Abstract
In this paper, a Haar wavelets based numerical method to solve a system of linear or nonlinear fractional differential equations has been proposed. Numerous nontrivial test examples along with practical problems from fluid dynamics and chemical engineering have been considered to illustrate applicability of the proposed method. We have derived a theoretical error bound which plays a crucial role whenever the exact solution of the system is not known and also it guarantees the convergence of approximate solution to exact solution.
Acknowledgement
The authors thankfully acknowledge the anonymous reviewers for their insightful comments and valuable suggestions that helped us to present the paper in a much stronger form.
References
Atanackovic, T.M., and B. Stankovic. 2002. “Dynamics of a Viscoelastic Rod of Fractional Derivative Type.” Zeitschrift fur Angewandte Mathematik und Mechanik 82: 377–386.10.1002/1521-4001(200206)82:6<377::AID-ZAMM377>3.0.CO;2-MSearch in Google Scholar
Babolian, E., and A. Shahsavaran. 2009. “Numerical Solution of Nonlinear Fredholm Integral Equations of the Second Kind Using Haar Wavelets.” Journal of Computational and Applied Mathematics 225: 87–95.10.1016/j.cam.2008.07.003Search in Google Scholar
Biazar, J., and H. Ebrahimi. 2012. “Chebyshev Wavelets Approach for Nonlinear Systems of Volterra Integral Equations.” Computers & Mathematcs with Applications 63: 608–616.10.1016/j.camwa.2011.09.059Search in Google Scholar
Chang, R.Y., and M.L. Wang. 1984. “Legendre Polynomials Approximation to Dynamical Linear State Space Equations with Initial and Boundary Value Conditions.” International Journal of Control 40: 215–232.10.1080/00207178408933269Search in Google Scholar
Chen, C.F., and C.H. Hsiao. 1997. “Haar Wavelet Method for Solving Lumped and Distributed Parameter Systems.” IEEE Proceedings-Control Theory and Applications 144: 87–94.10.1049/ip-cta:19970702Search in Google Scholar
Chen, Y., M. Yi, and C. Yu. 2012. “Error Analysis for Numerical Solution of Fractional Differential Equation by Haar Wavelets Method.” Journal of Computing Science 3: 367–373.10.1016/j.jocs.2012.04.008Search in Google Scholar
Cheng, C.F., and Y.T. Tsay. 1977. “Walsh Operational Matrices for Fractional Calculus and Their Application to Distributed Systems.” Journal of the Franklin Institute 303: 267–284.10.1016/0016-0032(77)90029-1Search in Google Scholar
Diethelm, K., N.J. Ford, and A.D. Freed. 2002. “A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations.” Nonlinear Dynamics 29: 3–22.10.1023/A:1016592219341Search in Google Scholar
Diethelm, K., N.J. Ford, and A.D. Freed. 2004. “Detailed Error Analysis for a Fractional Adams Method.” Numerical Algorithms 36: 31–52.10.1023/B:NUMA.0000027736.85078.beSearch in Google Scholar
Diethelm, K. 2010. The Analysis of Fractional Differential Equations. Berlin: Springer.10.1007/978-3-642-14574-2Search in Google Scholar
Diethelm, K. 2013. “A Fractional Calculus Based Model for the Simulation of an Outbreak of Dengue Fever.” Nonlinear Dynamics 71: 613–619.10.1007/s11071-012-0475-2Search in Google Scholar
Ding, Y., and H. Ye. 2009. “A Fractional-Order Differential Equation Model of HIV Infection of CD4+ T-Cells.” Mathematical and Computer Modelling 50: 386–392.10.1016/j.mcm.2009.04.019Search in Google Scholar
Edwards, J.T., N.J. Ford, and A.C. Simpson. 2002. “The Numerical Solution of Linear Multi-term Fractional Differential Equations: Systems of Equations.” Journal of Computational and Applied Mathematics 148: 401–418.10.1016/S0377-0427(02)00558-7Search in Google Scholar
Erturk, V.S., and S. Momani. 2008. “Solving Systems of Fractional Differential Equations Using Differential Transform Method.” Journal of Computational and Applied Mathematics 215: 142–151.10.1016/j.cam.2007.03.029Search in Google Scholar
Ezzat, M.A. 2011. “Magneto-Thermoelasticity with Thermoelectric Properties and Fractional Derivative Heat Transfer.” Physica B: Condensed Matter 406: 30–35.10.1016/j.physb.2010.10.005Search in Google Scholar
