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Journal articles on the topic 'Fractional-order ordinary differential equations'

Author: Grafiati

Published: 28 June 2021

Last updated: 7 February 2022

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1

Zhao, Ting Gang, Zi Lang Zhan, Jin Xia Huo, and Zi Guang Yang. "Legendre Collocation Solution to Fractional Ordinary Differential Equations." Applied Mechanics and Materials 687-691 (November 2014): 601–5. http://dx.doi.org/10.4028/www.scientific.net/amm.687-691.601.

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In this paper, we propose an efficient numerical method for ordinary differential equation with fractional order, based on Legendre-Gauss-Radau interpolation, which is easy to be implemented and possesses the spectral accuracy. We apply the proposed method to multi-order fractional ordinary differential equation. Numerical results demonstrate the effectiveness of the approach.

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2

Campos, L. M. B. C. "On the solution of some simple fractional differential equations." International Journal of Mathematics and Mathematical Sciences 13, no. 3 (1990): 481–96. http://dx.doi.org/10.1155/s0161171290000709.

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The differintegration or fractional derivative of complex orderν, is a generalization of the ordinary concept of derivative of ordern, from positive integerν=nto complex values ofν, including also, forν=−na negative integer, the ordinaryn-th primitive. Substituting, in an ordinary differential equation, derivatives of integer order by derivatives of non-integer order, leads to a fractional differential equation, which is generallyaintegro-differential equation. We present simple methods of solution of some classes of fractional differential equations, namely those with constant coefficients (standard I) and those with power type coefficients with exponents equal to the orders of differintegration (standard II). The fractional differential equations of standard I (II), both homogeneous, and inhomogeneous with exponential (power-type) forcing, can be solved in the Liouville (Riemann) systems of differintegration. The standard I (II) is linear with constant (non-constant) coefficients, and some results are also given for a class of non-linear fractional differential equations (standard III).

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3

Singh, Karanveer, and R. N. Prajapati. "Fractional differential equation with uncertainty." Journal of University of Shanghai for Science and Technology 23, no. 08 (August 7, 2021): 181–85. http://dx.doi.org/10.51201/jusst/21/08364.

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We consider a fractional order differential equation with uncertainty and introduce the concept of solution. It goes beyond ordinary first-order differential equations and differential equations with uncertainty.

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4

Vondra, Alexandr. "Geometry of second-order connections and ordinary differential equations." Mathematica Bohemica 120, no. 2 (1995): 145–67. http://dx.doi.org/10.21136/mb.1995.126226.

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5

Afuwape, Anthony Uyi, and M. O. Omeike. "Ultimate boundedness of some third order ordinary differential equations." Mathematica Bohemica 137, no. 3 (2012): 355–64. http://dx.doi.org/10.21136/mb.2012.142900.

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6

WU, CONG. "A GENERAL COMPARISON PRINCIPLE FOR CAPUTO FRACTIONAL-ORDER ORDINARY DIFFERENTIAL EQUATIONS." Fractals 28, no. 04 (June 2020): 2050070. http://dx.doi.org/10.1142/s0218348x2050070x.

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In this paper, we work on a general comparison principle for Caputo fractional-order ordinary differential equations. A full result on maximal solutions to Caputo fractional-order systems is given by using continuation of solutions and a newly proven formula of Caputo fractional derivatives. Based on this result and the formula, we prove a general fractional comparison principle under very weak conditions, in which only the Caputo fractional derivative is involved. This work makes up deficiencies of existing results.

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7

Huang, X., and X. Lu. "The Use of Fractional B-Splines Wavelets in Multiterms Fractional Ordinary Differential Equations." International Journal of Differential Equations 2010 (2010): 1–13. http://dx.doi.org/10.1155/2010/968186.

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We discuss the existence and uniqueness of the solutions of the nonhomogeneous linear differential equations of arbitrary positive real order by using the fractional B-Splines wavelets and the Mittag-Leffler function. The differential operators are taken in the Riemann-Liouville sense and the initial values are zeros. The scheme of solving the fractional differential equations and the explicit expression of the solution is given in this paper. At last, we show the asymptotic solution of the differential equations of fractional order and corresponding truncated error in theory.

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8

Gusu, Daba Meshesha, Dechasa Wegi, Girma Gemechu, and Diriba Gemechu. "Fractional Order Airy’s Type Differential Equations of Its Models Using RDTM." Mathematical Problems in Engineering 2021 (September 10, 2021): 1–21. http://dx.doi.org/10.1155/2021/3719206.

