抽象多项Riemann-Liouville分数阶微分方程

Abstract

Abstract:

In this paper, we investigate the following abstract multi-term fractional differential equation
D_t^αnu（t）+A_jD_t^αju(t)=AD_t^αu(t)+f(t),t∈(0,τ),
where n∈N\{1}, A and A₁,…,A_n-1 are closed linear operators on a complex Banach space E, 0≤α₁<…<α_n, 0≤α<α_n, 0<τ≤∞, f(t) is an E-valued function, and D_t^α denotes the Riemann-Liouville fractional derivative of order α ([5]). We introduce and systematically analyze several new types of k-regularized (C₁,C₂)-existence and uniqueness (propagation) families for (0.1), continuing in such a way our previous researches raised in [22, 24-25] and [34]. Plenty of various examples illustrates our abstract results.

Key words: Abstract multi-term fractional differential equations, Riemann-Liouville fractional derivatives, (a,k)-Regularized C-resolvent families, Well-posedness

CLC Number:

O175.2

Marko Kostić, Li Chenggang, Li Miao. Abstract Multi-Term Fractional Differential Equations with Riemann-Liouville Derivatives[J].Acta mathematica scientia,Series A, 2016, 36(4): 601-622.

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References

[1] Alvarez-Pardo E, Lizama C. Mild solutions for multi-term time-fractional differential equations with nonlocal initial conditions. Electron J Diff Equ, 2014, 39: 1-10
[2] Araya D, Lizama C. Existence of asymptotically almost automorphic solutions for a third order differential equation. Electron J Diff Equ, 2012, 53: 1-20
[3] Arendt W, Batty C J K, Hieber M, Neubrander F. Vector-valued Laplace Transforms and Cauchy Problems. Basel: Birkhäuser, 2001
[4] Ashyralyev A, Dal F, Pinar Z. A note on the fractional hyperbolic differential and difference equations. Applied Math Comp, 2011, 217: 4642-4664
[5] Bazhlekova E. Fractional Evolution Equations in Banach Spaces[D]. Eindhoven: Eindhoven University of Technology, 2001
[6] Bogacheva Yu V. Resolution's Problems of Initial Problems for Abstract Differential Equations with Fractional Derivatives[D]. Belgorod: Belgorod Univ, 2006
[7] Bogacheva Yu V, Glushak A V. About one abstract problem such as Cauchy with fractional derivatives//The International Conference on Differential Equations and Dynamical Systems. Suzdal, 2006, p244
[8] Chen C, Li M. On fractional resolvent operator functions. Semigroup Forum, 2010, 80: 121-142
[9] Chen C, Li M, Li F B. On boundary values of fractional resolvent families. J Math Anal Appl, 2011, 384: 453-467
[10] Conejero J A, Lizama C, Rodenas F. Chaotic behaviour of the solutions of the Moore-Gibson-Thompson equation. Preprint
[11] Diethelm K. The Analysis of Fractional Differential Equations. Berlin: Springer-Verlag, 2010
[12] Fujita Y. Integrodifferential equation which interpolates the heat equation and the wave equation (I). Osaka J Math, 1990, 27: 309-321
[13] Fujita Y. Integrodifferential equation which interpolates the heat equation and the wave equation (II). Osaka J Math, 1990, 27: 797-804
[14] Guidetti D, Karasozen B, Piskarev S. Approximation of abstract differential equations. J Math Sci, 2004, 122: 3013-3054
[15] Haase M. The Functional Calculi for Sectorial Operators. Basel: Birkhäuser Verlag, 2006
[16] Hilfer R. Applications of Fractional Calculus in Physics. Singapore: World Scientific Publ Co, 2000
[17] Jiang H, Liu F, Turner I, Burrage K. Analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain. J Math Anal Appl, 2012, 389: 1117-1127
[18] Kaltenbacher B, Lasiecka I. Exponential decay for low and higher energies in the third order linear Moore-Gibson-Thompson equation with variable viscosity. Palestine J Math, 2012, 1: 1-10
[19] Kilbas A A, Srivastava H M, Trujillo J J. Theory and Applications of Fractional Differential Equations. Amsterdam: Elsevier, 2006
[20] Kosti? M. Generalized Semigroups and Cosine Functions. Belgrade: Mathematical Institute SANU, 2011
[21] Kosti? M. Abstract Volterra Integro-Differential Equations. Boca Raton: CRC Press, 2015
