publication . Preprint . 2017

Existence of smooth solutions of multi-term Caputo-type fractional differential equations

Sin, Chung-Sik; Cheng, Shusen; Ri, Gang-Il; Kim, Mun-Chol;
Open Access English
• Published: 05 May 2017
Abstract
This paper deals with the initial value problem for the multi-term fractional differential equation. The fractional derivative is defined in the Caputo sense. Firstly the initial value problem is transformed into a equivalent Volterra-type integral equation under appropriate assumptions. Then new existence results for smooth solutions are established by using the Schauder fixed point theorem.
Subjects
free text keywords: Mathematics - Classical Analysis and ODEs, 34A12, 34A34, 45D05
20 references, page 1 of 2

[1] Bagley R L. Applications of generalized derivatives to viscoelasticity. PhD Dissertation, USA Air Force Institute of Technology, 1979.

[2] Deng J, Deng Z. Existence of solutions of initial value problems for nonlinear fractional differential equations. Appl Math Lett, 2014, 32: 6-12

[3] Deng J, Ma L. Existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations. Appl Math Lett, 2010, 23: 676-680

[4] Diethelm K. The Analysis of Fractional Differential Equations. Berlin: Springer, 2010

[5] Diethelm K. Smoothness properties of solutions of Caputo-type fractional differential equations. Fract Calc Appl Anal 2008, 10: 151-160 http://www.math.bas.bg/ fcaa [OpenAIRE]

[6] Diethelm K, Ford N J. Analysis of fractional differential equations. J Math Anal Appl 2002,265: 229-248 [OpenAIRE]

[7] Diethelm K, Ford N J. Multi-order fractional differential equations and their numerical solution. Appl Math Comp 2004, 154: 621-640

[8] Diethelm K, Ford N J, Freed A D. A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dynamics. 2002, 29: 3-22

[9] Hilfer R, Experimental implications of Bochner-Levy-Riesz diffusion. Fract Calc Appl Anal 2015, 18: 333-341 [OpenAIRE]

[10] Kilbas A A, Srivastava H M, Trujillo J J. Theory and applications of fractional differential Equations. Amsterdam: Elsevier Science, 2006

[11] Kosmatov N. Integral equations and initial value problems for nonlinear differential equations of fractional order. Nonlinear Anal, Theory Methods Appl 2009, 70: 2521-2529 [OpenAIRE]

[12] Li C, Deng W. Remarks on fractional derivatives, Appl Math Comput 2007, 187: 777-784

[13] Luchko Y, Gorenflo R. An operational method for solving fractional differential equations with the Caputo derivatives. Acta Math Vietnamica 1999, 24: 207-233

[14] Mainardi F. Fractional calculus and waves in linear viscoelasticity. World Scientific, 2010

[15] Mainardi F. An historical perspective on fractional calculus in linear viscoelasticity. Fract Calc Appl Anal 2012, 15: 712-717; DOI: 10.2478/s13540-012-0048-6; http://link.springer.com/article/10.2478/s13540-012-0048-6 [OpenAIRE]

20 references, page 1 of 2