Physical Sciences and Mathematics Commons^™

Numerical Simulations For Fractional Differential Equations Of Higher Order And A Wright-Type Transformation, Mariana Nacianceno, Tamer Oraby, Hansapani Rodrigo, Y. Sepulveda, Josef A. Sifuentes, Erwin Suazo, T. Stuck, J. Williams

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

In this work, a new relationship is established between the solutions of higher fractional differential equations and a Wright-type transformation. Solutions could be interpreted as expected values of functions in a random time process. As applications, we solve the fractional beam equation, fractional electric circuits with special functions as external sources, and derive d’Alembert’s formula for the fractional wave equation. Due to this relationship, we present two methods for simulating solutions of fractional differential equations. The two approaches use the interpretation of the Caputo derivative of a function as a Wright-type transformation of the higher derivative of the function. In …

Employing A Fractional Basis Set To Solve Nonlinear Multidimensional Fractional Differential Equations, Md. Habibur Rahman, Muhammad I. Bhatti, Nicholas Dimakis

Physics and Astronomy Faculty Publications and Presentations

Fractional-order partial differential equations have gained significant attention due to their wide range of applications in various fields. This paper employed a novel technique for solving nonlinear multidimensional fractional differential equations by means of a modified version of the Bernstein polynomials called the Bhatti-fractional polynomials basis set. The method involved approximating the desired solution and treated the resulting equation as a matrix equation. All fractional derivatives are considered in the Caputo sense. The resulting operational matrix was inverted, and the desired solution was obtained. The effectiveness of the method was demonstrated by solving two specific types of nonlinear multidimensional fractional …

A Novel Technique To Solve Fractional Differential Equations Using Fractional-Order B-Polynomial Basis Set, Md Habibur Rahman

Theses and Dissertations

This thesis uses B-Polynomial bases to solve both one-dimensional and multi-dimensional linear and nonlinear partial differential equations and linear and nonlinear fractional differential equations. The approach involves constructing an operational matrix from the terms of these equations using Caputo's fractional derivative of fractional B-polynomials. This leads to a semi-analytical solution derived from a matrix equation, and the results obtained using this method are compared to analytical and numerical solutions presented by other authors. The method is shown to be effective in calculating approximate solutions for various differential equations and provides a higher accuracy level than finite difference methods. This technique …

Vallée-Poussin Theorem For Equations With Caputo Fractional Derivative, Martin Bohner, Alexander Domoshnitsky, Seshadev Padhi, Satyam Narayan Srivastava

Mathematics and Statistics Faculty Research & Creative Works

In this paper, the functional differential equation (^CD_a^α+x)(t) + ^mΣ_i=0 (T_ix(ⁱ))(t) = f(t); t 2 [a; b]; with Caputo fractional derivative ^CD_a^α+ is studied. The operators Ti act from the space of continuous to the space of essentially bounded functions. They can be operators with deviations (delayed and advanced), integral operators and their various linear combinations and superpositions. Such equations could appear in various applications and in the study of systems of, for example, two fractional differential equations, when one of the components can be …

Nonoscillatory Solutions Of Higher-Order Fractional Differential Equations, Martin Bohner, Said R. Grace, Irena Jadlovská, Nurten Kılıç

Mathematics and Statistics Faculty Research & Creative Works

This paper deals with the asymptotic behavior of the nonoscillatory solutions of a certain forced fractional differential equations with positive and negative terms, involving the Caputo fractional derivative. The results obtained are new and generalize some known results appearing in the literature. Two examples are also provided to illustrate the results.

A Weak Fractional Calculus Theory And Numerical Methods For Fractional Differential Equations, Mitchell D. Sutton

Doctoral Dissertations

This dissertation is comprised of four integral parts. The first part comprises a self-contained new theory of weak fractional differential calculus in one-dimension. The crux of this new theory is the introduction of a weak fractional derivative notion which is a natural generalization of integer order weak derivatives; it also helps to unify multiple existing fractional derivative definitions.

