Nonlinear Fractional Differential Equations Research Articles

Advancing non-linear PDE's solutions: modified Yang transform method with Caputo-Fabrizio fractional operator

Abstract Nonlinear partial differential equations have a crucial rule in many physical processes. In this paper, a novel approach is used to study nonlinear partial differential equations of fractional order, which is named as Modified Yang Transform (MYT) method. This approach combines Yang transform with the Adomian decomposition method. The fractional order is considered in the Caputo-Fabrizio sense. Convergence analysis of the modified Yang transform to nonlinear fractional order partial differential equations is presented. Additionally, a solution framework for the solution of nonlinear partial differential equation is carried out and some examples are provided to highlight the application of the current method. To illustrate that how the solution behaves for various fractional orders, 2D and 3D graphs are plotted. Various tables are also provided to show the difference between exact and approximate solutions and the values are compared with other methods in the literature. Results and discussion sections are included for each example to explain the graphs, tables and their results.

Fixed Point and Stability Analysis of a Tripled System of Nonlinear Fractional Differential Equations with n-Nonlinear Terms

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 and Banach’s fixed point theorems, we derive sufficient conditions for the existence of at least one solution, as well as a unique solution. Furthermore, we apply Hyers–Ulam stability analysis to establish criteria for the stability of the system. To demonstrate the applicability of the main results, a detailed example is provided.

Solution analysis for non-linear fractional differential equations

In this research study, two new types of fractional derivatives, the Caputo-Fabrizio derivative that does not involve a singular kernel (like the Riemann–Liouville derivative), which helps in eliminating some drawbacks, such as non-locality, and the Atangana–Baleanu–Caputo (ABC) derivative that uses a non-singular and non-local kernel, offering different characteristics from other fractional derivatives and often leading to more accurate models in certain physical systems, are used. The primary goal of the research is to analyze a non-linear fractional differential equation involving the fractional derivatives of Caputo–Fabrizio and ABC admit at least one solution. Fixed point theory is a fundamental tool in mathematical analysis used to prove the existence of solutions to various equations. Hence, in this study, to achieve the desired results, we employ a novel fixed point theory known as the F-contraction type. The study also includes required conditions and inequalities that need to be satisfied to ensure that a solution exists. One of these conditions is based on the Lipschitz hypothesis. Therefore, we provide the required conditions and inequalities, based on the Lipschitz hypothesis, to show that solutions to our problem exist. Furthermore, we provide two illustrative examples to support our primary findings.

Parameter-dependent fractional boundary value problems: analysis and approximation of solutions

We study a parameter-dependent non-linear fractional differential equation, subject to Dirichlet boundary conditions. Using the fixed point theory, we restrict the parameter values to secure the existence and uniqueness of solutions, and analyse the monotonicity behaviour of the solutions. Additionally, we apply a numerical-analytic technique, coupled with the lower and upper solutions method, to construct a sequence of approximations to the boundary value problem and give conditions for its monotonicity. The theoretical results are confirmed by an example of the Antarctic Circumpolar Current equation in the fractional setting.

A Finite Difference-Based Adams-Type Approach for Numerical Solution of Nonlinear Fractional Differential Equations: A Fractional Lotka–Volterra Model as a Case Study

Abstract In this paper, we developed an efficient Adams-type predictor–corrector (PC) approach for the numerical solution of fractional differential equations (FDEs) with a power law kernel. The main idea of the proposed approach is to use a linear approximation to the nonlinear problem and then implement finite difference approximations of derivatives. Numerical comparisons with the fractional Adams method are made and simulation results are demonstrated to evaluate the approximation error of the proposed approach. The efficiency of this approach has been depicted by presenting numerical solutions of some test fractional calculus models. Numerical simulation of a fractional Lotka–Volterra model is provided, as a case study, using the proposed approach. The advantage of the proposed approach lies in its flexibility in providing approximate numerical solutions with high accuracy.

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.

On the Global Existence of Solution and Lyapunov Asymptotic Practical Stability for Nonlinear Impulsive Caputo Fractional Derivative via Comparison Principle

This paper examines the existence of maximal solution of the comparison differential system and also establishes sufficient conditions for the asymptotic practical stability of the trivial solution of a nonlinear impulsive Caputo fractional differential equations with fixed moments of impulse using the vector Lyapunov functions. First, it was discovered that the vector form of the Lyapunov function was majorized by the maximal solution of the comparison system. From the results obtained, it was also established that the main system is asymptotically practically stable in the sense of Lyapunov.