Fa, K.S. 2005. “A Falling Body Problem Through the Air in View of the Fractional Derivative Approach.” Physica A: Statistical Mechanics and its Applications 350: 199–206.10.1016/j.physa.2004.11.041Search in Google Scholar
Gejji, V.D., and H. Jafari. 2007. “Analysis of a System of Nonautonomous Fractional Differential Equations Involving Caputo Derivatives.” Journal on Mathematical Analysis and Applications 328: 1026–1033.10.1016/j.jmaa.2006.06.007Search in Google Scholar
Heydari, M.H., M.R. Hooshmandasl, F. Mohammadi, and C. Cattani. 2014. “Wavelets Method for Solving Systems of Nonlinear Singular Fractional Volterra Integro-Differential Equations.” Communications in Nonlinear Science and Numerical Simulation 19: 37–48.10.1016/j.cnsns.2013.04.026Search in Google Scholar
Hsiao, C.H., and W.J. Wang. 1999. “State Analysis of Time-varying Singular Nonlinear Systems via Haar Wavelets.” Mathematics and Computers in Simulation 51: 91–100.10.1016/S0378-4754(99)00107-XSearch in Google Scholar
Hwang, C., and Y.P. Shih. 1981. “Laguerre Operational Matrices for Fractional Calculus and Applications.” International Journal of Control 34: 577–584.10.1080/00207178108922549Search in Google Scholar
Jiwari, R. 2012. “Haar Wavelet Quasilinearization Approach for Numerical Simulation of Burgers’ Equation.” Computer Physics Communications 183: 2413–2423.10.1016/j.cpc.2012.06.009Search in Google Scholar
Jiwari, R. 2015. “A Hybrid Numerical Scheme for the Numerical Solution of the Burgers equation.” Computer Physics Communications 188: 59–67.10.1016/j.cpc.2014.11.004Search in Google Scholar
Khader, M.M., and M.M. Babatin. 2014. “Numerical Treatment for Solving Fractional SIRC Model and Influenza A.” Journal of Computational and Applied Mathematics 33: 543–556.10.1007/s40314-013-0079-6Search in Google Scholar
Khan, M. 2009. “The Rayleigh-Stokes Problem for an Edge in a Viscoelastic Fluid with a Fractional Derivative Model.” Nonlinear Analysis: Real World Applications 10: 3190–3195.10.1016/j.nonrwa.2008.10.002Search in Google Scholar
Khan, M., and S. Wang. 2009. “Flow of a Generalized Second-Grade Fluid Between Two Side Walls Perpendicular to a Plate with a Fractional Derivative Model.” Nonlinear Analysis: Real World Applications 10: 203–208.10.1016/j.nonrwa.2007.08.024Search in Google Scholar
Khan, N.A., A. Aray, and A. Mahmood. 2010. “Approximate Solution of Time-Fractional Chemical Engineering Equations: A Comparative Study.” International Journal of Chemical Reactor Engineering 8:. Article A19. DOI:. DOI:10.2202/1542-6580.2156.Search in Google Scholar
Li, Y., N. Sun, B. Zheng, Q. Wang, and Y. Zhang. 2014. “Wavelet Operational Matrix Method for Solving the Riccati Differential Equation.” Communications in Nonlinear Science and Numerical Simulation 19: 483–493.10.1016/j.cnsns.2013.05.022Search in Google Scholar
Mainardi, F. 1997. “Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics.”. In Carpinteri, A., and F. Mainardi (Eds.), In Fractals and Fractional Calculus in Continuum mechanics. 291–348. NewYork: Springer-Verlag.10.1007/978-3-7091-2664-6_7Search in Google Scholar
Miller, K.S., and B. Ross. 1993. An Introduction to the Fractional Calculus and Fractional Differential Equations. New York: John Wiley and Sons Inc.Search in Google Scholar
Mittal, R.C., and S. Pandit. 2017. “Sensitivity Analysis of Shock Wave Burgers Equation via a Novel Algorithm Based on scale-3 Haar Wavelets.” International Journal of Computer Mathematics. DOI:http://dx.doi.org/10.1080/00207160.2017.1293820.Search in Google Scholar
Mittal, R.C., and S. Pandit. 2017. “Numerical Simulation of Unsteady Squeezing Nanofluid and Heat Flow Between Two Parallel Plates Using Wavelets.” International Journal of Thermal Sciences 118: 410–422.10.1016/j.ijthermalsci.2017.04.019Search in Google Scholar
Momani, S., and K. Al-Khaled. 2005. “Numerical Solutions for Systems of Fractional Differential Equations by the Decomposition Method.” Applied Mathematics and Computing 162: 1351–1365.10.1016/j.amc.2004.03.014Search in Google Scholar