The second part of this work presents three new families of fractional Sobolev spaces and their accompanying theory in one-dimension. The new construction and theory are based on a newly developed notion of weak fractional derivatives, which are natural generalizations of the …

Existence And Transportation Inequalities For Fractional Stochastic Differential Equations, Abdelghani Ouahab, Mustapha Belabbas, Johnny Henderson, Fethi Souna

Turkish Journal of Mathematics

In this work, we establish the existence and uniqueness of solutions for a fractional stochastic differential equation driven by countably many Brownian motions on bounded and unbounded intervals. Also, we study the continuous dependence of solutions on initial data. Finally, we establish the transportation quadratic cost inequality for some classes of fractional stochastic equations and continuous dependence of solutions with respect Wasserstein distance.

Existence Results And Ulam-Hyers Stability To Impulsive Coupled System Fractional Differential Equations, Hadjer Belbali, Maamar Benbachir

Turkish Journal of Mathematics

In this paper, the existence and uniqueness of the solutions to impulsive coupled system of fractional differential equations with Caputo--Hadamard are investigated. Furthermore, Ulam's type stability of the proposed coupled system is studied. The approach is based on a Perov type fixed point theorem for contractions.

Solving Fractional Differential Equations Using Collocation Method Based On Hybrid Of Block-Pulse Functions And Taylor Polynomials, Yao Lu, Yinggan Tang

Turkish Journal of Mathematics

In this paper, a novel approach is proposed to solve fractional differential equations (FDEs) based on hybrid functions. The hybrid functions consist of block-pulse functions and Taylor polynomials. The exact formula for the Riemann--Liouville fractional integral of the hybrid functions is derived via Laplace transform. The FDE under consideration is converted into an algebraic equation with undetermined coefficients by using this formula. A set of linear or nonlinear equations are obtained through collocating the algebraic equation at Newton-Cotes nodes. The numerical solution of the FDE is achieved by solving the linear or nonlinear equations. Error analysis is performed on the …

Existence Results For A Class Of Boundary Value Problems For Fractional Differential Equations, Abdülkadi̇r Doğan

Turkish Journal of Mathematics

By application of some fixed point theorems, that is, the Banach fixed point theorem, Schaefer's and the Leray-Schauder fixed point theorem, we establish new existence results of solutions to boundary value problems of fractional differential equations. This paper is motivated by Agarwal et al. (Georgian Math. J. 16 (2009) No.3, 401-411).

Existence And Stability Results Of Nonlinear Fractional Differential Equations With Nonlinear Integral Boundary Condition On Time Scales, Vipin Kumar, Muslim Malik

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we establish the existence and uniqueness of the solution to a nonlinear fractional differential equation with nonlinear integral boundary conditions on time scales.We used the fixed point theorems due to Banach, Schaefer’s, nonlinear alternative of Leray Schauder’s type and Krasnoselskii’s to establish these results. In addition, we study Ulam-Hyer’s (UH) type stability result. At the end, we present two examples to show the effectiveness of the obtained analytical results.

A New Method To Solve Fractional Differential Equations: Inverse Fractional Shehu Transform Method, Ali Khalouta, Abdelouahab Kadem

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we propose a new method called the inverse fractional Shehu transform method to solve homogenous and non-homogenous linear fractional differential equations. Fractional derivatives are described in the sense of Riemann-Liouville and Caputo. Illustrative examples are given to demonstrate the validity, efficiency and applicability of the presented method. The solutions obtained by the proposed method are in complete agreement with the solutions available in the literature.