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In this paper, we propose a novel reduced differential transform method (RDTM) to compute analytical and semianalytical approximate solutions of fractional order Airy’s ordinary differential equations and fractional order Airy’s and Airy’s type partial differential equations subjected to certain initial conditions. The performance of the proposed method was analyzed and compared with a convergent series solution form with easily computable coefficients. The behavior of approximated series solutions at different values of fractional order α and its modeling in 2-dimensional and 3-dimensional spaces are compared with exact solutions using MATLAB graphical method analysis. Moreover, the physical and geometrical interpretations of the computed graphs are given in detail within 2- and 3-dimensional spaces. Accordingly, the obtained approximate solutions of fractional order Airy’s ordinary differential equations and fractional order Airy’s and Airy’s type partial differential equations subjected to certain initial conditions exactly fit with exact solutions. Hence, the proposed method reveals reliability, effectiveness, efficiency, and strengthening of computed mathematical results in order to easily solve fractional order Airy’s type differential equations.

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9

Khan, Hassan, Shoaib Barak, Poom Kumam, and Muhammad Arif. "Analytical Solutions of Fractional Klein-Gordon and Gas Dynamics Equations, via the (G′/G)-Expansion Method." Symmetry 11, no. 4 (April 19, 2019): 566. http://dx.doi.org/10.3390/sym11040566.

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In this article, the ( G ′ / G ) -expansion method is used for the analytical solutions of fractional-order Klein-Gordon and Gas Dynamics equations. The fractional derivatives are defined in the term of Jumarie’s operator. The proposed method is based on certain variable transformation, which transforms the given problems into ordinary differential equations. The solution of resultant ordinary differential equation can be expressed by a polynomial in ( G ′ / G ) , where G = G ( ξ ) satisfies a second order linear ordinary differential equation. In this paper, ( G ′ / G ) -expansion method will represent, the travelling wave solutions of fractional-order Klein-Gordon and Gas Dynamics equations in the term of trigonometric, hyperbolic and rational functions.

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10

Cevikel, Adem. "New exact solutions of the space-time fractional KdV-burgers and nonlinear fractional foam drainage equation." Thermal Science 22, Suppl. 1 (2018): 15–24. http://dx.doi.org/10.2298/tsci170615267c.

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The fractional differential equations have been studied by many authors and some effective methods for fractional calculus were appeared in literature, such as the fractional sub-equation method and the first integral method. The fractional complex transform approach is to convert the fractional differential equations into ordinary differential equations, making the solution procedure simple. Recently, the fractional complex transform has been suggested to convert fractional order differential equations with modified Riemann-Liouville derivatives into integer order differential equations, and the reduced equations can be solved by symbolic computation. The present paper investigates for the applicability and efficiency of the exp-function method on some fractional non-linear differential equations.

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11

Alam, Md Nur, Aly R. Seadawy, and Dumitru Baleanu. "Closed-form wave structures of the space-time fractional Hirota–Satsuma coupled KdV equation with nonlinear physical phenomena." Open Physics 18, no. 1 (September 24, 2020): 555–65. http://dx.doi.org/10.1515/phys-2020-0179.

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AbstractThe present paper applies the variation of (G^{\prime} /G)-expansion method on the space-time fractional Hirota–Satsuma coupled KdV equation with applications in physics. We employ the new approach to receive some closed form wave solutions for any nonlinear fractional ordinary differential equations. First, the fractional derivatives in this research are manifested in terms of Riemann–Liouville derivative. A complex fractional transformation is applied to transform the fractional-order ordinary and partial differential equation into the integer order ordinary differential equation. The reduced equations are then solved by the method. Some novel and more comprehensive solutions of these equations are successfully constructed. Besides, the intended approach is simplistic, conventional, and able to significantly reduce the size of computational work associated with other existing methods.

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12

Tóthová, Mária, and Oleg Palumbíny. "On monotone solutions of the fourth order ordinary differential equations." Czechoslovak Mathematical Journal 45, no. 4 (1995): 737–46. http://dx.doi.org/10.21136/cmj.1995.128553.

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13

Yilmazer, Resat, and Okkes Ozturk. "Explicit Solutions of Singular Differential Equation by Means of Fractional Calculus Operators." Abstract and Applied Analysis 2013 (2013): 1–6. http://dx.doi.org/10.1155/2013/715258.

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Recently, several authors demonstrated the usefulness of fractional calculus operators in the derivation of particular solutions of a considerably large number of linear ordinary and partial differential equations of the second and higher orders. By means of fractional calculus techniques, we find explicit solutions of second-order linear ordinary differential equations.

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14

Zheng, Bin. "Exact Solutions for Some Fractional Partial Differential Equations by the Method." Mathematical Problems in Engineering 2013 (2013): 1–13. http://dx.doi.org/10.1155/2013/826369.

