Fractional Ordinary Differential Equations Research Articles

Exact Solution of Non-Homogeneous Fractional Differential System Containing 2n Periodic Terms under Physical Conditions

This paper solves a generalized class of first-order fractional ordinary differential equations (1st-order FODEs) by means of Riemann–Liouville fractional derivative (RLFD). The principal incentive of this paper is to generalize some existing results in the literature. An effective approach is applied to solve non-homogeneous fractional differential systems containing 2n periodic terms. The exact solutions are determined explicitly in a straightforward manner. The solutions are expressed in terms of entire functions with fractional order arguments. Features of the current solutions are discussed and analyzed. In addition, the existing solutions in the literature are recovered as special cases of our results.

Constructing RKM-Method for Solving Fractional Ordinary Differential Equations of Fifth-Order with Applications

This paper sheds the light on the vital role that fractional ordinary differential equations(FrODEs) play in the mathematical modeling and in real life, particularly in the physical conditions. Furthermore, if the problem is handled directly by using numerical method, it is a far more powerful and efficient numerical method in terms of computational time, number of function evaluations, and precision. In this paper, we concentrate on the derivation of the direct numerical methods for solving fifth-order FrODEs in one, two, and three stages. Additionally, it is important to note that the RKM-numerical methods with two- and three-stages for solving fifth-order ODEs are convenient, for solving class's fifth-order FrODEs. Numerical examples have been analyzed to demonstrate the efficacy of the new methods in comparison to the analytical method. Therefore, the numerical compression is carried out to confirm the efficiency and precision of the modified numerical methods. Significantly, the study demonstrates that the numerical outcomes of the proposed derived and modified numerical applied methods proved to be brilliant. Finally, based on the findings of the study, it could be said that the numerical outcomes of the test-problems using proposed and modified methods agree well with the analytical solutions. Hence, we can conclude that the proposed numerical methods that are derived or modified in the analytic study of this paper are quite efficient.

State feedback controller design of fractional ordinary differential equations coupled with a fractional reaction–advection–diffusion equation with delay

The boundary control problem of fractional ordinary differential equations coupled with a time fractional reaction–advection–diffusion equation with delay is studied in this paper. To ensure the asymptotic stability of the system we studied, a state feedback boundary controller is proposed. By backstepping method, we transform the fractional coupled system into a chosen target system under a controller. Furthermore, we obtain the existence and uniqueness of the state solution of the considered system. A Lyapunov functional is constructed to show the asymptotic stability of the fractional coupled system by the special fractional Halanay inequality. The asymptotic stability criterion of the fractional coupled system is described by Linear Matrix Inequalities (LIMs). Which can be easily solved and verified. Finally, the applicability of our theoretical results is showed by a numerical simulation.

Approximate symmetry of time-fractional partial differential equations with a small parameter

In the sense of Riemann–Liouville fractional derivative, an approximate symmetry method of (1+1)-dimensional time-fractional partial differential equations (PDEs) with a small parameter is proposed. By first inserting the power series expansion of dependent variable in the small parameter into the fractional PDEs and then separating the equation with respect to the small parameter to get a coupled system of fractional PDEs, we define the pth-order approximate symmetries by the Lie symmetries of the first p+1 coupled fractional PDEs. After performing symmetry reductions for the coupled system of fractional PDEs, we find that zero-order reduced equation coincides with the fractional ordinary differential equation corresponding to the fractional PDE without containing the small parameter while the solutions of higher-order reduced equation can be obtained by solving linear variable coefficient fractional ordinary differential equations. We first exemplify the whole procedure of computing approximate symmetry by studying a time-fractional KdV equation with a small parameter. Specifically, we obtain the higher-order approximate symmetries and perform approximate symmetry reductions by means of an optimal system of one-dimensional approximate Lie subalgebras. Moreover, we construct an explicit power series solution and show the dynamic behaviors of the truncated power series solutions by the evolutional figures. Then in order to make the proposed approximate symmetry method be further convincing, we study an anomalous diffusion equation with a small parameter and construct an analytical solution in terms of the Mittag-Leffler function.

A Method to solve ordinary fractional differential equations using Elzaki and Sumudu transform

The main objective of the paper is to solve ordinary fractional differential equations using Elzaki and Sumudu transform. Moreover some ordinary fractional differential equations are solved by using the presented method. Using different types of fractional differential operators existing methods have been extended and applied for ordinary fractional differential equations.

