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Relevant bibliographies by topics / Fractional-order ordinary differential equations

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Academic literature on the topic 'Fractional-order ordinary differential equations'

Author: Grafiati

Published: 28 June 2021

Last updated: 7 February 2022

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

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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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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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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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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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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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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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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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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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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Dissertations / Theses on the topic "Fractional-order ordinary differential equations"

1

Woods, Patrick Daniel. "Localisation in reversible fourth-order ordinary differential equations." Thesis, University of Bristol, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.299269.

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Jenab, Bita. "Asymptotic theory of second-order nonlinear ordinary differential equations." Thesis, University of British Columbia, 1985. http://hdl.handle.net/2429/24690.

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The asymptotic behaviour of nonoscillatory solutions of second order nonlinear ordinary differential equations is studied. Necessary and sufficient conditions are given for the existence of positive solutions with specified asymptotic behaviour at infinity. Existence of nonoscillatory solutions is established using the Schauder-Tychonoff fixed point theorem. Techniques such as factorization of linear disconjugate operators are employed to reveal the similar nature of asymptotic solutions of nonlinear differential equations to that of linear equations. Some examples illustrating the asymptotic theory of ordinary differential equations are given.
Science, Faculty of
Mathematics, Department of
Graduate

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Sun, Xun. "Twin solutions of even order boundary value problems for ordinary differential equations and finite difference equations." [Huntington, WV : Marshall University Libraries], 2009. http://www.marshall.edu/etd/descript.asp?ref=1014.

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Boutayeb, Abdesslam. "Numerical methods for high-order ordinary differential equations with applications to eigenvalue problems." Thesis, Brunel University, 1990. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.278244.

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Gray, Michael Jeffery Henderson Johnny L. "Uniqueness implies uniqueness and existence for nonlocal boundary value problems for third order ordinary differential equations." Waco, Tex. : Baylor University, 2006. http://hdl.handle.net/2104/4185.

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Koike, Tatsuya. "On the exact WKB analysis of second order linear ordinary differential equations with simple poles." 京都大学 (Kyoto University), 2000. http://hdl.handle.net/2433/181093.

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Granström, Frida. "Symmetry methods and some nonlinear differential equations : Background and illustrative examples." Thesis, Karlstads universitet, Institutionen för matematik och datavetenskap (from 2013), 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:kau:diva-48020.

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Differential equations, in particular the nonlinear ones, are commonly used in formulating most of the fundamental laws of nature as well as many technological problems, among others. This makes the need for methods in finding closed form solutions to such equations all-important. In this thesis we study Lie symmetry methods for some nonlinear ordinary differential equations (ODE). The study focuses on identifying and using the underlying symmetries of the given first order nonlinear ordinary differential equation. An extension of the method to higher order ODE is also discussed. Several illustrative examples are presented.
Differentialekvationer, framförallt icke-linjära, används ofta vid formulering av fundamentala naturlagar liksom många tekniska problem. Därmed finns det ett stort behov av metoder där det går att hitta lösningar i sluten form till sådana ekvationer. I det här arbetet studerar vi Lie symmetrimetoder för några icke-linjära ordinära differentialekvationer (ODE). Studien fokuserar på att identifiera och använda de underliggande symmetrierna av den givna första ordningens icke-linjära ordinära differentialekvationen. En utvidgning av metoden till högre ordningens ODE diskuteras också. Ett flertal illustrativa exempel presenteras.

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Charoenphon, Sutthirut. "Green's Functions of Discrete Fractional Calculus Boundary Value Problems and an Application of Discrete Fractional Calculus to a Pharmacokinetic Model." TopSCHOLAR®, 2014. http://digitalcommons.wku.edu/theses/1327.