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We apply the method to seek exact solutions for several fractional partial differential equations including the space-time fractional (2 + 1)-dimensional dispersive long wave equations, the (2 + 1)-dimensional space-time fractional Nizhnik-Novikov-Veselov system, and the time fractional fifth-order Sawada-Kotera equation. The fractional derivative is defined in the sense of modified Riemann-liouville derivative. Based on a certain variable transformation, these fractional partial differential equations are transformed into ordinary differential equations of integer order. With the aid of mathematical software, a variety of exact solutions for them are obtained.

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15

Tuluce Demiray, Seyma, Yusuf Pandir, and Hasan Bulut. "Generalized Kudryashov Method for Time-Fractional Differential Equations." Abstract and Applied Analysis 2014 (2014): 1–13. http://dx.doi.org/10.1155/2014/901540.

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In this study, the generalized Kudryashov method (GKM) is handled to find exact solutions of time-fractional Burgers equation, time-fractional Cahn-Hilliard equation, and time-fractional generalized third-order KdV equation. These time-fractional equations can be turned into another nonlinear ordinary differantial equation by travelling wave transformation. Then, GKM has been implemented to attain exact solutions of time-fractional Burgers equation, time-fractional Cahn-Hilliard equation, and time-fractional generalized third-order KdV equation. Also, some new hyperbolic function solutions have been obtained by using this method. It can be said that this method is a generalized form of the classical Kudryashov method.

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16

Wang, JinRong, and Xuezhu Li. "E α -Ulam type stability of fractional order ordinary differential equations." Journal of Applied Mathematics and Computing 45, no. 1-2 (October 12, 2013): 449–59. http://dx.doi.org/10.1007/s12190-013-0731-8.

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17

Umarov, Sabir, Ravshan Ashurov, and YangQuan Chen. "On a method of solution of systems of fractional pseudo-differential equations." Fractional Calculus and Applied Analysis 24, no. 1 (January 29, 2021): 254–77. http://dx.doi.org/10.1515/fca-2021-0011.

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Abstract This paper is devoted to the general theory of linear systems of fractional order pseudo-differential equations. Single fractional order differential and pseudo-differential equations are studied by many authors and several monographs and handbooks have been published devoted to its theory and applications. However, the state of systems of fractional order ordinary and partial or pseudo-differential equations is still far from completeness, even in the linear case. In this paper we develop a new method of solution of general systems of fractional order linear pseudo-differential equations and prove existence and uniqueness theorems in the special classes of distributions, as well as in the Sobolev spaces.

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18

Ludu, Andrei. "Nonlocal Symmetries for Time-Dependent Order Differential Equations." Symmetry 10, no. 12 (December 19, 2018): 771. http://dx.doi.org/10.3390/sym10120771.

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A new type of ordinary differential equation is introduced and discussed: time-dependent order ordinary differential equations. These equations are solved via fractional calculus by transforming them into Volterra integral equations of second kind with singular integrable kernel. The solutions of the time-dependent order differential equation represent deformations of the solutions of the classical (integer order) differential equations, mapping them into one-another as limiting cases. This equation can also move, remove or generate singularities without involving variable coefficients. An interesting symmetry of the solution in relation to the Riemann zeta function and Harmonic numbers is observed.

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19

Zhang, Hai, Jinde Cao, and Wei Jiang. "General Solution of Linear Fractional Neutral Differential Difference Equations." Discrete Dynamics in Nature and Society 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/489521.

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This paper is concerned with the general solution of linear fractional neutral differential difference equations. The exponential estimates of the solution and the variation of constant formula for linear fractional neutral differential difference equations are derived by using the Gronwall integral inequality and the Laplace transform method, respectively. The obtained results extend the corresponding ones of integer order linear ordinary differential equations and delay differential equations.

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20

Zhou, Xiaojun, and Chuanju Xu. "Numerical Solution of the Coupled System of Nonlinear Fractional Ordinary Differential Equations." Advances in Applied Mathematics and Mechanics 9, no. 3 (January 17, 2017): 574–95. http://dx.doi.org/10.4208/aamm.2015.m1054.

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AbstractIn this paper, we consider the numerical method that is proposed and analyzed in [J. Cao and C. Xu, J. Comput. Phys., 238 (2013), pp. 154–168] for the fractional ordinary differential equations. It is based on the so-called block-by-block approach, which is a common method for the integral equations. We extend the technique to solve the nonlinear system of fractional ordinary differential equations (FODEs) and present a general technique to construct high order schemes for the numerical solution of the nonlinear coupled system of fractional ordinary differential equations (FODEs). By using the present method, we are able to construct a high order schema for nonlinear system of FODEs of the orderα,α>0. The stability and convergence of the schema is rigorously established. Under the smoothness assumptionf,g∈C4[0,T], we prove that the numerical solution converges to the exact solution with order 3+αfor 0<α≤1 and order 4 forα>1. Some numerical examples are provided to confirm the theoretical claims.