[22] Kosti? M, Li C G, Li M. On a class of abstract time-fractional equations on locally convex spaces. Abstr Appl Anal, 2012, Article ID: 131652
[23] Kosti? M. Generalized well-posedness of hyperbolic Volterra equations of non-scalar type. Ann Acad Rom Sci Ser Math Appl, 2014, 6: 19-45
[24] Kosti? M, Li C G, Li M. Abstract multi-term fractional differential equations. Krag J Math, 2014, 38: 51-71
[25] Kosti? M. On the existence and uniqueness of solutions of certain classes of abstract multi-term fractional differential equations. Funct Anal Appr Comp, 2014, 6: 13-33
[26] Kosti? M. Abstract differential operators generating fractional resolvent families. Acta Math Sin (English Ser), 2014, 30: 1989-1998
[27] Kosti? M. (a,k)-Regularized (C₁,C₂)-existence and uniqueness families. Bull Cl Sci Math Nat Sci Math, 2013, 38: 9-26
[28] Kosti? M. Degenerate multi-term fractional differential equations in locally convex spaces. Preprint
[29] Kosti? M. Hypercyclic and topologically mixing properties of degenerate multi-term fractional differential equations. Diff Eqns Dyn Sys, 2015, 3: DOI: 10.1007/s12591-015-0238-x
[30] Langlands T A M, Henry B I, Wearne S L. Fractional cable equation models for anomalous electrodiffusion in nerve cells: finite domain solutions. SIAM J Appl Math, 2011, 71: 1168-1203
[31] Langlands T A M, Henry B I, Wearne S L. Fractional cable equation models for anomalous electrodiffusion in nerve cells: infinite domain solutions. J Math Biol, 2009, 59: 761-808
[32] Langlands T A M, Henry B I, Wearne S L. Anomalous subdiffusion with multispecies linear reaction dynamics. Phys Rev E, 2008, 77(3): 021111
[33] Langlands T A M, Henry B I, Wearne S L. Anomalous diffusion with linear reaction dynamics: from continuous time random walks to fractional reaction-diffusion equations. Phys Rev E, 2006, 74(3): 031116
[34] Li C G, Kosti? M, Li M, Piskarev S. On a class of time-fractional differential equations. Fract Calc Appl Anal, 2012, 15: 639-668
[35] Li F B, Li M, Zheng Q. Fractional evolution equations governed by coercive differential operators. Abstr Appl Anal, 2009, Article ID: 438690
[36] Li K X, Peng J G, Jia J X. Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives. J Funct Anal, 2012, 263: 476-510
[37] Li M, Chen C, Li F B. On the fractional powers of generators of fractional resolvent families. J Funct Anal, 2010, 259: 2702-2726
[38] Lizama C, Poblete F. On a functional equation associated with (a,k)-regularized resolvent families. Abstr Appl Anal, 2012, Article ID: 495487
[39] Mainardi F. Fractional Calculus and Waves in Linear Viscoelasticity. London: Imperial College Press, 2010
[40] Mainardi F, Gorenflo R. On Mittag-Leffler-type functions in fractional evolution processes. J Comp Appl Math, 2000, 118: 283-299
[41] Martinez C, Sanz M. The Theory of Fractional Powers of Operators. Amsterdam: Elsevier, 2001
[42] Orsingher E, Beghin L. Time-fractional telegraph equations and telegraph processes with brownian time. Probab Theory Relat Fields, 2004, 128: 141-160
[43] Podlubny I. Fractional Differential Equations. New York: Academic Press, 1999
[44] Prüss J. Evolutionary Integral Equations and Applications. Basel: Birkhaüser Verlag, 1993
[45] Saxena R K, Mathai A M, Haubold H J. Reaction-diffusion systems and nonlinear waves. Astrophys Space Sci, 2006, 305: 297-303
[46] Schneider W R, Wyss W. Fractional diffusion and wave equations. J Math Phys, 1989, 30: 134-144
[47] Wyss W. Fractional diffusion equation. J Math Phys, 1986, 27: 2782-2785
[48] Xiao T J, Liang J. Higher order abstract Cauchy problems: their existence and uniqueness families. J London Math Soc, 2003, 67(2): 149-164
[49] Xiao T J, Liang J. The Cauchy Problem for Higher-Order Abstract Differential Equations. Berlin: Springer-Verlag, 1998
[50] Yang Q. Novel Analytical and Numerical Methods for Solving Fractional Dynamical Systems[D]. Brisbane: Queensland University of Technology, 2010

Abstract Multi-Term Fractional Differential Equations with Riemann-Liouville Derivatives

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