Fractional Ordinary Differential Equations Research Articles

Symmetry Analysis and Wave Solutions of Time Fractional Kupershmidt Equation

This study employs the Lie symmetry technique to explore the symmetry features of the time fractional Kupershmidt equation. Specifically, we use the Lie symmetry technique to derive the symmetry generators for this equation, which incorporates a conformal fractional derivative. We use the symmetry generators to transform the fractional partial differential equation into a fractional ordinary differential equation, thereby simplifying the analysis. The obtained reduced equation is of fourth order nonlinear ordinary differential equation. To find the wave solutions, F/G‐expansion process has been used to obatin different types of solutions of the time‐fractional Kuperschmidt equation. The obtained wave solutions are hyperbolic and trigonometric in nature. We then use Maple software to visually depict these wave solutions for specific parameter values, providing insights into the behaviour of the system under investigation. Peak and kink wave solutions are achieved for the given problem.

A MODERN TRAVELING WAVE SOLUTION FOR CAPUTO-FRACTIONAL KLEIN–GORDON EQUATIONS

This research paper introduces a novel approach to deriving traveling wave solutions (TWSs) for the Caputo-fractional Klein–Gordon equations. This research presents a distinct methodological advancement by introducing TWSs of a particular time-fractional partial differential equation, utilizing a non-local fractional operator, specifically the Caputo derivative. To achieve our goal, a novel transformation is considered, that converts a time-fractional partial differential equation into fractional ordinary differential equations, enabling analytical solutions through various analytical methods. This paper employs the homotopy analysis method to achieve the target objectives. To demonstrate the efficiency and applicability of the proposed transform and method, two examples are discussed and analyzed in figures.

New results for fractional ordinary differential equations in fuzzy metric space

<abstract><p>In this paper, we primarily focused on the existence and uniqueness of the initial value problem for fractional order fuzzy ordinary differential equations in a fuzzy metric space. First, definitions and relevant properties of the Gamma function and Beta function within a fuzzy metric space were provided. Second, by employing the principle of fuzzy compression mapping and Choquet integral of fuzzy numerical functions, we established the existence and uniqueness of solutions to initial value problems for fuzzy ordinary differential equations. Finally, several examples were presented to demonstrate the validity of our obtained results.</p></abstract>

A third-order numerical method for solving fractional ordinary differential equations

In this paper, we developed a novel numerical method for solving general nonlinear fractional ordinary differential equations (FODEs). First, we transformed the nonlinear FODEs into the equivalent Volterra integral equations. We then developed a time-stepping algorithm for the numerical solution of the Volterra integral equations based on the third-order Taylor expansion for approximating the integrands in the Volterra integral equations on a chosen mesh with the mesh parameter $ h $. This approximation led to implicit nonlinear algebraic equations in the unknowns at each given mesh point, and an iterative algorithm based on Newton's method was developed to solve the resulting implicit equations. A convergence analysis of this numerical scheme showed that the error between the exact solution and numerical solution at each mesh point is $ \mathcal{O}(h^{3}) $, independent of the fractional order. Finally, four numerical examples were solved to verify the theoretical results and demonstrate the effectiveness of the proposed method.

Solution homogeneous and non-homogeneous fractional ordinary differential equations by ZMA-transform

In this view, a modern vision to discuss homogeneous and non-homogeneous fractional ordinary differential equations (HFODEs)- (NHFODEs) with initial condition (IC) by employment ZMA-transform (ZMAT). Out of exhibit basic properties for theorems of ZMA-transform (ZMAT) and fractional calculus(FC). Moreover presented ZMA-transform (ZMAT) of Riemann-Liovill derivative the consequent application of ZMA-transform (ZMAT) of fractional derivatives and to understand the task more, some applications has been solved.

Existence and uniqueness of nonlinear fractional differential equations with the Caputo and the Atangana-Baleanu derivatives: Maximal, minimal and Chaplygin approaches

<p>This work provided a detailed theoretical analysis of fractional ordinary differential equations with Caputo and the Atangana-Baleanu fractional derivative. The work started with an extension of Tychonoff's fixed point and the Perron principle to prove the global existence with extra conditions due to the properties of the fractional derivatives used. Then, a detailed analysis of the existence of maximal and minimal solutions was presented for both cases. Then, using Chaplygin's approach with extra conditions, we also established the existence and uniqueness of the solutions of these equations. The Abel and the Bernoulli equations were considered as illustrative examples and were solved using the fractional middle point method.</p>

LYAPUNOV-TYPE INEQUALITY FOR CERTAIN HALF-LINEAR LOCAL FRACTIONAL ORDINARY DIFFERENTIAL EQUATIONS

In this paper, we establish a Lyapunov-type inequality for the half-linear local fractional ordinary differential equation based on the formulation of the local fractional derivative. In addition, we apply the inequality to investigate the non-existence and uniqueness of solutions for related homogeneous and non-homogeneous local fractional boundary value problems.