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21

Kloeden, Peter E., and Arnulf Jentzen. "Pathwise convergent higher order numerical schemes for random ordinary differential equations." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 463, no. 2087 (August 21, 2007): 2929–44. http://dx.doi.org/10.1098/rspa.2007.0055.

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Random ordinary differential equations (RODEs) are ordinary differential equations (ODEs) with a stochastic process in their vector field. They can be analysed pathwise using deterministic calculus, but since the driving stochastic process is usually only Hölder continuous in time, the vector field is not differentiable in the time variable, so traditional numerical schemes for ODEs do not achieve their usual order of convergence when applied to RODEs. Nevertheless deterministic calculus can still be used to derive higher order numerical schemes for RODEs via integral versions of implicit Taylor-like expansions. The theory is developed systematically here and applied to illustrative examples involving Brownian motion and fractional Brownian motion as the driving processes.

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22

Duan, Jun-Sheng, Zhong Wang, Yu-Lu Liu, and Xiang Qiu. "Eigenvalue problems for fractional ordinary differential equations." Chaos, Solitons & Fractals 46 (January 2013): 46–53. http://dx.doi.org/10.1016/j.chaos.2012.11.004.

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23

Tuluce Demiray, Seyma, Hasan Bulut, and Fethi Bin Muhammad Belgacem. "Sumudu Transform Method for Analytical Solutions of Fractional Type Ordinary Differential Equations." Mathematical Problems in Engineering 2015 (2015): 1–6. http://dx.doi.org/10.1155/2015/131690.

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We make use of the so-called Sumudu transform method (STM), a type of ordinary differential equations with both integer and noninteger order derivative. Firstly, we give the properties of STM, and then we directly apply it to fractional type ordinary differential equations, both homogeneous and inhomogeneous ones. We obtain exact solutions of fractional type ordinary differential equations, both homogeneous and inhomogeneous, by using STM. We present some numerical simulations of the obtained solutions and exhibit two-dimensional graphics by means of Mathematica tools. The method used here is highly efficient, powerful, and confidential tool in terms of finding exact solutions.

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24

Zhang, Xiangzhi, and Yufeng Zhang. "Some Similarity Solutions and Numerical Solutions to the Time-Fractional Burgers System." Symmetry 11, no. 1 (January 18, 2019): 112. http://dx.doi.org/10.3390/sym11010112.

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In the paper, we discuss some similarity solutions of the time-fractional Burgers system (TFBS). Firstly, with the help of the Lie-point symmetry and the corresponding invariant variables, we transform the TFBS to a fractional ordinary differential system (FODS) under the case where the time-fractional derivative is the Riemann–Liouville type. The FODS can be approximated by some integer-order ordinary differential equations; here, we present three such integer-order ordinary differential equations (called IODE-1, IODE-2, and IODE-3, respectively). For IODE-1, we obtain its similarity solutions and numerical solutions, which approximate the similarity solutions and the numerical solutions of the TFBS. Secondly, we apply the numerical analysis method to obtain the numerical solutions of IODE-2 and IODE-3.

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25

Kassim, Mohammed D., Khaled M. Furati, and Nasser-Eddine Tatar. "ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS." Mathematical Modelling and Analysis 21, no. 5 (September 20, 2016): 610–29. http://dx.doi.org/10.3846/13926292.2016.1198279.

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It is known that, under certain conditions, solutions of some ordinary differential equations of first, second or even higher order are asymptotic to polynomials as time goes to infinity. We generalize and extend some of the existing results to differential equations of non-integer order. Reasonable conditions and appropriate underlying spaces are determined ensuring that solutions of fractional differential equations with nonlinear right hand sides approach power type functions as time goes to infinity. The case of fractional differential problems with fractional damping is also considered. Our results are obtained by using generalized versions of GronwallBellman inequality and appropriate desingularization techniques.

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26

Kaplan, Melike, Murat Koparan, and Ahmet Bekir. "Regarding on the exact solutions for the nonlinear fractional differential equations." Open Physics 14, no. 1 (January 1, 2016): 478–82. http://dx.doi.org/10.1515/phys-2016-0056.

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AbstractIn this work, we have considered the modified simple equation (MSE) method for obtaining exact solutions of nonlinear fractional-order differential equations. The space-time fractional equal width (EW) and the modified equal width (mEW) equation are considered for illustrating the effectiveness of the algorithm. It has been observed that all exact solutions obtained in this paper verify the nonlinear ordinary differential equations which was obtained from nonlinear fractional-order differential equations under the terms of wave transformation relationship. The obtained results are shown graphically.

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27

Xu, Fei. "A Class of Integer Order and Fractional Order Hyperchaotic Systems via the Chen System." International Journal of Bifurcation and Chaos 26, no. 06 (June 15, 2016): 1650109. http://dx.doi.org/10.1142/s0218127416501091.

