We use waveform relaxation (WR) method to solve numerically fractional neutral functional differential equations and mainly consider the convergence of the numerical method with the help of a generalized Volterra-integral operator associated with the Mittag–Leffler function. We first give some properties of the integral operator. Using the proposed properties, we establish the convergence condition of the numerical method. Finally, we provide a new way to prove the convergence of waveform relaxation method for integer-order neutral functional differential equation, which is a special case of fractional neutral functional differential equation. Compared to the existing proof in the literature, our proof is concise and original.
Issue Section:
Research Papers
References
1.
Benson
, D. A.
, Wheatcraft
, S. W.
, and Meerschaert
, M. M.
, 2000
, “Application of a Fractional Advection-Dispersion Equation
,” Water Resour. Res.
, 36
(6
), pp. 1403
–1412
.2.
Hilfer
, R.
, 2000
, Applications of Fractional Calculus in Physics
, World Scientific
, Singapore
.3.
Metzler
, R.
, and Klafter
, J.
, 2000
, “The Random Walk's Guide to Anomalous Diffusion: A Fractional Dynamics Approach
,” Phys. Rep.
, 339
(1
), pp. 1
–77
.4.
Orsingher
, E.
, and Beghin
, L.
, 2004
, “Time-Fractional Telegraph Equations and Telegraph Processes With Brownian Time
,” Probab. Theory Relat. Fields
, 128
(1), pp. 141
–160
.5.
Niu
, M.
, and Xie
, B.
, 2012
, “Impacts of Gaussian Noises on the Blow-Up Times of Nonlinear Stochastic Partial Differential Equations
,” Nonlinear Anal. Real World Appl.
, 13
(3
), pp. 1346
–1352
.6.
Ahmad
, B.
, Nieto
, J. J.
, Alsaedi
, A.
, and El-Shahed
, M.
, 2012
, “A Study of Nonlinear Langevin Equation Involving Two Fractional Orders in Different Intervals
,” Nonlinear Anal. Real World Appl.
, 13
(2
), pp. 599
–606
.7.
Wang
, J. R.
, and Zhou
, Y.
, 2011
, “A Class of Fractional Evolution Equations and Optimal Controls
,” Nonlinear Anal. Real World Appl.
, 12
(1
), pp. 262
–272
.8.
Wang
, J. R.
, Zhou
, Y.
, and Wei
, W.
, 2012
, “Fractional Schrödinger Equations With Potential and Optimal Controls
,” Nonlinear Anal. Real World Appl.
, 13
(6
), pp. 2755
–2766
.9.
Povstenko
, Y. Z.
, 2010
, “Signaling Problem for Time-Fractional Diffusion-Wave Equation in a Half-Space in the Case of Angular Symmetry
,” Nonlinear Dyn.
, 59
(4
), pp. 593
–605
.10.
Mainardi
, F.
, 1997
, Fractals and Fractional Calculus Continuum Mechanics
, Springer Verlag
, Wien, Germany, New York
.11.
Malinowska
, A. B.
, and Torres
, D. F. M.
, 2012
, “Towards a Combined Fractional Mechanics and Quantization
,” Fractional Calculus Appl. Anal.
, 15
(3), pp. 407
–417
.12.
Hall
, M. G.
, and Barrick
, T. R.
, 2008
, “From Diffusion-Weighted MRI to Anomalous Diffusion Imaging
,” Magn. Reson. Med.
, 59
(3
), pp. 447
–455
.13.
Denton
, Z.
, and Vatsala
, A. S.
, 2010
, “Fractional Integral Inequalities and Applications
,” Comput. Math. Appl.
, 59
(3
), pp. 1087
–1094
.14.
Diethelm
, K.
, and Neville
, J. F.
, 2002
, “Analysis of Fractional Differential Equations
,” J. Math. Anal. Appl.
, 265
(2
), pp. 229
–248
.15.
Kilbas
, A. A.
, Srivastava
, H. M.
, and Trujillo
, J. J.
, 2006
, Theory and Applications of Fractional Differential Equations
(North-Holland Mathematics Studies), Vol. 204
, Elsevier Science B.V.
, Amsterdam, The Netherlands
.16.
Kosmatov
, N.
, 2009
, “Integral Equations and Initial Value Problems for Nonlinear Differential Equations of Fractional Order
,” Nonlinear Anal. Theory Methods Appl.
, 70
(7
), pp. 2521
–2529
.17.
Lakshmikantham
, V.
, and Vatsala
, A. S.
, 2008
, “Basic Theory of Fractional Differential Equations
,” Nonlinear Anal. Theory Methods Appl.
, 69
(8
), pp. 2677
–2682
.18.
Mirzaee
, F.
, Bimesl
, S.
, and Tohidi
, E.
, 2015
, “Solving Nonlinear Fractional Integro-Differential Equations of Volterra Type Using Novel Mathematical Matrices
,” ASME J. Comput. Nonlinear Dyn.
, 10
(6
), p. 061016
.19.
Firoozjaee
, M. A.
, Yousefi
, S. A.
, Jafari
, H.
, and Baleanu
, D.
, 2015
, “On a Numerical Approach to Solve Multi-Order Fractional Differential Equations With Initial/Boundary Conditions
,” ASME J. Comput. Nonlinear Dyn.
, 10
(6
), p. 061025
.20.
Saha Ray
, S.
, and Sahoo
, S.
, 2015
, “Traveling Wave Solutions to Riesz Time-Fractional Camassa–Holm Equation in Modeling for Shallow-Water Waves
,” ASME J. Comput. Nonlinear Dyn.
, 10
(6
), p. 061026
.21.