A Robust and Versatile Numerical Framework for Modeling Complex Fractional Phenomena: Applications to Riccati and Lorenz Systems

The fractional differential quadrature method (FDQM) with generalized Caputo derivatives is used in this paper to show a new numerical way to solve fractional Riccati equations and fractional Lorenz systems. Unlike previous FDQM applications that have primarily focused on linear problems, our work pioneers the use of this method for nonlinear fractional initial value problems. By combining Lagrange interpolation polynomials and discrete singular convolution (DSC) shape functions with the generalized Caputo operator, we effectively transform nonlinear fractional equations into algebraic systems. An iterative method is then utilized to address the nonlinearity. Our numerical results, obtained using MATLAB, demonstrate the exceptional accuracy and efficiency of this approach, with convergence rates reaching 10−8. Comparative analysis with existing methods highlights the superior performance of the DSC shape function in terms of accuracy, convergence speed, and reliability. Our results highlight the versatility of our approach in tackling a wider variety of intricate nonlinear fractional differential equations.

Study of the parametric effect of the wave profiles of the time-space fractional soliton neuron model equation arising in the topic of neuroscience

Time-space fractional nonlinear problems (T-SFNLPs) play a crucial role in the study of nonlinear wave propagation. Time-space nonlinearity is prevalent across various fields of applied science, nonlinear dynamics, mathematical physics, and engineering, including biosciences, neurosciences, plasma physics, geochemistry, and fluid mechanics. In this context, we examine the time-space fractional soliton neuron model (TSFSNM), which holds significant importance in neuroscience. This model explains how action potentials are initiated and propagated by axons, based on a thermodynamic theory of nerve pulse transmission. The signals passing through the cell membrane (CM) are proposed to take the form of solitary sound pulses, which can be represented as solitons. To investigate these soliton solutions, nonlinear fractional differential equations (NLFDEs) are transformed into corresponding partial differential equations (PDEs) using a fractional complex transform (FCT). The Kudryashov method is then applied to determine the wave profiles for the TSFSNM equation. We present 3D, 2D, contour, and density plots of the TSFSNM equation, and further analyze how fractional and time-space parameters influence these wave profiles through additional graphical representations. Kink, singular kink and different types of soliton solutions are successfully recovered through the Kudryashov method. The outcomes of various studies show that the applied method is highly efficient and well-suited for tackling problems in applied sciences and mathematical physics. Graphical representations, coupled with numerical data, reinforce the validity and accuracy of the technique. The proposed method is a convenient and powerful tool for handling the solution of nonlinear equations, making it particularly effective in exploring complex wave phenomena in diverse scientific fields.

Application of Sumudu Transform and New Iterative Method to Solve Certain Nonlinear Ordinary Caputo Fractional Differential Equations

By combining the new iteration method (NIM) and Sumudu transform (ST) methods, together known as the new iteration ST method (NISTM), a semi-analytical or series solution for various fractional differential equations, in which new iteration method (NIM) is used to decompose the nonlinear operator prior to performing the ST, is produced in this script. Using the Caputo sense to account for the fractional derivative, this method computes series solutions for a variety of nonlinear fractional differential equation instances. While in many convergence issues a solution is achieved for just a small number of series terms, our series solution merges with the exact solution differential equation of fractional order in many example problems.

On the Existence of Maximal Solution and Lyapunov Practical Stability of Nonlinear Impulsive Caputo Fractional Derivative via Comparison Principle

This paper examines the existence of maximal solution of the comparison differential system and also establishes sufficient conditions for the practical stability of the trivial solution of a nonlinear impulsive Caputo fractional differential equations with fixed moments of impulse using the vector Lyapunov functions. First, it was discovered that the vector form of the Lyapunov function was majorized by the maximal solution of the comparison system. From the results obtained, it was established that the main system is practically stable in the sense of Lyapunov.

A Study on Positive Solutions to Nonlinear Fractional Differential Equations

The purpose of this article concerns a type of nonlinear boundary value problem of fractional differential equations with the Caputo derivative. Using the fixed point index theory of condensing mapping in cones, the existence results of positive solutions for such a problem are obtained.

The discrete fractional Karamata theorem and its applications

Abstract We establish a fractional extension of the discrete Karamata integration theorem for two types of fractional sum operators. This result, along with other properties of regularly varying sequences and the tools such as comparison theorems and a fixed point theorem in FK-space, is then used to study asymptotic properties of solutions to a nonlinear fractional difference equation. We also establish and utilize a fractional extension of the Stolz–Cesáro theorem, i.e., of the discrete l’Hospital rule. Both, the discrete fractional Karamata theorem and the discrete fractional l’Hospital rule, are believed to find applications in a broader context within the discrete fractional calculus.