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In this article, we investigate the generation of a class of hyperchaotic systems via the Chen chaotic system using both integer order and fractional order differential equation systems. Based on the Chen chaotic system, we designed a system with four nonlinear ordinary differential equations. For different parameter sets, the trajectory of the system may diverge or display a hyperchaotic attractor with double wings. By linearizing the ordinary differential equation system with divergent trajectory and designing proper switching controls, we obtain a chaotic attractor. Similar phenomenon has also been observed in linearizing the hyperchaotic system. The corresponding fractional order systems are also considered. Our investigation indicates that, switching control can be applied to either linearized chaotic or nonchaotic differential equation systems to create chaotic attractor.

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28

N. Hasan, Nabaa, and Fadhel S.Fadhel. "Variational Iteration Method for Solving System of Fractional Order Ordinary Differential Equations." IOSR Journal of Mathematics 10, no. 6 (2014): 48–54. http://dx.doi.org/10.9790/5728-10624854.

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29

Cao, Junying, Lizhen Chen, and Ziqiang Wang. "A high-order numerical scheme for the impulsive fractional ordinary differential equations." International Journal of Computer Mathematics 95, no. 12 (November 20, 2017): 2433–57. http://dx.doi.org/10.1080/00207160.2017.1398322.

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30

Reutskiy, Sergiy, and Zhuo-Jia Fu. "A semi-analytic method for fractional-order ordinary differential equations: Testing results." Fractional Calculus and Applied Analysis 21, no. 6 (December 19, 2018): 1598–618. http://dx.doi.org/10.1515/fca-2018-0084.

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Abstract The paper presents the testing results of a semi-analytic collocation method, using five benchmark problems published in a paper by Xue and Bai in Fract. Calc. Appl. Anal., Vol. 20, No 5 (2017), pp. 1305–1312, DOI: 10.1515/fca-2017-0068.

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31

Atabakzadeh, M. H., M. H. Akrami, and G. H. Erjaee. "Chebyshev operational matrix method for solving multi-order fractional ordinary differential equations." Applied Mathematical Modelling 37, no. 20-21 (November 2013): 8903–11. http://dx.doi.org/10.1016/j.apm.2013.04.019.

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32

Javeed, Shumaila, Dumitru Baleanu, Asif Waheed, Mansoor Shaukat Khan, and Hira Affan. "Analysis of Homotopy Perturbation Method for Solving Fractional Order Differential Equations." Mathematics 7, no. 1 (January 3, 2019): 40. http://dx.doi.org/10.3390/math7010040.

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The analysis of Homotopy Perturbation Method (HPM) for the solution of fractional partial differential equations (FPDEs) is presented. A unified convergence theorem is given. In order to validate the theory, the solution of fractional-order Burger-Poisson (FBP) equation is obtained. Furthermore, this work presents the method to find the solution of FPDEs, while the same partial differential equation (PDE) with ordinary derivative i.e., for α = 1 , is not defined in the given domain. Moreover, HPM is applied to a complicated obstacle boundary value problem (BVP) of fractional order.

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33

Matuszak, Zenon. "Fractional Bloch's Equations Approach to Magnetic Relaxation." Current Topics in Biophysics 37, no. 1 (February 2, 2015): 9–22. http://dx.doi.org/10.2478/ctb-2014-0069.

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It is the goal of this paper to present general strategy for using fractional operators to model the magnetic relaxation in complex environments revealing time and spacial disorder. Such systems have anomalous temporal and spacial response (non-local interactions and long memory) compared to systems without disorder. The systems having no memory can be modeled by linear differential equations with constant coefficients (exponential relaxation); the differential equations governing the systems with memory are known as Fractional Order Differential Equations (FODE). The relaxation of the spin system is best described phenomenologically by so-called Bloch's equations, which detail the rate of change of the magnetization M of the spin system. The Ordinary Order Bloch's Equations (OOBE) are a set of macroscopic differential equations of the first order describing the magnetization behavior under influence of static, varying magnetic fields and relaxation. It is assumed that spins relax along the z axis and in the x-y plane at different rates, designated as R1 and R2 (R1=1/T1,R2=1/T2) respectively, but following first order kinetics. To consider heterogeneity, complex structure, and memory effects in the relaxation process the Ordinary Order Bloch's Equations were generalized to Fractional Order Bloch's Equations (FOBE) through extension of the time derivative to fractional (non-integer) order. To investigate systematically the influence of “fractionality” (power order of derivative) on the dynamics of the spin system a general approach was proposed. The OOBE and FOBE were successively solved using analytical (Laplace transform), semi-analytical (ADM - Adomian Decomposition Method) and numerical methods (Grunwald- Letnikov method for FOBE). Solutions of both OOBE and FOBE systems of equations were obtained for various sets of experimental parameters used in spin !! NMR and EPR spectroscopies. The physical meaning of the fractional relaxation in magnetic resonance is shortly discussed.