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Fractional calculus has been used as a research tool in the fields of pharmacology, biology, chemistry, and other areas [3]. The main purpose of this thesis is to calculate Green's functions of fractional difference equations, and to model problems in pharmacokinetics. We claim that the discrete fractional calculus yields the best prediction performance compared to the continuous fractional calculus in the application of a one-compartmental model of drug concentration. In Chapter 1, the Gamma function and its properties are discussed to establish a theoretical basis. Additionally, the basics of discrete fractional calculus are discussed using particular examples for further calculations. In Chapter 2, we use these basic results in the analysis of a linear fractional difference equation. Existence of solutions to this difference equation is then established for both initial conditions (IVP) and two-point boundary conditions (BVP). In Chapter 3, Green's functions are introduced and discussed, along with examples. Instead of using Cauchy functions, the technique of finding Green's functions by a traditional method is demonstrated and used throughout this chapter. The solutions of the BVP play an important role in analysis and construction of the Green's functions. Then, Green's functions for the discrete calculus case are calculated using particular problems, such as boundary value problems, discrete boundary value problems (DBVP) and fractional boundary value problems (FBVP). Finally, we demonstrate how the Green's functions of the FBVP generalize the existence results of the Green's functions of DVBP. In Chapter 4, different compartmental pharmacokinetic models are discussed. This thesis limits discussion to the one-compartmental model. The Mathematica FindFit command and the statistical computational techniques of mean square error (MSE) and cross-validation are discussed. Each of the four models (continuous, continuous fractional, discrete and discrete fractional) is used to compute the MSE numerically with the given data of drug concentration. Then, the best fit and the best model are obtained by inspection of the resulting MSE. In the last Chapter, the results are summarized, conclusions are drawn, and directions for future work are stated.

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Šustková, Apolena. "Řešení obyčejných diferenciálních rovnic neceločíselného řádu metodou Adomianova rozkladu." Master's thesis, Vysoké učení technické v Brně. Fakulta strojního inženýrství, 2021. http://www.nusl.cz/ntk/nusl-445455.

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This master's thesis deals with solving fractional-order ordinary differential equations by the Adomian decomposition method. A part of the work is therefore devoted to the theory of equations containing differential operators of non-integer order, especially the Caputo operator. The next part is devoted to the Adomian decomposition method itself, its properties and implementation in the case of Chen system. The work also deals with bifurcation analysis of this system, both for integer and non-integer case. One of the objectives is to clarify the discrepancy in the literature concerning the fractional-order Chen system, where experiments based on the use of the Adomian decomposition method give different results for certain input parameters compared with numerical methods. The clarification of this discrepancy is based on recent theoretical knowledge in the field of fractional-order differential equations and their systems. The conclusions are supported by numerical experiments, own code implementing the Adomian decomposition method on the Chen system was used.

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Shu, Yupeng. "Numerical Solutions of Generalized Burgers' Equations for Some Incompressible Non-Newtonian Fluids." ScholarWorks@UNO, 2015. http://scholarworks.uno.edu/td/2051.

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The author presents some generalized Burgers' equations for incompressible and isothermal flow of viscous non-Newtonian fluids based on the Cross model, the Carreau model, and the Power-Law model and some simple assumptions on the flows. The author numerically solves the traveling wave equations for the Cross model, the Carreau model, the Power-Law model by using industrial data. The author proves existence and uniqueness of solutions to the traveling wave equations of each of the three models. The author also provides numerical estimates of the shock thickness as well as maximum strain $\varepsilon_{11}$ for each of the fluids.

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Books on the topic "Fractional-order ordinary differential equations"

1

Paris, R. B. Asymptotics of high-order ordinary differential equations. Boston, (Mass.): Pitman Advanced, 1985.

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D, Wood A., ed. Asymptotics of high-order ordinary differential equations. Boston: Pitman Pub., 1986.

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3

Zhukova, Galina. Differential equations. ru: INFRA-M Academic Publishing LLC., 2020. http://dx.doi.org/10.12737/1072180.

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The textbook presents the theory of ordinary differential equations constituting the subject of the discipline "Differential equations". Studied topics: differential equations of first, second, arbitrary order; differential equations; integration of initial and boundary value problems; stability theory of solutions of differential equations and systems. Introduced the basic concepts, proven properties of differential equations and systems. The article presents methods of analysis and solutions. We consider the applications of the obtained results, which are illustrated on a large number of specific tasks. For independent quality control mastering the course material suggested test questions on the theory, exercises and tasks. It is recommended that teachers, postgraduates and students of higher educational institutions, studying differential equations and their applications.