Oldham, K.B., and J. Spanier. 1974. Fractional Calculus: Theory and Applications, Differentiation and Integration to Arbitrary Order. New York-London: Academic Press Inc.Search in Google Scholar
Pandey, R.K., O.P. Singh, and V.K. Singh. 2010. “A Stable Algorithm for Numerical Evaluation of Hankel Transforms Using Haar Wavelets.” Numerical Algorithms 53: 451–466.10.1007/s11075-009-9313-0Search in Google Scholar
Pandit, S., and M. Kumar. 2014. “Haar Wavelet Approach for Numerical Solution of Two Parameters Singularly Purturbed Boundary Value Problems.” Applied Mathematics & Information Sciences 8: 2965–2974.10.12785/amis/080634Search in Google Scholar
Paraskevopoulos, P.N. 1983. “Chebyshev Series Approach to System Identification, Analysis and Optimal Control.” Journal of the Franklin Institute 316: 135–157.10.1016/0016-0032(83)90082-0Search in Google Scholar
Paraskevopoulos, P.N., P.D. Sparcis, and S.G. Monroursos. 1985. “The Fourier Series Operational Matrix of Integration.” International Journal of Systems Science 16: 171–176.10.1080/00207728508926663Search in Google Scholar
Podlubny, I. 1999. Fractional Differential Equations. California, USA: Academic Press.Search in Google Scholar
Razzaghi, M., and S. Yousefi. 2001. “The Legendre Wavelets Operational Matrix of Integration.” International Journal of Systems Science 32: 495–502.10.1080/00207720120227Search in Google Scholar
Rehman, M.u., and R.A. Khan. 2011. “The Legendre Wavelet Method for Solving Fractional Differential Equations.” Communications in Nonlinear Science and Numerical Simulation 16: 4163–4173.10.1016/j.cnsns.2011.01.014Search in Google Scholar
Rehman, M.U., and U. Saeed. 2015. “Gegenbauer Wavelets Operational Matrix Method for Fractional Differential Equations.” Journal of the Korean Mathematical Society 52: 1069–1096.10.4134/JKMS.2015.52.5.1069Search in Google Scholar
Saeedi, H., M.M. Moghadam, N. Mollahasani, and G.N. Chuev. 2011. “A CAS Wavelet Method for Solving Nonlinear Fredholm Integro-Differential Equations of Fractional Order.” {16: 1154–1163. Communications in Nonlinear Science and Numerical Simulation.10.1016/j.cnsns.2010.05.036Search in Google Scholar
Saeed, U., and M.U. Rehman. 2013. “Haar Wavelet-Quasilinearization Technique for Fractional Nonlinear Differential Equations.” Applied Mathematics and Computation 220: 630–648.10.1016/j.amc.2013.07.018Search in Google Scholar
Saeed, U., and M.U. Rehman. 2015. “Haar Wavelet Picard Method for Fractional Nonlinear Partial Differential Equations.” Applied Mathematics and Computing 264: 310–322.10.1016/j.amc.2015.04.096Search in Google Scholar
Samko, S.G., A.A. Kilbas, and O.I. Marichev. 1993. Fractional Integrals and Derivatives. Yverdon, Switzerland: Gordon and Breach Science Publishers.Search in Google Scholar
Aziz Siraj-ul-Islam, I., and B. Šarler. 2010. “The Numerical Solution of Second-Order Boundary-Value Problems by Collocation Method with the Haar Wavelets.” Mathematical and Computer Modelling 52: 1577–1590.10.1016/j.mcm.2010.06.023Search in Google Scholar
Siraj-ul-Islam, I Aziz, and M. Fayyaz. 2013. “A New Approach for Numerical Solution of Integro-Differential Equations Via Haar Wavelets.” International Journal of Computer Mathematics 90: 1971–1989.10.1080/00207160.2013.770481Search in Google Scholar
Siraj-ul-Islam, I Aziz 2013. “New Algorithms for the Numerical Solution of Nonlinear Fredholm and Volterra Integral Equations Using Haar Wavelets.” Journal of Computational and Applied Mathematics 239: 333–345.10.1016/j.cam.2012.08.031Search in Google Scholar
Siraj-ul-Islam, Aziz I., and A.S. Al-Fhaid. 2014. “An Improved Method Based on Haar Wavelets for Numerical Solution of Nonlinear Integral and Integro-Differential Equations of First and Higher Orders.” Journal of Computational and Applied Mathematics 260: 449–469.10.1016/j.cam.2013.10.024Search in Google Scholar
Wang, Y., and Q. Fan. 2012. “The Second Kind Chebyshev Wavelet Method for Solving Fractional Differential Equations.” Applied Mathematics and Computation 218: 8592–8601.10.1016/j.amc.2012.02.022Search in Google Scholar
