We present an efficient numerical method for solving fractional pantograph differential equations by applying Taylor wavelets. We give an exact formula for the Riemann-Liouville fractional integral of the Taylor wavelets. We then apply the given formula to reduce our problem to the problem of solving a system of algebraic equations. Several examples are included to show the the effectiveness of our numerical method and in comparison with previous methods.
AbdulazizOHashimIMomaniS (2008) Solving systems of fractional differential equations by homotopy-perturbation method. Physics Letters A372(4): 451–459.
2.
AielloWGFreedmanHWuJ (1992) Analysis of a model representing stage-structured population growth with state-dependent time delay. SIAM Journal on Applied Mathematics52(3): 855–869.
3.
AnapaliAMustafaG (2015) Numerical approach for solving fractional pantograph equation. International Journal of Computer Applications113(9): 45–52.
4.
BaillieRT (1996) Long memory processes and fractional integration in econometrics. Journal of Econometrics73(1): 5–59.
5.
BellenAZennaroM (2013) Numerical Methods for Delay Differential Equations. Oxford: Oxford University Press.
6.
BeylkinGCoifmanRRokhlinV (1991) Fast wavelet transforms and numerical algorithms i. Communications on Pure and Applied Mathematics44(2): 141–183.
7.
BhrawyATharwatMYildirimA (2013) A new formula for fractional integrals of chebyshev polynomials: Application for solving multi-term fractional differential equations. Applied Mathematical Modelling37(6): 4245–4252.
8.
BrunnerHHuangQXieH (2010) Discontinuous galerkin methods for delay differential equations of pantograph type. SIAM Journal on Numerical Analysis48(5): 1944–1967.
9.
CaputoM (1967) Linear models of dissipation whose q is almost frequency independent—ii. Geophysical Journal International13(5): 529–539.
10.
ChenXWangL (2010) The variational iteration method for solving a neutral functional-differential equation with proportional delays. Computers & Mathematics with Applications59(8): 2696–2702.
11.
Daftardar-GejjiVJafariH (2007) Solving a multi-order fractional differential equation using adomian decomposition. Applied Mathematics and Computation189(1): 541–548.
EnghetaN (1996) On fractional calculus and fractional multipoles in electromagnetism. IEEE Transactions on Antennas and Propagation44(4): 554–566.
14.
Ezz-EldienSS (2018) On solving systems of multi-pantograph equations via spectral tau method. Applied Mathematics and Computation321: 63–73.
15.
HallMGBarrickTR (2008) From diffusion-weighted mri to anomalous diffusion imaging. Magnetic Resonance in Medicine: An Official Journal of the International Society for Magnetic Resonance in Medicine59(3): 447–455.
16.
HeydariMHooshmandaslMRMohammadiF (2014) Legendre wavelets method for solving fractional partial differential equations with dirichlet boundary conditions. Applied Mathematics and Computation234: 267–276.
17.
IsahAPhangCPhangP (2017) Collocation method based on genocchi operational matrix for solving generalized fractional pantograph equations. International Journal of Differential Equations2017: 1–10.
18.
IsikORTurkogluT (2016) A rational approximate solution for generalized pantograph-delay differential equations. Mathematical Methods in the Applied Sciences39(8): 2011–2024.
19.
KeshavarzEOrdokhaniYRazzaghiM (2014) Bernoulli wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations. Applied Mathematical Modelling38(24): 6038–6051.
20.
KeshavarzEOrdokhaniYRazzaghiM (2018) The taylor wavelets method for solving the initial and boundary value problems of bratu-type equations. Applied Numerical Mathematics128: 205–216.
21.
KulishVVLageJL (2002) Application of fractional calculus to fluid mechanics. Journal of Fluids Engineering124(3): 803–806.
22.
LiYZhaoW (2010) Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations. Applied Mathematics and Computation216(8): 2276–2285.
23.
LuenbergerDG (1997) Optimization by Vector Space Methods. New York, USA: John Wiley & Sons.
24.
MainardiF (1997) Fractional calculus, fractals and fractional calculus in continuum mechanics. Wien: Springer-Verlag.