Symmetries and exact solution of certain nonlinear fractional ordinary differential equations

Symmetries and exact solution of certain nonlinear fractional ordinary differential equations

Asymptotic stability of an epidemiological fractional reaction-diffusion model

Abstract The aim of this article is to study the known susceptible-infectious (SI) epidemic model using fractional order reaction-diffusion fractional partial differential equations [FPDEs] in order to better describe the dynamics of a reaction-diffusion SI with a nonlinear incidence rate describing the infection dynamics of the HIV/AIDS virus. We initially examined the nonnegativity, global existence, and boundedness for solutions of the proposed system. After determining that the proposed model has two steady states, we derived sufficient conditions for the global and local asymptotic stability of the equilibrium of the proposed system and their relationship to basic reproduction in the case of fractional ordinary differential equations and FPDEs by analyzing the eigenvalues and using the appropriately chosen Lyapunov function. Finally, we used numerical examples to illustrate our theoretical results.

Parametric resonance of fractional viscoelastic webs under time-dependent tension

The focus of this paper is on the parametric resonance of fractional viscoelastic webs under time-dependent tension. The novelty of this work lies in the introduction of a fractional order model for viscoelastic webs and the time-dependent tension in roll-to-roll manufacturing. The time-dependent tension is induced by the web treatment processes in roll-to-roll mass production. The rheological properties of viscoelastic webs are represented by the fractional Kelvin-Voigt relation. Considering the geometric nonlinearity, the governing equation is derived based on the Hamilton principle, and then the resulting equation is reduced to a fractional ordinary differential equation by using the Bubnov-Galerkin method. The multiple-scale method is applied to analyze the occurrence condition and dynamic responses of parametric resonance, and the solution stability is demarcated by the Routh-Hurwitz criterion. The effects of the tension variation coefficient, viscoelastic parameter and initial tension on the resonance responses are discussed. The results reveal that the increase in tension variation coefficient and initial tension not only widen the unstable zone of the response curve, but also enhance the resonance amplitude. Moreover, the resonance amplitude decreases with a larger fractional order. This investigation aims to explore the cause of the instability phenomena and thus avoid the unexpected failure of flexible fabrication.

Investigation of Solutions of 𝜷 −conformable Fractional Ordinary Differential Equation With Artificial Neural Network

İn this study, we present a method in order to get initial value fractional differential equations with artificial neural networks. On the basis of the function approach of feedforward neural networks, this method is a general method that is written in an implicit analytical form and results in the creation of a differentiable solution. The first part of the created trial solution which is stated as the sum of the two parts, with no controllable parameters, gives the initial conditions. The second part, unaffected by the initial conditions, consists of a feedforward neural network with controllable parameters (weights). The applicability of this approach is demonstrated in systems of both fractional single ODEs and fractional coupled ODEs.

Similarity Solution for a System of Fractional-Order Coupled Nonlinear Hirota Equations with Conservation Laws

The analysis of differential equations using Lie symmetry has been proved a very robust tool. It is also a powerful technique for reducing the order and nonlinearity of differential equations. Lie symmetry of a differential equation allows a dynamic framework for the establishment of invariant solutions of initial value and boundary value problems, and for the deduction of laws of conservations. This article is aimed at applying Lie symmetry to the fractional-order coupled nonlinear complex Hirota system of partial differential equations. This system is reduced to nonlinear fractional ordinary differential equations (FODEs) by using symmetries and explicit solutions. The reduced equations are exhibited in the form of an Erdelyi–Kober fractional (E-K) operator. The series solution of the fractional-order system and its convergence is investigated. Noether’s theorem is used to devise conservation laws.

Ordinary fractional differential equations to show results of oscillations, using the graphical and numerical approximation in the Julia programming language

The present study developed at a documentary and explanatory level, which aims to assume through an illustrative approach in the Julia programming language, the representation of oscillating results in solutions of fractional ordinary differential equations, among these topics the solutions stand out in: Systems of differential equations equations, differential equations of partial fractions and the differential equations of fractional lags. In this sense, this scientific article assumes the recent foundations in this field and serves as foundations for future research contributions.

