technique has benefits in terms of accuracy and computational cost efficiency, which may be used to solve nonlinear fractional reaction-diffusion equations.

A Novel Analytical Approximate Solution for Strongly Nonlinear Third-Order Jerk Equations Using a Modified Iteration Method

Nonlinear jerk equations, characterized by a third-order time derivative, play a crucial role in modeling various physical phenomena across disciplines like mechanics, circuits, and biology. Accurately solving these equations is essential for understanding and predicting the behavior of such systems. However, obtaining analytical solutions for nonlinear jerk equations can be challenging, necessitating the development of robust and accurate approximation methods. This work explores, for the first time, the application of the modified iteration approach to solve third-order jerk equations. By comparing the obtained approximate solutions with both exact and existing analytical solutions for established engineering problems, we demonstrate the superior accuracy and rapid convergence of the proposed method. The significantly reduced error percentages highlight the effectiveness of the modified iteration approach in providing precise solutions for nonlinear jerk equations, paving the way for its application in a wide range of oscillation problems within nonlinear sciences and engineering.

Exact solutions of a third order nonlinear Schrodinger equation with time variable coefficients

Exact solutions of a third order nonlinear Schrodinger equation with time variable coefficients

Comparison of Runge-Kutta Methods for Solving Nonlinear Equations

This research explores the numerical solutions of nonlinear ordinary differential equations using four distinct methods: Heun method, the Midpoint method, Ralston's method, and the fourth-order Runge-Kutta method (RK4). Given the complexity and prevalence of nonlinear equations in various scientific fields, effective numerical techniques are essential for obtaining accurate solutions. This research contributes to the understanding of numerical methods in solving complex differential equations, aiding practitioners in selecting the most appropriate approach for their needs.

Empowering Discovery Learning and Direct Instruction with Computer-Assisted to Enhance Mathematical Performance

In the current educational context, the low academic performance of students in mathematics indicates the need for the implementation of more effective teaching methods to support the understanding of complex concepts. The effectiveness of learning can be enhanced through computer-assisted instruction, particularly in discovery learning (DL-CA) and direct instruction (DI-CA) utilizing web-based media. This study employs a quantitative approach with statistical analysis to examine the significant impact of these teaching methods on students' mathematical performance. Discovery learning and direct instruction were applied to two groups studying numerical methods, focusing on the topic of nonlinear equation solutions using the bisection method, with each group consisting of 33 students. The statistical test results indicated that there was no significant difference in mathematical performance between the DL-CA and DI-CA groups; both methods showed comparable effectiveness. However, both methods had a significant effect on the N-Gain of the learning outcomes in both groups. The improvement in mathematical performance for the DL-CA group was better than that for the DI-CA group, with more students showing high improvement (11 students) compared to the DI-CA group (1 student). Additionally, in the low improvement category, the DL-CA group had fewer students (2 students) than the DI-CA group (8 students). This suggests that both discovery learning and direct instruction have almost equivalent reliability in achieving students' mathematical abilities, but DL-CA is more effective in enhancing performance compared to DI-CA.

Numerical solution of nonlinear partial differential equations using shifted Legendre collocation method

In life important applications are modeled mathematically by nonlinear partial differential equations. Primarily, the objective is to transform such equations into a system of algebraic equations to get their solutions. The Legendre collocation method is demonstrated by the differentiation operational matrix via shifted Legendre polynomials in such a transformation. The validity and effectiveness of the prospective method are manifested by illustrative examples, an error analysis, residual analysis, and comparison with others.

The extended Adomian decomposition method and its application to the rotating shallow water system for the numerical pulsrodon solutions

The rotating shallow water system is an important physical model, which has been widely used in many scientific areas, such as fluids, hydrodynamics, geophysics, oceanic and atmospheric dynamics. In this paper, we extend the application of the Adomian decomposition method from the single equation to the coupled system to investigate the numerical solutions of the rotating shallow water system with an underlying circular paraboloidal basin. By introducing some special initial values, we obtain interesting approximate pulsrodon solutions corresponding to pulsating elliptic warm-core rings, which take the form of realistic series solutions. Numerical results reveal that the numerical pulsrodon solutions can quickly converge to the exact solutions derived by Rogers and An, which fully shows the efficiency and accuracy of the proposed method. Note that the method proposed can be effectively used to construct numerical solutions of many nonlinear mathematical physics equations. The results obtained provide some potential theoretical guidance for experts to study the related phenomena in geography, oceanic and atmospheric science.