MillerKSRossB (1993) An Introduction to the Fractional Calculus and Fractional Differential Equations. New York, USA: Wiley-Interscience.
27.
MomaniSOdibatZ (2007) Numerical comparison of methods for solving linear differential equations of fractional order. Chaos, Solitons & Fractals31(5): 1248–1255.
28.
NematiSLimaPSedaghatS (2018) An effective numerical method for solving fractional pantograph differential equations using modification of hat functions. Applied Numerical Mathematics131: 174–189.
29.
NoorMAShahFANoorKI, et al. (2011) Variational iteration technique for finding multiple roots of nonlinear equations. Scientific Research and Essays6(6): 1344–1350.
30.
OckendonJRTaylerAB (1971) The dynamics of a current collection system for an electric locomotive. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences322(1551): 447–468.
31.
OldhamKB (2010) Fractional differential equations in electrochemistry. Advances in Engineering Software41(1): 9–12.
32.
PovstenkoY (2010) Signaling problem for time-fractional diffusion-wave equation in a half-space in the case of angular symmetry. Nonlinear Dynamics59(4): 593–605.
33.
RahimkhaniPOrdokhaniYBabolianE (2017a) A new operational matrix based on bernoulli wavelets for solving fractional delay differential equations. Numerical Algorithms74(1): 223–245.
34.
RahimkhaniPOrdokhaniYBabolianE (2017b) Numerical solution of fractional pantograph differential equations by using generalized fractional-order bernoulli wavelet. Journal of Computational and Applied Mathematics309: 493–510.
35.
RahimkhaniPOrdokhaniYBabolianE (2018) Müntz-legendre wavelet operational matrix of fractional-order integration and its applications for solving the fractional pantograph differential equations. Numerical Algorithms77(4): 1283–1305.
36.
RazzaghiMYousefiS (2001) The legendre wavelets operational matrix of integration. International Journal of Systems Science32(4): 495–502.
37.
RossikhinYAShitikovaMV (1997) Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Applied Mechanics Reviews50(1): 15–67.
38.
SaadatmandiA (2014) Bernstein operational matrix of fractional derivatives and its applications. Applied Mathematical Modelling38(4): 1365–1372.
39.
SaadatmandiADehghanM (2011) A legendre collocation method for fractional integro-differential equations. Journal of Vibration and Control17(13): 2050–2058.
40.
SaeediHMoghadamMMMollahasaniN, et al. (2011) A cas wavelet method for solving nonlinear fredholm integro-differential equations of fractional order. Communications in Nonlinear Science and Numerical Simulation16(3): 1154–1163.
41.
SedaghatSOrdokhaniYDehghanM (2012) Numerical solution of the delay differential equations of pantograph type via chebyshev polynomials. Communications in Nonlinear Science and Numerical Simulation17(12): 4815–4830.
SuarezLShokoohA (1997) An eigenvector expansion method for the solution of motion containing fractional derivatives. Journal of Applied Mechanics64(3): 629–635.
44.
TohidiEBhrawyAErfaniK (2013) A collocation method based on bernoulli operational matrix for numerical solution of generalized pantograph equation. Applied Mathematical Modelling37(6): 4283–4294.
45.
WangW-SLiS-F (2007) On the one-leg -methods for solving nonlinear neutral functional differential equations. Applied Mathematics and Computation193(1): 285–301.
46.
XieL-JZhouC-LXuS (2018) A new computational approach for the solutions of generalized pantograph-delay differential equations. Computational and Applied Mathematics37(2): 1756–1783.
YangYHuangY (2013) Spectral-collocation methods for fractional pantograph delay-integrodifferential equations. Advances in Mathematical Physics2013: 1–14.
49.
YuZ-H (2008) Variational iteration method for solving the multi-pantograph delay equation. Physics Letters A372(43): 6475–6479.
50.
YüzbaşıŞ.SezerM (2013) An exponential approximation for solutions of generalized pantograph-delay differential equations. Applied Mathematical Modelling37(22): 9160–9173.
51.
ZhuLFanQ (2012) Solving fractional nonlinear fredholm integro-differential equations by the second kind chebyshev wavelet. Communications in Nonlinear Science and Numerical Simulation17(6): 2333–2341.