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4

Boutayeb, Abdesslam. Numerical methods for high-order ordinary differential equations with applications to eigenvalue problems. Uxbridge: Brunel University, 1990.

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5

Hartley, T. T. A solution to the fundamental linear fractional order differential equation. [Cleveland, Ohio]: National Aeronautics and Space Administration, Lewis Research Center, 1998.

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6

Spectral analysis, differential equations, and mathematical physics: A festschrift in honor of Fritz Gesztesy's 60th birthday. Providence, Rhode Island: American Mathematical Society, 2013.

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7

1954-, Sickel Winfried, ed. Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations. Berlin: Walter de Gruyter, 1996.

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Hartley, T. T. Fractional system identification: An approach using continuous order-distributions. Cleveland, Ohio: National Aeronautics and Space Administration, Glenn Research Center, 1999.

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Hartley, T. T. Insights into the fractional order initial value problem via semi-infinite systems. [Cleveland, Ohio]: National Aeronautics and Space Administration, Lewis Research Center, 1998.

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Kalinin, Sergey, and Larisa Pankratova. Ordinary differential equations of the first order. Science and Innovation Center Publishing House, 2020. http://dx.doi.org/10.12731/978-5-907208-23-0.

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Book chapters on the topic "Fractional-order ordinary differential equations"

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Kubica, Adam, Katarzyna Ryszewska, and Masahiro Yamamoto. "Fractional Ordinary Differential Equations." In Time-Fractional Differential Equations, 47–71. Singapore: Springer Singapore, 2020. http://dx.doi.org/10.1007/978-981-15-9066-5_3.

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Luchko, Yuri. "Operational method for fractional ordinary differential equations." In Fractional Differential Equations, edited by Anatoly Kochubei and Yuri Luchko, 91–118. Berlin, Boston: De Gruyter, 2019. http://dx.doi.org/10.1515/9783110571660-005.

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Walter, Wolfgang. "First Order Systems. Equations of Higher Order." In Ordinary Differential Equations, 105–57. New York, NY: Springer New York, 1998. http://dx.doi.org/10.1007/978-1-4612-0601-9_4.

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Adkins, William A., and Mark G. Davidson. "First Order Differential Equations." In Ordinary Differential Equations, 1–100. New York, NY: Springer New York, 2012. http://dx.doi.org/10.1007/978-1-4614-3618-8_1.

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Gazizov, Rafail K., Alexey A. Kasatkin, and Stanislav Yu Lukashchuk. "Symmetries and group invariant solutions of fractional ordinary differential equations." In Fractional Differential Equations, edited by Anatoly Kochubei and Yuri Luchko, 65–90. Berlin, Boston: De Gruyter, 2019. http://dx.doi.org/10.1515/9783110571660-004.

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Adkins, William A., and Mark G. Davidson. "Second Order Linear Differential Equations." In Ordinary Differential Equations, 331–81. New York, NY: Springer New York, 2012. http://dx.doi.org/10.1007/978-1-4614-3618-8_5.

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Walter, Wolfgang. "First Order Equations: Some Integrable Cases." In Ordinary Differential Equations, 9–52. New York, NY: Springer New York, 1998. http://dx.doi.org/10.1007/978-1-4612-0601-9_2.

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Walter, Wolfgang. "Theory of First Order Differential Equations." In Ordinary Differential Equations, 53–104. New York, NY: Springer New York, 1998. http://dx.doi.org/10.1007/978-1-4612-0601-9_3.

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Adkins, William A., and Mark G. Davidson. "Second Order Constant Coefficient Linear Differential Equations." In Ordinary Differential Equations, 203–73. New York, NY: Springer New York, 2012. http://dx.doi.org/10.1007/978-1-4614-3618-8_3.

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Goodwine, Bill. "First-Order Ordinary Differential Equations." In Engineering Differential Equations, 57–90. New York, NY: Springer New York, 2010. http://dx.doi.org/10.1007/978-1-4419-7919-3_2.