A fourth-order fractional Adams-type implicit–explicit method for nonlinear fractional ordinary differential equations with weakly singular solutions

Based on piecewise cubic interpolation polynomials, we develop a high-order numerical method with nonuniform meshes for nonlinear fractional ordinary differential equations (FODEs) with weakly singular solutions. This method is a fractional variant of the classical Adams-type implicit–explicit method widely used for integer order ordinary differential equations. We rigorously prove that the method is unconditionally convergent under the local Lipschitz condition of the nonlinear function, and when the mesh parameter is properly selected, it can achieve optimal fourth-order convergence for weakly singular solutions. We also prove the stability of the method, and discuss the applicability of the method to multi-term nonlinear FODEs and systems of multi-order nonlinear FODEs with weakly singular solutions. Numerical results are given to confirm the theoretical convergence results.

Comparison principles for systems of Caputo fractional order ordinary differential equations

The comparison principle for systems of Caputo fractional order ordinary differential equations still remains blank. By applying the Caputo solution continuation and Vainikko’s definition, and reconstructing the framework of system comparison, here we state and prove general comparison principles for systems of Caputo fractional order ordinary differential equations, and illustrate them by examples with numerical simulations.

Application of the Optimal Homotopy Asymptotic Approach for Solving Two-Point Fuzzy Ordinary Differential Equations of Fractional Order Arising in Physics

This work focuses on solving and analyzing two-point fuzzy boundary value problems in the form of fractional ordinary differential equations (FFOBVPs) using a new version of the approximation analytical approach. FFOBVPs are useful in describing complex scientific phenomena that include heritable characteristics and uncertainty, and obtaining exact or close analytical solutions for these equations can be challenging, especially in the case of nonlinear problems. To address these difficulties, the optimal homotopy asymptotic method (OHAM) was studied and extended in a new form to solve FFOBVPs. The OHAM is known for its ability to solve both linear and nonlinear fractional models and provides a straightforward methodology that uses multiple convergence control parameters to optimally manage the convergence of approximate series solutions. The new form of the OHAM presented in this work incorporates the concepts of fuzzy sets theory and some fractional calculus principles to include fuzzy analysis in the method. The steps of fuzzification and defuzzification are used to transform the fuzzy problem into a crisp problem that can be solved using the OHAM. The method is demonstrated by solving and analyzing linear and nonlinear FFOBVPs at different values of fractional derivatives. The results obtained using the new form of the fuzzy OHAM are analyzed and compared to those found in the literature to demonstrate the method’s efficiency and high accuracy in the fuzzy domain. Overall, this work presents a feasible and efficient approach for solving FFOBVPs using a new form of the OHAM with fuzzy analysis.

The pulse narrowing nonlinear transmission lines model within the local fractional calculus on the Cantor sets

PurposeThe fractal and fractional calculus have obtained considerable attention in the electrical and electronic engineering since they can model many complex phenomena that the traditional integer-order calculus cannot. The purpose of this paper is to develop a new fractional pulse narrowing nonlinear transmission lines model within the local fractional calculus for the first time and derive a novel method, namely, the direct mapping method, to seek for the nondifferentiable (ND) exact solutions.Design/methodology/approachBy defining some special functions via the Mittag–Leffler function on the Cantor sets, a novel approach, namely, the direct mapping method is derived via constructing a group of the nonlinear local fractional ordinary differential equations. With the aid of the direct mapping method, four groups of the ND exact solutions are obtained in just one step. The dynamic behaviors of the ND exact solutions on the Cantor sets are also described through the 3D graphical illustration.FindingsIt is found that the proposed method is simple but effective and can construct four sets of the ND exact solutions in just one step. In addition, one of the ND exact solutions becomes the exact solution of the classic pulse narrowing nonlinear transmission lines model for the special case 9 = 1, which strongly proves the correctness and effectiveness of the method. The ideas in the paper can be used to study the other fractal partial differential equations (PDEs) within the local fractional derivative (LFD) arising in electrical and electronic engineering.Originality/valueThe fractional pulse narrowing nonlinear transmission lines model within the LFD is proposed for the first time in this paper. The proposed method in the work can be used to study the other fractal PDEs arising in electrical and electronic engineering. The findings in this work are expected to shed a light on the study of the fractal PDEs arising in electrical and electronic engineering.