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34

Zhang, Xumei, and Junying Cao. "A high order numerical method for solving Caputo nonlinear fractional ordinary differential equations." AIMS Mathematics 6, no. 12 (2021): 13187–209. http://dx.doi.org/10.3934/math.2021762.

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<abstract><p>In this paper, we construct a high order numerical scheme for Caputo nonlinear fractional ordinary differential equations. Firstly, we use the piecewise Quadratic Lagrange interpolation method to construct a high order numerical scheme for Caputo nonlinear fractional ordinary differential equations, and then analyze the local truncation error of the high order numerical scheme. Secondly, based on the local truncation error, the convergence order of $ 3-\theta $ order is obtained. And the convergence are strictly analyzed. Finally, the numerical simulation of the high order numerical scheme is carried out. Through the calculation of typical problems, the effectiveness of the numerical algorithm and the correctness of theoretical analysis are verified.</p></abstract>

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35

Kossowski, Marek, and Gerard Thompson. "Submersive second order ordinary differential equations." Mathematical Proceedings of the Cambridge Philosophical Society 110, no. 1 (July 1991): 207–24. http://dx.doi.org/10.1017/s0305004100070262.

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The objectives of this paper are to define and to characterize submersive second order ordinary differential equations (ODE) and to examine several situations in which such ODE occur. This definition and characterization is in terms of tangent bundle geometry as developed in [4, 6, 7, 10, 11, 14]. From this viewpoint second order ODE are identified with a special vector field on the tangent bundle. The ODE are said to be submersive when this vector field and the canonical vertical endomorphism [14] define a foliation, relative to which the vector field passes to the local quotient.

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36

Swinnerton-Dyer, Peter, and Thomas Wagenknecht. "Some third-order ordinary differential equations." Bulletin of the London Mathematical Society 40, no. 5 (May 29, 2008): 725–48. http://dx.doi.org/10.1112/blms/bdn046.

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37

Saeed, Umer. "Sine–cosine wavelets operational matrix method for fractional nonlinear differential equation." International Journal of Wavelets, Multiresolution and Information Processing 17, no. 04 (July 2019): 1950026. http://dx.doi.org/10.1142/s0219691319500267.

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In this paper, we present a solution method for fractional nonlinear ordinary differential equations. We propose a method by utilizing the sine–cosine wavelets (SCWs) in conjunction with quasilinearization technique. The fractional nonlinear differential equations are transformed into a system of discrete fractional differential equations by quasilinearization technique. The operational matrices of fractional order integration for SCW are derived and utilized to transform the obtained discrete system into systems of algebraic equations and the solutions of algebraic systems lead to the solution of fractional nonlinear differential equations. Convergence analysis and procedure of implementation for the proposed method are also considered. To illustrate the reliability and accuracy of the method, we tested the method on fractional nonlinear Lane–Emden type equation and temperature distribution equation.

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38

Yan, Li-Mei, and Feng-Sheng Xu. "Generalized exp-function method for non-linear space-time fractional differential equations." Thermal Science 18, no. 5 (2014): 1573–76. http://dx.doi.org/10.2298/tsci1405573y.

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A generalized exp-function method is proposed to solve non-linear space-time fractional differential equations. The basic idea of the method is to convert a fractional partial differential equation into an ordinary equation with integer order derivatives by fractional complex transform. To illustrate the effectiveness of the method, space-time fractional asymmetrical Nizhnik-Novikor-Veselov equation is considered. The fractional derivatives in the present paper are in Jumarie?s modified Riemann-Liouville sense.

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39

Nakpim, Warisa. "Third-order ordinary differential equations equivalent to linear second-order ordinary differential equations via tangent transformations." Journal of Symbolic Computation 77 (November 2016): 63–77. http://dx.doi.org/10.1016/j.jsc.2016.01.006.

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40

Guner, Ozkan, Ahmet Bekir, and Halis Bilgil. "A note on exp-function method combined with complex transform method applied to fractional differential equations." Advances in Nonlinear Analysis 4, no. 3 (August 1, 2015): 201–8. http://dx.doi.org/10.1515/anona-2015-0019.

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AbstractIn this article, the fractional derivatives in the sense of modified Riemann–Liouville and the exp-function method are used to construct exact solutions for some nonlinear partial fractional differential equations via the nonlinear fractional Liouville equation and nonlinear fractional Zoomeron equation. These nonlinear fractional equations can be turned into another nonlinear ordinary differential equation by complex transform method. This method is efficient and powerful in solving wide classes of nonlinear fractional order equations. The exp-function method appears to be easier and more convenient by means of a symbolic computation system.