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Conference papers on the topic "Fractional-order ordinary differential equations"

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Damasceno, Berenice C., and Luciano Barbanti. "Ordinary fractional differential equations are in fact usual entire ordinary differential equations on time scales." In 10TH INTERNATIONAL CONFERENCE ON MATHEMATICAL PROBLEMS IN ENGINEERING, AEROSPACE AND SCIENCES: ICNPAA 2014. AIP Publishing LLC, 2014. http://dx.doi.org/10.1063/1.4904589.

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Gazizov, Rafail K., Alexey A. Kasatkin, and Stanislav Yu Lukashchuk. "Linearly autonomous symmetries of the ordinary fractional differential equations." In 2014 International Conference on Fractional Differentiation and its Applications (ICFDA). IEEE, 2014. http://dx.doi.org/10.1109/icfda.2014.6967419.

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BONHEURE, D., J. M. GOMES, and L. SANCHEZ. "POSITIVE SOLUTIONS OF A SECOND ORDER SINGULAR ORDINARY DIFFERENTIAL EQUATION." In Proceedings of the International Conference on Differential Equations. WORLD SCIENTIFIC, 2005. http://dx.doi.org/10.1142/9789812702067_0028.

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Podlubny, Igor, Tomas Skovranek, and Blas M. Vinagre Jara. "Matrix Approach to Discretization of Ordinary and Partial Differential Equations of Arbitrary Real Order: The Matlab Toolbox." In ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2009. http://dx.doi.org/10.1115/detc2009-86944.

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The method developed recently by Podlubny et al. (I. Podlubny, Fractional Calculus and Applied Analysis, vol. 3, no. 4, 2000, pp. 359–386; I. Podlubny et al., Journal of Computational Physics, vol. 228, no. 8, 1 May 2009, pp. 3137–3153) makes it possible to immediately obtain the discretization of ordinary and partial differential equations by replacing the derivatives with their discrete analogs in the form of triangular strip matrices. This article presents a Matlab toolbox that implements the matrix approach and allows easy and convenient discretization of ordinary and partial differential equations of arbitrary real order. The basic use of the functions implementing the matrix approach to discretization of derivatives of arbitrary real order (so-called fractional derivatives, or fractional-order derivatives), and to solution of ordinary and partial fractional differential equations, is illustrated by examples with explanations.

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Schwarz, Fritz. "Janet bases of 2nd order ordinary differential equations." In the 1996 international symposium. New York, New York, USA: ACM Press, 1996. http://dx.doi.org/10.1145/236869.240354.

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Takahashi, Masatomo. "On completely integrable first order ordinary differential equations." In Proceedings of the Australian-Japanese Workshop. WORLD SCIENTIFIC, 2007. http://dx.doi.org/10.1142/9789812706898_0018.

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Zainuddin, Nooraini, and Zarina Bibi Ibrahim. "Block method for third order ordinary differential equations." In PROCEEDINGS OF THE 24TH NATIONAL SYMPOSIUM ON MATHEMATICAL SCIENCES: Mathematical Sciences Exploration for the Universal Preservation. Author(s), 2017. http://dx.doi.org/10.1063/1.4995919.

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KLOKOV, Y. A., and F. SADYRBAEV. "SHARP CONDITIONS FOR THE SUPERLINEARITY OF THE SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS." In Proceedings of the International Conference on Differential Equations. WORLD SCIENTIFIC, 2005. http://dx.doi.org/10.1142/9789812702067_0030.

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Yarman, Ahmad Fauzan, Armiati, and Lufri. "Hypothetical Learning Trajectory for First-Order Ordinary Differential Equations." In 2nd International Conference Innovation in Education (ICoIE 2020). Paris, France: Atlantis Press, 2020. http://dx.doi.org/10.2991/assehr.k.201209.245.

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Yap, Lee Ken, and Fudziah Ismail. "Ninth order block hybrid collocation method for second order ordinary differential equations." In PROGRESS IN APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING PROCEEDINGS. AIP Publishing LLC, 2016. http://dx.doi.org/10.1063/1.4940254.

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