On new aspects of Chebyshev polynomials for space-time fractional diffusion process

Abstract Chebyshev collocation scheme and Finite difference method plays central roles for solving fractional differential equations (FDE). Therefore purpose of this paper is to solve fractional mathematical problem of diffusion by Chebyshev collocation method which turns the original problems into the system of fractional ordinary differential and algebraic equations by imposing orthogonality property. This system is solved by implementing Finite difference method. The numerical illustrations confirm that the combination of these two methods allow us to establish one of the best truncated solution in the series form.

Tumor treatment with chemo-virotherapy and MEK inhibitor: A mathematical model of Caputo fractional differential operator

Mitogen-activated protein kinase (MEK) inhibitors and oncolytic virotherapy are identified as promising cancer therapies that can enhance the efficacy of other cancer treatments. A few studies demonstrate that cancer cells proliferate when exposed to virotherapy with MEK inhibitors in an integer order model or without them in a fractional order model. None of them are intended to investigate tumor cell growth under the combined treatment strategy of chemo-virotherapy with a MEK inhibitor in a fractional order model. In this paper, a mathematical model based on fractional order ordinary differential equations (ODEs) is developed for the mutual interactions among tumor cells, as well as a therapeutic combination of chemotherapy, oncolytic viruses and the functional consequence of MEK inhibitor, to investigate how virotherapy could enhance chemotherapy under the action of MEK inhibitor. The numerical results show that virus burst size and MEK inhibitors have a noticeable impact on regulating the trend of tumor cell proliferation. While virotherapy responses to tumor cell proliferation are undoubtedly quicker than chemotherapeutic treatment responses, MEK intensity clearly affects the success of the treatment regimen. The results of this study can contribute to the development of a therapeutic strategy that combines MEK inhibitor functional monitoring with tumor cell growth control.

A New Approach for Solving Nonlinear Fractional Ordinary Differential Equations

Recently, researchers have been interested in studying fractional differential equations and their solutions due to the wide range of their applications in many scientific fields. In this paper, a new approach called the Hussein–Jassim (HJ) method is presented for solving nonlinear fractional ordinary differential equations. The new method is based on a power series of fractional order. The proposed approach is employed to obtain an approximate solution for the fractional differential equations. The results of this study show that the solutions obtained from solving the fractional differential equations are highly consistent with those obtained by exact solutions.

On Using Piecewise Fractional Differential Operator to Study a Dynamical System

This research work is devoted to undertaking a dynamical system representing SARS-CoV-19 disease under the concept of piecewise fractional-order derivative using the Caputo concept since long-memory and short-memory terms are not well explained by ordinary fractional differential equations. It has been found that for such disruption, piecewise operators of fractional derivatives have been found useful in many cases. Therefore, we study a compartmental model of susceptible and infected individuals under the concept of piecewise derivative. We establish the existence theory of the considered model by using some Banach and Schauder fixed-point theorems. Keeping the importance of stability, a pertinent result related to the said area is also developed. The said concept of stability is based on the concept given by Ulam and Hyers. Further, to derive the numerical results, we use the Euler method to develop a numerical scheme for the considered model. Using real available data, we have presented various graphical presentations of two compartments against different fractional orders and various values of isolation parameters. The crossover behaviors in the dynamics can be clearly observed, which is explained by the piecewise operators, not the usual fractional-order derivative.

Fractional Modeling of Non-Newtonian Casson Fluid between Two Parallel Plates

In this manuscript, fractional modeling of non-Newtonian Casson fluid squeezed between two parallel plates is performed under the influence of magneto-hydro-dynamic and Darcian effects. The Casson fluid model is fractionally transformed through mixed similarity transformations. As a result, partial differential equations (PDEs) are transformed to a fractional ordinary differential equation (FODE). In the current modeling, the continuity equation is satisfied while the momentum equation of the integral order Casson fluid is recovered when the fractional parameter is taken as α = 1 . A modified homotopy perturbation algorithm is used for the solution and analysis of highly nonlinear and fully fractional ordinary differential equations. Obtained solutions and errors are compared with existing integral order results from the literature. Graphical analysis is also performed at normal and radial velocity components for different fluid and fractional parameters. Analysis reveals that a few parameters are showing different behavior in a fractional environment as compared to existing integer-order cases from the literature. These findings affirm the importance of fractional calculus in terms of more generalized analysis of physical phenomena.