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Islam, Md Rafiul, Angela Peace, Daniel Medina, and Tamer Oraby. "Integer Versus Fractional Order SEIR Deterministic and Stochastic Models of Measles." International Journal of Environmental Research and Public Health 17, no. 6 (March 18, 2020): 2014. http://dx.doi.org/10.3390/ijerph17062014.

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In this paper, we compare the performance between systems of ordinary and (Caputo) fractional differential equations depicting the susceptible-exposed-infectious-recovered (SEIR) models of diseases. In order to understand the origins of both approaches as mean-field approximations of integer and fractional stochastic processes, we introduce the fractional differential equations (FDEs) as approximations of some type of fractional nonlinear birth and death processes. Then, we examine validity of the two approaches against empirical courses of epidemics; we fit both of them to case counts of three measles epidemics that occurred during the pre-vaccination era in three different locations. While ordinary differential equations (ODEs) are commonly used to model epidemics, FDEs are more flexible in fitting empirical data and theoretically offer improved model predictions. The question arises whether, in practice, the benefits of using FDEs over ODEs outweigh the added computational complexities. While important differences in transient dynamics were observed, the FDE only outperformed the ODE in one of out three data sets. In general, FDE modeling approaches may be worth it in situations with large refined data sets and good numerical algorithms.

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Aleroev, Temirkhan. "On One Problems of Spectral Theory for Ordinary Differential Equations of Fractional Order." Axioms 8, no. 4 (October 18, 2019): 117. http://dx.doi.org/10.3390/axioms8040117.

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The present paper is devoted to the spectral analysis of operators induced by fractional differential equations and boundary conditions of Sturm-Liouville type. It should be noted that these operators are non-self-adjoint. The spectral structure of such operators has been insufficiently explored. In particular, a study of the completeness of systems of eigenfunctions and associated functions has begun relatively recently. In this paper, the completeness of the system of eigenfunctions and associated functions of one class of non-self-adjoint integral operators corresponding boundary value problems for fractional differential equations is established. The proof is based on the well-known Theorem of M.S. Livshits on the spectral decomposition of linear non-self-adjoint operators, as well as on the sectoriality of the fractional differentiation operator. The results of Dzhrbashian-Nersesian on the asymptotics of the zeros of the Mittag-Leffler function are used.

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Zhang, Jingna, Haobo Gong, Sadia Arshad, and Jianfei Huang. "A superlinear numerical scheme for multi-term fractional nonlinear ordinary differential equations." International Journal of Modeling, Simulation, and Scientific Computing 11, no. 02 (April 2020): 2050015. http://dx.doi.org/10.1142/s1793962320500154.

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In this paper, we present a superlinear numerical method for multi-term fractional nonlinear ordinary differential equations (MTFNODEs). First, the presented problem is equivalently transformed into its integral form with multi-term Riemann–Liouville integrals. Second, the compound product trapezoidal rule is used to approximate the fractional integrals. Then, the unconditional stability and convergence with the order [Formula: see text] of the proposed scheme are strictly established, where [Formula: see text] and [Formula: see text] are the maximum and the second maximum fractional indexes in the considered MTFNODEs, respectively. Finally, two numerical examples are provided to support the theoretical results.

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Hemeda, A. A. "Modified Homotopy Perturbation Method for Solving Fractional Differential Equations." Journal of Applied Mathematics 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/594245.

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The modified homotopy perturbation method is extended to derive the exact solutions for linear (nonlinear) ordinary (partial) differential equations of fractional order in fluid mechanics. The fractional derivatives are taken in the Caputo sense. This work will present a numerical comparison between the considered method and some other methods through solving various fractional differential equations in applied fields. The obtained results reveal that this method is very effective and simple, accelerates the rapid convergence of the series solution, and reduces the size of work to only one iteration.

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Duran, Serbay. "Exact Solutions for Time-Fractional Ramani and Jimbo—Miwa Equations by Direct Algebraic Method." Advanced Science, Engineering and Medicine 12, no. 7 (July 1, 2020): 982–88. http://dx.doi.org/10.1166/asem.2020.2663.

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In this study, the analytical solutions of some nonlinear time-fractional partial differential equations are investigated by the direct algebraic method. The nonlinear fractional partial differential equation (NLfPDE) which is based on the fractional derivative (fd) in the sense of modified Riemann-Liouville derivative is transformed to the nonlinear non-fractional ordinary differential equation. The hyperbolic and rational functions which are contained solutions are obtained for the sixth-order time-fractional Ramani equation and time-fractional Jimbo—Miwa equation (JME) with the help of this technique. In addition, this method can be applied to higher order and higher dimensional NLfPDEs. Finally, three dimensional simulations of some solutions are given.