Mathematical Analysis of Fractional Diabetes Model via an Efficient Computational Technique

Diabetes is referred to a chronic metabolic disease signalized by elevated levels of blood glucose (also known as blood sugar level), which results over time in serious damage to the heart, blood vessels, eyes, kidneys, and nerves in the body. A mathematical assessment of the diabetes model using the Caputo fractional order derivative operator is given in this research paper. The concept of a Caputo fractional order derivative is a novel class of non-integer order derivative that has many applications in real-life scenarios. The proposed model is represented by a set of fractional ordinary differential equations. The authors employed the Sumudu Transform Homotopy Perturbation Method (STHPM) for finding the series solutions of the model being studied. By giving various numerical values to the respective model parameters, graphical analysis is also performed. It is observed in the numerical discussion that a decrease in both fractional order and leads to decrease in the number of diabetic people.

Analytical methods in fractional biological population modeling: Unveiling solitary wave solutions

<abstract><p>We examine a biological population model of fractional order (FBPM) in this paper using the Riccati-Bernoulli sub-ODE approach. Many scenarios in computational biology make use of this fundamental fractional model. Of particular note is that our study's FBPM uses fractional derivatives to track changes in the density populations. The study is concerned with the construction of new solitary wave solutions for the FBPM, a system of two nonlinear fractional ordinary differential equations. In this investigation, we use the conformable derivative as the fractional derivative. The Backlund transformation is the foundation of the solution process. We create a variety of families of soliton wave solutions and explain different physical behaviours that are inherent in the problems we explore. In particular, we apply the suggested methods to investigate rational, periodic, and hyperbolic solutions. The solutions found in various classes provide insightful information about the underlying physical mechanisms. To sum up, our current methods are superior instruments for analyzing different families of solutions in fractional-order issues.</p></abstract>

Convergence of the L1 Two-Term Equation Scheme

Fractional derivatives have found application in modeling various processes in different fields of science. Finite difference schemes are a main approach for numerical solution of models using fractional differential equations. In this paper we investigate the convergence and order of the numerical solution of two-term ordinary fractional differential equation which uses the L1 approximation of the fractional derivative. Inequalities for the weights of L1 approximation are derived and used to prove the convergence of the L1 scheme for the two-term equation. Conditions for the parameter of the two-term equation and error estimates of L1 scheme are obtained. Experimental results for the order and error of the L1 scheme are presented in the paper.

Optimizing option pricing: Exact and approximate solutions for the time-fractional Ivancevic model

This research investigates the time fractional Ivancevic option pricing model and presents two distinct solution methods: the invariant subspace method for obtaining exact solutions and the residual power series method for generating approximate solutions. In the invariant subspace method, we employ an algebraic approach to solve the time fractional Ivancevic option pricing model. By constructing an appropriate invariant subspace, we derive a system of fractional ordinary differential equations that characterizes the exact solution. This method provides a rigorous and efficient technique for generating various exact solutions. We also propose the residual power series method for obtaining approximate solutions. This method involves expressing the solution as an expansion of power functions and iteratively solving the residual equation to refine the approximation. By truncating the series expansion, we obtain an approximate solution that captures the essential features of the model while balancing computational efficiency and accuracy. Different graphs in 2-dimensions and 3-dimensions are presented to pioneer the obtained solutions. Our findings demonstrate the effectiveness of the invariant subspace method in generating exact solutions and shedding light on the model's dynamics. The residual power series method, on the other hand, provides a flexible and computationally efficient approach for obtaining reliable approximate solutions. The integration of exact and approximate methodologies presents a thorough structure for evaluating and valuing options in the time fractional Ivancevic option pricing model. This comprehensive approach yields valuable understandings for financial professionals and researchers alike.

Remarks on the Solution of Fractional Ordinary Differential Equations Using Laplace Transform Method

In this work we used the Laplace transform method to solve linear fractional-order differential equation, fractional ordinary differential equations with constant and variable coefficients. The solutions were expressed in terms of Mittag-Leffler functions, and then written in a compact simplified form. As a special case for simplicity, the order of the derivative determined the order of the solution that was obtained. This paper presented several case studies involving the implementation of Fractional Order calculus-based models, whose results demonstrate the importance of Fractional Order Calculus.

Approximation of Caputo Fractional Derivative and Numerical Solutions of Fractional Differential Equations

In this paper, we consider an approximation of the Caputo fractional derivative and its asymptotic expansion formula, whose generating function is the polylogarithm function. We prove the convergence of the approximation and derive an estimate for the error and order. The approximation is applied for the construction of finite difference schemes for the two-term ordinary fractional differential equation and the time fractional Black–Scholes equation for option pricing. The properties of the approximation are used to prove the convergence and order of the finite difference schemes and to obtain bounds for the error of the numerical methods. The theoretical results for the order and error of the methods are illustrated by the results of the numerical experiments.