Mathematical Analysis of a Navier–Stokes Model with a Mittag–Leffler Kernel

In this paper, we establish the existence and uniqueness results of the fractional Navier–Stokes (N-S) evolution equation using the Banach fixed-point theorem, where the fractional order β is in the form of the Atangana–Baleanu–Caputo fractional order. The iterative method combined with the Laplace transform and Sumudu transform is employed to find the exact and approximate solutions of the fractional Navier–Stokes equation of a one-dimensional problem of unsteady flow of a viscous fluid in a tube. In the domains of science and engineering, these methods work well for solving a wide range of linear and nonlinear fractional partial differential equations and provide numerical solutions in terms of power series, with terms that are simple to compute and that quickly converge to the exact solution. After obtaining the solutions using these methods, we use Mathematica software Version 13.0.1.0 to present them graphically. We create two- and three-dimensional plots of the obtained solutions at various values of β and manipulate other variables to visualize and model relationships between the variables. We observe that as the fractional order β becomes closer to the integer order 1, the solutions approach the exact solution. Lastly, we plot a 2D graph of the first-, second-, third-, and fourth-term approximations of the series solution and observe from the graph that as the number of iterations increases, the approximate solutions become close to the series solution of the fourth-term approximation.

Schauder’s fixed point theorem approach for stability analysis of nonlinear fractional difference equations

Analyzing the stability of solutions is a key qualitative aspect of discrete fractional calculus, with many applications. This paper explores a type of discrete equation that incorporates a Hilfer-like fractional difference. We use Picard’s iteration method and Schauder’s fixed point theorem to establish results concerning the existence and uniqueness of solutions. Additionally, we examine both attractive stability and Ulam–Hyers stability for the proposed system. To illustrate our findings, we provide three examples that demonstrate how the main results are verified.

Existence Results for Differential Equations with Tempered Ψ–Caputo Fractional Derivatives

The method of the equivalent system of fractional integral equations is used to prove the existence results of a unique solution for initial value problems corresponding to various classes of nonlinear fractional differential equations involving the tempered Ψ–Caputo fractional derivative. These include equations with their right side depending on ordinary as well as fractional-order derivatives, or fractional integrals of the solution.

Existence of solution to Hadamard–Caputo fractional differential equation with time delay

This article focuses on the study of the existence of solution to Hadamard–Caputo fractional nonlinear differential equation with time delay. For time delay, two different cases of finite delay and infinite delay are considered. Based on the weighted function solution spaces, the existence and uniqueness of solution to the differential equations are verified by Leray–Schauder selection theorem and Banach fixed point theorem respectively. Finally, examples are given to illustrate the abstract results of this article.

Stability of Composition Caputa– Katugampola Fractional Differential Nonlinear Control System with Delay Riemann −Katugampola

work used the Composition Caputo-Katugampola Fractional Derivatives technique to tackle nonlinear problems and delay fractional differential equations. The fractional derivative is defined using the Caputo and Riemann-Katugampola Fractional Derivatives Method. Proposed method In comparison to other digital technologies, this one is simple, effective, and uncomplicated. Ensure authenticity and correctness proposed method Some exemplary problems have been solved

Stability of Composition Caputa– Katugampola Fractional Differential Nonlinear Control System with Delay Riemann −Katugampola

work used the Composition Caputo-Katugampola Fractional Derivatives technique to tackle nonlinear problems and delay fractional differential equations. The fractional derivative is defined using the Caputo and Riemann-Katugampola Fractional Derivatives Method. Proposed method In comparison to other digital technologies, this one is simple, effective, and uncomplicated. Ensure authenticity and correctness proposed method Some exemplary problems have been solved

Existence theory for ψ -Caputo fractional differential equations

UDC 517.9 The target of this paper is to handle a nonlocal boundary-value problem for a specific kind of nonlinear fractional differential equations encapsuling a collective fractional derivative known as the ψ -Caputo fractional operator. The applied fractional operator generated by the kernel is of the following kind: k ( t , s ) = ψ ( t ) - ψ ( s ) . The existence of the solutions of the above-mentioned equations is tackled by using Mönch's fixed-point theorem combined with the technique of measures of noncompactness. In addition, we discuss the problem of stability within the scope of the Ulam–Hyers stability criteria for the main fractional system. Finally, an example is given to illustrate the viability of the reported results.