Existence and Stability for Fractional Differential Equations with a ψ–Hilfer Fractional Derivative in the Caputo Sense

This article aims to explore the existence and stability of solutions to differential equations involving a ψ-Hilfer fractional derivative in the Caputo sense, which, compared to classical ψ-Hilfer fractional derivatives (in the Riemann–Liouville sense), provide a clear physical interpretation when dealing with initial conditions. We discovered that the ψ-Hilfer fractional derivative in the Caputo sense can be represented as the inverse operation of the ψ-Riemann–Liouville fractional integral, and used this property to prove the existence of solutions for linear differential equations with a ψ-Hilfer fractional derivative in the Caputo sense. Additionally, we applied Mönch’s fixed-point theorem and knowledge of non-compactness measures to demonstrate the existence of solutions for nonlinear differential equations with a ψ-Hilfer fractional derivative in the Caputo sense, and further discussed the Ulam–Hyers–Rassias stability and semi-Ulam–Hyers–Rassias stability of these solutions. Finally, we illustrated our results through case studies.

A finite-difference-based Adams-type approach for numerical solution of nonlinear fractional differential equations: A fractional Lotka-Volterra model as a case study

A finite-difference-based Adams-type approach for numerical solution of nonlinear fractional differential equations: A fractional Lotka-Volterra model as a case study

A novel efficient analytical technique and its application to fractional Cauchy reaction–diffusion equations

In this research paper, we suggest a novel analytical method to obtain the approximate solutions of linear and nonlinear differential equations of arbitrary order. The proposed technique is a good combination of Kharrat–Toma transform and q-homotopy analysis method. Additionally, we also find out the solution of Cauchy reaction–diffusion equation associated with regularized version of Hilfer–Prabhakar fractional derivative. The Cauchy reaction–diffusion equation is broadly used to describe the dynamical processes in physics, geology, biology, ecology, etc. Some characteristic properties of the considered model and existence as well as the uniqueness of the solutions are also discussed. The outcomes of the considered model are exhibited graphically to show the accuracy and efficiency of the suggested method.

Fractional Consistent Riccati Expansion Method and Soliton-Cnoidal Solutions for the Time-Fractional Extended Shallow Water Wave Equation in (2 + 1)-Dimension

In this work, a fractional consistent Riccati expansion (FCRE) method is proposed to seek soliton and soliton-cnoidal solutions for fractional nonlinear evolutional equations. The method is illustrated by the time-fractional extended shallow water wave equation in the (2 + 1)-dimension, which includes a lot of KdV-type equations as particular cases, such as the KdV equation, potential KdV equation, Boiti–Leon–Manna–Pempinelli (BLMP) equation, and so on. A rich variety of exact solutions, including soliton solutions, soliton-cnoidal solutions, and three-wave interaction solutions, have been obtained. Comparing with the fractional sub-equation method, G′/G-expansion method, and exp-function method, the proposed method gives new results. The method presented here can also be applied to other fractional nonlinear evolutional equations.

Novel results from quadratically nonlinear elastic wave models using Murnaghan’s potential

AbstractIn this article, we study one, two and three-dimensional nonlinear elastic wave equations using quadratically nonlinear Murnaghan potential. We employ two effective methods for obtaining approximate series solutions the Adomian decomposition and the variational iteration method. These methods have the advantage of not requiring any physical parametric assumptions in the problem. Finally, these methods can generate expansion solutions for linear and nonlinear differential equations without perturbation, linearization, or discretization. The results obtained using the adopted methods along various initial and boundary conditions are in excellent agreement with the numerical results on MATLAB, which show the reliability of our methods to these problems. We came to the conclusion that our methods are accurate and simple to use.

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.