Similarity solution of fractional viscoelastic magnetohydrodynamic fluid flow over a permeable stretching sheet with suction/injection

A novel easy‐to‐use similarity technique for spatial fractional partial differential equations is introduced to solve the flow and heat transfer of fractional viscoelastic magnetohydrodynamic (MHD) fluid over a permeable stretching sheet. The proposed method represents a new technique for deriving similarity transformations, which allows the conversion of the fractional partial differential equations to fractional ordinary differential equations. Moreover, using the Grünwald–Letnikov formula, the fourth‐order Runge–Kutta method is adopted here to solve the obtained similarity equations, using the shooting method. The transformed governing equations were found to be dependent on six physical parameters, namely, fractional parameter, magnetic field parameter, generalized local Reynolds number, generalized Prandtl number, the wall suction/injection parameter, and the wall stretching exponential. The results reveal that maximums of the skin friction coefficient and local Nusselt number are obtained for fractional‐order derivative of about 0.5. Moreover, application of a magnetic field on electrically conducting fractional fluid thins the velocity boundary layer and thickens the thermal boundary layer; therefore, the skin friction coefficient increases whereas the local Nusselt number decreases. Furthermore, an increase in any parameters of the local Reynolds number, suction parameter, or the wall stretching exponential leads to thinner boundary layers and higher values of skin friction coefficient and local Nusselt number, whereas imposing injection has opposite effects. Finally, increasing the generalized Prandtl number leads to a thinner thermal boundary layer and higher local Nusselt number for the fractional viscoelastic fluid. It would be of interest to extend the proposed method to other practical problems in the same field.

The Numerical Methods of Fractional Differential Equations

Differential equations with non-integer order derivatives have demonstrated are suitable models for a variety of physical events in several fields including diffusion processes and damping laws, fluid mechanics neural networks. In this study, i will discuss two numerical methods Diethelm's method and Adams-Bashforth-Moulton method for solving fractional ordinary differential equations (ODEs) with initial conditions.

Time-fractional Davey–Stewartson equation: Lie point symmetries, similarity reductions, conservation laws and traveling wave solutions

As a celebrated nonlinear water wave equation, the Davey–Stewartson equation is widely studied by researchers, especially in the field of mathematical physics. On the basis of the Riemann–Liouville fractional derivative, the time-fractional Davey–Stewartson equation is investigated in this paper. By application of the Lie symmetry analysis approach, the Lie point symmetries and symmetry groups are obtained. At the same time, the similarity reductions are derived. Furthermore, the equation is converted to a system of fractional partial differential equations and a system of fractional ordinary differential equations in the sense of Riemann–Liouville fractional derivative. By virtue of the symmetry corresponding to the scalar transformation, the equation is converted to a system of fractional ordinary differential equations in the sense of Erdélyi–Kober fractional integro-differential operators. By using Noether’s theorem and Ibragimov’s new conservation theorem, the conserved vectors and the conservation laws are derived. Finally, the traveling wave solutions are achieved and plotted.

Collocation-Based Approximation for a Time-Fractional Sub-Diffusion Model

We consider the numerical solution of a one-dimensional time-fractional diffusion problem, where the order of the Caputo time derivative belongs to (0, 1). Using the technique of the method of lines, we first develop from the original problem a system of fractional ordinary differential equations. Using an integral equation reformulation of this system, we study the regularity properties of the exact solution of the system of fractional differential equations and construct a piecewise polynomial collocation method to solve it numerically. We also investigate the convergence and the convergence order of the proposed method. To conclude, we present the results of some numerical experiments.

Group Analysis Explicit Power Series Solutions and Conservation Laws of the Time-Fractional Generalized Foam Drainage Equation

In this study, the classical Lie symmetry method is successfully applied to investigate the symmetries of the time-fractional generalized foam drainage equation with the Riemann–Liouville derivative. With the help of the obtained Lie point symmetries, the equation is reduced to nonlinear fractional ordinary differential equations (NLFODEs) which contain the Erdélyi–Kober fractional differential operator. The equation is also studied by applying the power series method, which enables us to obtain extra solutions. The obtained power series solution is further examined for convergence. Conservation laws for this equation are obtained with the aid of the new conservation theorem and the fractional generalization of the Noether operators.

On the solution of fractional differential equations using Atangana’s beta derivative and its applications in chaotic systems

In this research, we examine the solution of ordinary fractional differential equations using Atangana’s beta derivative. Our approach is divided into two parts. First, we establish conditions under which the fractional differential equation has a unique solution. Next, we develop a numerical technique based on the Adams–Bashforth method and demonstrate its convergence. We then apply the numerical method to a test example to evaluate its efficacy. Finally, we analyze three nonlinear chaotic and hyperchaotic fractional dynamical systems, calculating the strange attractors for various fractional orders. We analyzed the fractional chaotic systems and showed that by changing the order of the fractional derivative, some properties of the system change, such as the area of the chaos attracting region.

Naimark Problem for a Fractional Ordinary Differential Equation

Naimark Problem for a Fractional Ordinary Differential Equation