Converse Theorems In Lyapunov's Second Method And Applications For Fractional Order Systems, Javier Gallegos, Manuel Duarte-Mermoud

Turkish Journal of Mathematics

We establish a characterization of the Lyapunov and Mittag-Leffler stability through (fractional) Lyapunov functions, by proving converse theorems for Caputo fractional order systems. A hierarchy for the Mittag-Leffler order convergence is also proved which shows, in particular, that fractional differential equation with derivation order lesser than one cannot be exponentially stable. The converse results are then applied to show that if an integer order system is (exponentially) stable, then its corresponding fractional system, obtained from changing its differentiation order, is (Mittag-Leffler) stable. Hence, available integer order control techniques can be disposed to control nonlinear fractional systems. Finally, we provide examples …

On Oscillatory And Nonoscillatory Behavior Of Solutions For A Class Of Fractional Orderdifferential Equations, Arjumand Seemab, Mujeeb Ur Rehman

Turkish Journal of Mathematics

This work aims to develop oscillation criterion and asymptotic behavior of solutions for a class of fractional order differential equation: $D^{\alpha}_{0}u(t)+\lambda u(t)=f(t,u(t)),~~t> 0,$ $D^{\alpha-1}_{0}u(t) _{t=0}=u_{0},~~\lim_{t\to 0}J^{2-\alpha}_{0}u(t)=u_{1}$ where $D^{\alpha}_{0}$ denotes the Riemann--Liouville differential operator of order $\alpha$ with $1

Existence Of Solution For Some Two-Point Boundary Value Fractional Differential Equations, Kenneth Ifeanyi Isife

Turkish Journal of Mathematics

Using a fixed point theorem, we establish the existence of a solution for a class of boundary value fractional differential equation. Secondly, we will adopt the method of successive approximations to obtain an approximate solution to our problem. Furthermore, using the Laplace transform technique, an explicit solution to a particular case of our problem is obtained. Finally, some examples are given to illustrate our results.

A New Hybrid Method For Solving Nonlinear Fractional Differential Equations, R. Delpasand, M. M. Hosseini, F. M. Maalek Ghaini

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, numerical solution of initial and boundary value problems for nonlinear fractional differential equations is considered by pseudospectral method. In order to avoid solving systems of nonlinear equations resulting from the method, the residual function of the problem is constructed, as well as a suggested unconstrained optimization model solved by PSOGSA algorithm. Furthermore, the research inspects and discusses the spectral accuracy of Chebyshev polynomials in the approximation theory. The following scheme is tested for a number of prominent examples, and the obtained results demonstrate the accuracy and efficiency of the proposed method.

A New Analytic Numeric Method Solution For Fractional Modified Epidemiological Model For Computer Viruses, Ali H. Handam, Asad A. Freihat

Applications and Applied Mathematics: An International Journal (AAM)

Computer viruses are an extremely important aspect of computer security, and understanding their spread and extent is an important component of any defensive strategy. Epidemiological models have been proposed to deal with this issue, and we present one such here. We consider the modified epidemiological model for computer viruses (SAIR) proposed by J. R. C. Piqueira and V. O. Araujo. This model includes an antidotal population compartment (A) representing nodes of the network equipped with fully effective anti-virus programs. The multi-step generalized differential transform method (MSGDTM) is employed to compute an approximation to the solution of the model of fractional …

The Shifted Jacobi Polynomial Integral Operational Matrix For Solving Riccati Differential Equation Of Fractional Order, A. Neamaty, B. Agheli, R. Darzi

Applications and Applied Mathematics: An International Journal (AAM)

In this article, we have applied Jacobi polynomial to solve Riccati differential equation of fractional order. To do so, we have presented a general formula for the Jacobi operational matrix of fractional integral operator. Using the Tau method, the solution of this problem reduces to the solution of a system of algebraic equations. The numerical results for the examples presented in this paper demonstrate the efficiency of the present method.

Series Solutions Of Multi-Term Fractional Differential Equations, Yousef Ibrahim Al-Srihin

Theses

In this thesis, we introduce a new series solutions for multi-term fractional differential equations of Caputo’s type. The idea is similar to the well-known Taylor Series method, but we overcome the difficulty of computing iterated fractional derivatives, which do not commuted in general. To illustrate the efficiency of the new algorithm, we apply it for several types of multi-term fractional differential equations and compare the results with the ones obtained by the well-known Adomian decomposition method (ADM).