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Çenesiz, Y., and A. Kurt. "New Fractional Complex Transform for Conformable Fractional Partial Differential Equations." Journal of Applied Mathematics, Statistics and Informatics 12, no. 2 (December 1, 2016): 41–47. http://dx.doi.org/10.1515/jamsi-2016-0007.

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Abstract Conformable fractional complex transform is introduced in this paper for converting fractional partial differential equations to ordinary differential equations. Hence analytical methods in advanced calculus can be used to solve these equations. Conformable fractional complex transform is implemented to fractional partial differential equations such as space fractional advection diffusion equation and space fractional telegraph equation to obtain the exact solutions of these equations.

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Xu, Qinwu, and Zhoushun Zheng. "Spectral Collocation Method for Fractional Differential/Integral Equations with Generalized Fractional Operator." International Journal of Differential Equations 2019 (January 1, 2019): 1–14. http://dx.doi.org/10.1155/2019/3734617.

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Generalized fractional operators are generalization of the Riemann-Liouville and Caputo fractional derivatives, which include Erdélyi-Kober and Hadamard operators as their special cases. Due to the complicated form of the kernel and weight function in the convolution, it is even harder to design high order numerical methods for differential equations with generalized fractional operators. In this paper, we first derive analytical formulas for α-th (α>0) order fractional derivative of Jacobi polynomials. Spectral approximation method is proposed for generalized fractional operators through a variable transform technique. Then, operational matrices for generalized fractional operators are derived and spectral collocation methods are proposed for differential and integral equations with different fractional operators. At last, the method is applied to generalized fractional ordinary differential equation and Hadamard-type integral equations, and exponential convergence of the method is confirmed. Further, based on the proposed method, a kind of generalized grey Brownian motion is simulated and properties of the model are analyzed.

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Ul Hassan, Qazi Mahmood, Jamshad Ahmad, and Muhammad Shakeel. "A Novel Analytical Technique to Obtain Kink Solutions for Higher Order Nonlinear Fractional Evolution Equations." Abstract and Applied Analysis 2014 (2014): 1–11. http://dx.doi.org/10.1155/2014/213482.

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We use the fractional derivatives in Caputo’s sense to construct exact solutions to fractional fifth order nonlinear evolution equations. A generalized fractional complex transform is appropriately used to convert this equation to ordinary differential equation which subsequently resulted in a number of exact solutions.

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Khajehsaeid, Hesam. "A Comparison Between Fractional-Order and Integer-Order Differential Finite Deformation Viscoelastic Models: Effects of Filler Content and Loading Rate on Material Parameters." International Journal of Applied Mechanics 10, no. 09 (November 2018): 1850099. http://dx.doi.org/10.1142/s1758825118500990.

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Elastomers or rubber-like materials exhibit nonlinear viscoelastic behavior such as creep and relaxation upon mechanical loading. Differential constitutive models and hereditary integrals are the main frameworks followed in the literature for modeling the viscoelastic behavior at finite deformations. Regular differential operators can be replaced by fractional-order derivatives in the standard models in order to make fractional viscoelastic models. In the present paper, the relaxation behavior of elastomers is formulated both in terms of ordinary (integer-order) and fractional differential viscoelastic models. The derived constitutive equations are fitted to several experimental data to compare their efficiency in modeling the stress relaxation phenomenon. Specifically, a fractional viscoelastic model with one fractional dashpot (FD) is compared with two ordinary models including respectively one and two ordinary dashpots (OD). The models are compared in fitting accuracy, number of required material parameters and also variation of parameters from one compound to another to clarify the effects of filler content and deformation rate. It is shown that, the results of the ordinary model with one OD is not good at all. The fractional model with one FD and the ordinary model with two ODs provide good fittings for all compounds whereas the former uses only three parameters and the latter uses five material parameters. For the fractional model, the order of the Maxwell element and the associated relaxation time approximately remain the same for different compounds of each material at certain loading rates, but it is not the case for the ordinary differential models.

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Abdelrahman, Mahmoud A. E. "A note on Riccati-Bernoulli Sub-ODE method combined with complex transform method applied to fractional differential equations." Nonlinear Engineering 7, no. 4 (December 19, 2018): 279–85. http://dx.doi.org/10.1515/nleng-2017-0145.

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AbstractIn this paper, the fractional derivatives in the sense of modified Riemann–Liouville and the Riccati-Bernoulli Sub-ODE method are used to construct exact solutions for some nonlinear partial fractional differential equations via the nonlinear fractional Zoomeron equation and the (3 + 1) dimensional space-time fractional mKDV-ZK equation. These nonlinear fractional equations can be turned into another nonlinear ordinary differential equation by complex transform method. This method is efficient and powerful in solving wide classes of nonlinear fractional order equations. The Riccati-Bernoulli Sub-ODE method appears to be easier and more convenient by means of a symbolic computation system.

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