Zhi, S., and C. Yong-yan. 2011. “A spectral Collocation Method Based on Haar Wavelets for Poisson Equations and Biharmonic Equations.” Mathematical and Computer Modelling 54: 2858–2868.10.1016/j.mcm.2011.07.006Search in Google Scholar
Zhu, L., and Q. Fan. 2012. “Solving Fractional Nonlinear Fredholm Integro-Differential Equations by the Second Kind Chebyshev Wavelet.” Communications in Nonlinear Science and Numerical Simulation 17: 2333–2341.10.1016/j.cnsns.2011.10.014Search in Google Scholar
Zhu, L., and Q. Fan. 2013. “Numerical Solution of Nonlinear Fractional-Order Volterra integro-Differential Equations by SCW.” Communications in Nonlinear Science and Numerical Simulation 18: 1203–1213.10.1016/j.cnsns.2012.09.024Search in Google Scholar
© 2017 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Editorial
- Special Issue in Honor of J. Alberto Ochoa-Tapia
- Research Articles
- The Scientific Achievements of J. Alberto Ochoa-Tapia
- Volume Averaged Equations for Mass Transport and Reaction for In-Situ Combustion
- Non-Isothermal Effectiveness Factor for Catalytic Particles with Non-Fickian Diffusion
- Finite Time Estimation for Switched Nonlinear Systems: Application to Stirred Tank Bioreactor
- Numerical Solution for a System of Fractional Differential Equations with Applications in Fluid Dynamics and Chemical Engineering
- Characterization and Evaluation of Sorbent Materials Obtained from Orange Peel as an Alternative of Sustainable Materials for Water Treatment
- Metal complexes supported on activated carbon as catalysts for the hydrogenation of anthracene
- Iron Supported on Ion Exchange Resin as Source of Iron for Fenton Reagent: A Heterogeneous or a Homogeneous Fenton Reagent Generation?
- Stokes Flow Inside Topographically Patterned Microchannel Using Boundary Element Method
- Degradation and Mineralization of a Cationic Dye by a Sequential Photo-Sono Catalytic Process
- Preparation of Mo/HZSM-5/Bentonite Catalyst for Methane Aromatization in a Fluidized Bed Reactor
- On the Understanding of the Adsorption of 2-Phenylethanol on Polyurethane-Keratin based Membranes
- La-, Mn- and Fe-Doped Zirconia Washcoats Deposited on Monolithic Reactors via Sol-Gel Method: Characterization and Evaluation of their Mass Transfer Phenomena and Kinetics in Trichloroethylene Combustion
- State Estimation Based on Nonlinear Observer for Hydrogen Production in a Photocatalytic Anaerobic Bioreactor
Articles in the same Issue
- Editorial
- Special Issue in Honor of J. Alberto Ochoa-Tapia
- Research Articles
- The Scientific Achievements of J. Alberto Ochoa-Tapia
- Volume Averaged Equations for Mass Transport and Reaction for In-Situ Combustion
- Non-Isothermal Effectiveness Factor for Catalytic Particles with Non-Fickian Diffusion
- Finite Time Estimation for Switched Nonlinear Systems: Application to Stirred Tank Bioreactor
- Numerical Solution for a System of Fractional Differential Equations with Applications in Fluid Dynamics and Chemical Engineering
- Characterization and Evaluation of Sorbent Materials Obtained from Orange Peel as an Alternative of Sustainable Materials for Water Treatment
- Metal complexes supported on activated carbon as catalysts for the hydrogenation of anthracene
- Iron Supported on Ion Exchange Resin as Source of Iron for Fenton Reagent: A Heterogeneous or a Homogeneous Fenton Reagent Generation?
- Stokes Flow Inside Topographically Patterned Microchannel Using Boundary Element Method
- Degradation and Mineralization of a Cationic Dye by a Sequential Photo-Sono Catalytic Process
- Preparation of Mo/HZSM-5/Bentonite Catalyst for Methane Aromatization in a Fluidized Bed Reactor
- On the Understanding of the Adsorption of 2-Phenylethanol on Polyurethane-Keratin based Membranes
- La-, Mn- and Fe-Doped Zirconia Washcoats Deposited on Monolithic Reactors via Sol-Gel Method: Characterization and Evaluation of their Mass Transfer Phenomena and Kinetics in Trichloroethylene Combustion
- State Estimation Based on Nonlinear Observer for Hydrogen Production in a Photocatalytic Anaerobic Bioreactor