Existence And Uniqueness Of Solutions For A Class Of Non-Linear Boundary Value Problems Of Fractional Order, Arwa Abdulla Omar Salem Ba Abdulla

Theses

In this thesis, we extend the maximum principle and the method of upper and lower solutions to study a class of nonlinear fractional boundary value problems with the Caputo fractional derivative 1

Solution Of The Sir Models Of Epidemics Using Msgdtm, Asad A. Freihat, Ali H. Handam

Applications and Applied Mathematics: An International Journal (AAM)

Stochastic compartmental (e.g., SIR) models have proven useful for studying the epidemics of childhood diseases while taking into account the variability of the epidemic dynamics. Here, we use the multi-step generalized differential transform method (MSGDTM) to approximate the numerical solution of the SIR model and numerical simulations are presented graphically.

Fractional Calculus For Nanoscale Flow And Heat Transfer, Hong-Yan Liu, Ji-Huan He, Zheng-Biao Li

Ji-Huan He

Purpose – Academic and industrial researches on nanoscale flows and heat transfers are an area of increasing global interest, where fascinating phenomena are always observed, e.g. admirable water or air permeation and remarkable thermal conductivity. The purpose of this paper is to reveal the phenomena by the fractional calculus. Design/methodology/approach – This paper begins with the continuum assumption in conventional theories, and then the fractional Gauss’ divergence theorems are used to derive fractional differential equations in fractal media. Fractional derivatives are introduced heuristically by the variational iteration method, and fractal derivatives are explained geometrically. Some effective analytical approaches to fractional …

Fractal Approach To Heat Transfer In Silkworm Cocoon Hierarchy, Dong-Dong Fei, Fu-Juan Liu, Qiu-Na Cui, Ji-Huan He

Ji-Huan He

Silkworm cocoon has a complex hierarchic structure with discontinuity. In this paper, heat transfer through the silkworm cocoon is studied using fractal theory. The fractal approach has been successfully applied to explain the fascinating phenomenon of cocoon survival under extreme temperature environment. A better understanding of heat transfer mechanisms for the cocoon could be beneficial to the design of biomimetic clothes for special applications.

Local Fractional Variational Iteration Method For Fractal Heat Transfer In Silk Cocoon Hierarchy, Ji-Huan He

Ji-Huan He

A local fractional equation is established for fractal heat transfer in silk cocoon hierarchy, and the local fractional variational iteration method is adopted to solve the equation analytically. The result can well explain the intriguing phenomenon for pupa's survival at extremes of weather from negative 40 degrees to 50 degrees.

Converting Fractional Differential Equations Into Partial Differential Equations, Ji-Huan He, Zheng-Biao Li

Ji-Huan He

A transform is suggested in this paper to convert fractional differential equations with the modified Riemann-Liouville derivative into partial differential equations, and it is concluded that the fractional order in fractional differential equations is equivalent to the fractal dimension.

Application Of The Fractional Complex Transform To Fractional Differential Equations, Zheng-Biao Li, Ji-Huan He

Ji-Huan He

The fractional complex transform is used to analytically deal with fractional differential equations. Two examples are given to elucidate the solution procedure, showing it is extremely accessible to nonmathematicians

A Short Remark On Fractional Variational Iteration Method, Ji-Huan He

Ji-Huan He

This Letter compares the classical variational iteration method with the fractional variational iteration method. The fractional complex transform is introduced to convert a fractional differential equation to its differential partner, so that its variational iteration algorithm can be simply constructed

A New Fractal Derivation, Ji-Huan He

Ji-Huan He

A new fractal derive is defined, which is very easy for engineering applications to discontinuous problems, two simple examples are given to elucidate to establish governing equations with fractal derive and how to solve such equations, respectively.