Lakshmikantham
, V.
, 2008
, “Theory of Fractional Functional Differential Equations
,” Nonlinear Anal. Theory Methods Appl.
, 69
(10
), pp. 3337
–3343
.22.
Maraaba
, T. A.
, Jarad
, F.
, and Baleanu
, D.
, 2008
, “On the Existence and the Uniqueness Theorem for Fractional Differential Equations With Bounded Delay Within Caputo Derivatives
,” Sci. China Ser. A Math.
, 51
(10
), pp. 1775
–1786
.23.
Podlubny
, I.
, 1999
, Fractional Differential Equations
, Academic Press
, New York
.24.
Agarwal
, R. P.
, Zhou
, Y.
, and He
, Y. Y.
, 2010
, “Existence of Fractional Neutral Functional Differential Equations
,” Comput. Math. Appl.
, 59
(3
), pp. 1095
–1100
.25.
Jankowski
, T.
, 2013
, “Initial Value Problems for Neutral Fractional Differential Equations Involving a Riemann–Liouville Derivative
,” Appl. Math. Comput.
, 209
(14), pp. 7772
–7779
.26.
Vijayakumar
, V.
, Selvakumar
, A.
, and Murugesu
, R.
, 2014
, “Controllability for a Class of Fractional Neutral Integro-Differential Equations With Unbounded Delay
,” Appl. Math. Comput.
, 232
, pp. 303
–312
.27.
Gautam
, G. R.
, and Dabas
, J.
, 2015
, “Mild Solutions for Class of Neutral Fractional Functional Differential Equations With Not Instantaneous Impulses
,” Appl. Math. Comput.
, 259
, pp. 480
–489
.28.
Losada
, J.
, Nieto
, J. J.
, and Pourhadi
, E.
, 2015
, “On the Attractivity of Solutions for a Class of Multi-Term Fractional Functional Differential Equations
,” J. Comput. Appl. Math.
, 312
, pp. 2
–12
.29.
Brzdȩk
, J.
, and Eghbali
, N.
, 2016
, “On Approximate Solutions of Some Delayed Fractional Differential Equations
,” Appl. Math. Lett.
, 54
, pp. 31
–35
.30.
Lelarasmee
, E.
, Ruehli
, A.
, and Sangiovanni-Vincentelli
, A.
, 1982
, “The Waveform Relaxation Method for Time Domain Analysis of Large Scale Integrated Circuits
,” IEEE Trans. Comput. Aided Des.
, 1
(3
), pp. 131
–145
.31.
Zubik-Kowal
, B.
, and Vandewalle
, S.
, 1999
, “Waveform Relaxation for Functional-Differential Equations
,” SIAM J. Sci. Comput.
, 21
(1
), pp. 207
–226
.32.
Jackiewicz
, Z.
, Kwapisz
, M.
, and Lo
, E.
, 1997
, “Waveform Relaxation Methods for Functional Differential Systems of Neutral Type
,” J. Math. Anal. Appl.
, 207
(1
), pp. 255
–285
.33.
Bartoszewski
, Z.
, and Kwapisz
, M.
, 2000
, “On Error Estimates for Waveform Relaxation Methods for Delay-Differential Equations
,” SIAM J. Numer. Anal.
, 38
(2
), pp. 639
–659
.34.
Bartoszewski
, Z.
, and Kwapisz
, M.
, 2004
, “Delay Dependent Estimations for Waveform Relaxation Methods for Neutral Differential-Functional Systems
,” Comput. Math. Appl.
, 48
(12
), pp. 1877
–1892
.35.
Bartoszewski
, Z.
, and Kwapisz
, M.
, 1999
, On the “Convergence of Waveform Relaxation Methods for Differential-Functional Systems of Equations
,” J. Math. Anal. Appl.
, 235
(2
), pp. 478
–496
.36.
Ding
, X. L.
, and Jiang
, Y. L.
, 2013
, “Waveform Relaxation Methods for Fractional Functional Differential Equations
,” Fractional Calculus Appl. Anal.
, 16
(3), pp. 573
–594
.37.
Kilbas
, A. A.
, Saigo
, M.
, and Saxena
, R. K.
, 2004
, “Generalized Mittag–Leffler Function and Generalized Fractional Calculus Operators
,” Integr. Transforms Spec. Funct.
, 15
(1
), pp. 31
–49
.38.
Shukla
, A. K.
, and Prajapati
, J. C.
, 2007
, “On a Generalization of Mittag–Leffler Function and Its Properties
,” J. Math. Anal. Appl.
, 336
(2
), pp. 797
–811
.39.
Ye
, H. P.
, Gao
, J. M.
, and Ding
, Y. S.
, 2007
, “A Generalized Gronwall Inequality and Its Application to a Fractional Differential Equation
,” J. Math. Anal. Appl.
, 328
(2
), pp. 1075
–1081
.40.
Kilbas
, A. A.
, Saigo
, M.
, and Saxena
, R. K.
, 2002
, “Solution of Volterra Integro-Differential Equations With Generalized Mittag–Leffler Function in the Kernels
,” J. Integr. Equations Appl.
, 14
(4
), pp. 377
–396
.41.
Ding
, X. L.
, and Jiang
, Y. L.
, 2012
, “Semilinear Fractional Differential Equations Based on a New Integral Operator Approach
,” Commun. Nonlinear Sci. Numer. Simul.
, 17
(12
), pp. 5143
–5150
.42.
Samko
, S.
, Kilbas
, A. A.
, and Marichev
, O. I.
, 1993
, Fractional Integrals and Derivatives: Theory and Applications
, Gordon and Breach Science Publishers
, Switzerland
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