×

Numerical analysis of the fractional-order nonlinear system of Volterra integro-differential equations. (English) Zbl 07420392

Summary: This paper presents the nonlinear systems of Volterra-type fractional integro-differential equation solutions through a Chebyshev pseudospectral method. The proposed method is based on the Caputo fractional derivative. The results that we get show the accuracy and reliability of the present method. Different nonlinear systems have been solved; the solutions that we get are compared with other methods and the exact solution. Also, from the presented figures, it is easy to conclude that the CPM error converges quickly as compared to other methods. Comparing the exact solution and other techniques reveals that the Chebyshev pseudospectral method has a higher degree of accuracy and converges quickly towards the exact solution. Moreover, it is easy to implement the suggested method for solving fractional-order linear and nonlinear physical problems related to science and engineering.

MSC:

65R20 Numerical methods for integral equations
45J05 Integro-ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Loverro, A., Fractional calculus: history, definitions and applications for the engineer (2004), Rapport technique, Univeristy of Notre Dame: Department of Aerospace and Mechanical Engineering
[2] Bagley, R. L.; Torvik, P. J., Fractional calculus in the transient analysis of viscoelastically damped structures, AIAA Journal, 23, 6, 918-925 (1985) · Zbl 0562.73071 · doi:10.2514/3.9007
[3] Baillie, R. T.; Bollerslev, T.; Mikkelsen, H. O., Fractionally integrated generalized autoregressive conditional heteroskedasticity, Journal of Econometrics, 74, 1, 3-30 (1996) · Zbl 0865.62085 · doi:10.1016/S0304-4076(95)01749-6
[4] Mainardi, F., Fractional calculus: some basic problems in continuum and statistical mechanics (2012), https://arxiv.org/abs/1201.0863
[5] Chow, T. S., Fractional dynamics of interfaces between soft-nanoparticles and rough substrates, Physics Letters A, 342, 1-2, 148-155 (2005) · doi:10.1016/j.physleta.2005.05.045
[6] Rossikhin, Y. A.; Shitikova, M. V., Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Applied Mechanics Reviews, 50, 1, 15-67 (1997) · doi:10.1115/1.3101682
[7] Li, C.; Dao, X.; Guo, P., Fractional derivatives in complex planes, Nonlinear Analysis: Theory, Methods & Applications, 71, 5-6, 1857-1869 (2009) · Zbl 1173.26305 · doi:10.1016/j.na.2009.01.021
[8] Guariglia, E., Fractional calculus, zeta functions and Shannon entropy, Open Mathematics, 19, 1, 87-100 (2021) · Zbl 1475.11151 · doi:10.1515/math-2021-0010
[9] Sunthrayuth, P.; Aljahdaly, N. H.; Ali, A.; Shah, R.; Mahariq, I.; Tchalla, A. M., Φ-Haar Wavelet Operational Matrix Method for Fractional Relaxation-Oscillation Equations Containing Φ-Caputo Fractional Derivative, Journal of Function Spaces (2021) · Zbl 07420406
[10] Guariglia, E.; Silvestrov, S., Fractional-Wavelet Analysis of Positive definite Distributions and Wavelets on \(\mathcal{D}^\prime\left( \mathbb{C}\right)\), Engineering Mathematics II. Engineering Mathematics II, Springer Proceedings in Mathematics & Statistics, 337-353 (2016), Cham. Switzerland: Springer, Cham. Switzerland · Zbl 1365.65294
[11] Chen, Y.; Yan, Y.; Zhang, K., On the local fractional derivative, Journal of Mathematical Analysis and Applications, 362, 1, 17-33 (2010) · Zbl 1196.26011 · doi:10.1016/j.jmaa.2009.08.014
[12] Alessa, N.; Tamilvanan, K.; Loganathan, K.; Selvi, K. K., Hyers-Ulam Stability of Functional Equation Deriving from Quadratic Mapping in Non-Archimedean -Normed Spaces, Journal of Function Spaces, 2021 (2021) · Zbl 1466.39018 · doi:10.1155/2021/9953214
[13] Arfan, M.; Mahariq, I.; Shah, K.; Abdeljawad, T.; Laouini, G.; Mohammed, P. O., Numerical computations and theoretical investigations of a dynamical system with fractional order derivative, Alexandria Engineering Journal (2021)
[14] Kaur, D.; Agarwal, P.; Rakshit, M.; Chand, M., Fractional calculus involving (p, q)-Mathieu type series, Applied Mathematics and Nonlinear Sciences, 5, 2, 15-34 (2020) · Zbl 1524.26013 · doi:10.2478/amns.2020.2.00011
[15] Deng, W. H.; Li, C. P., Chaos synchronization of the fractional Lu system, Physica A: Statistical Mechanics and its Applications, 353, 61-72 (2005) · doi:10.1016/j.physa.2005.01.021
[16] Oldham, K. B., The reformulation of an infinite sum via semiintegration, SIAM Journal on Mathematical Analysis, 14, 5, 974-981 (1983) · Zbl 0527.26006 · doi:10.1137/0514076
[17] Hartley, T. T.; Lorenzo, C. F.; Killory Qammer, H., Chaos in a fractional order Chua’s system, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 42, 8, 485-490 (1995) · doi:10.1109/81.404062
[18] Baskin, E.; Iomin, A., Electro-chemical manifestation of nanoplasmonics in fractal media, Central European Journal of Physics, 11, 6, 676-684 (2013) · doi:10.2478/s11534-013-0266-5
[19] Povstenko, Y. Z., Thermoelasticity that uses fractional heat conduction equation, Journal of Mathematical Sciences, 162, 2, 296-305 (2009) · doi:10.1007/s10958-009-9636-3
[20] Bhrawy, A. H.; Zaky, M. A., A method based on the Jacobi tau approximation for solving multi-term time- space fractional partial differential equations, Journal of Computational Physics, 281, 876-895 (2015) · Zbl 1352.65386 · doi:10.1016/j.jcp.2014.10.060
[21] Amin, R.; Mahariq, I.; Shah, K.; Awais, M.; Elsayed, F., Numerical solution of the second order linear and nonlinear integro-differential equations using Haar wavelet method, Arab Journal of Basic and Applied Sciences, 28, 1, 11-19 (2021)
[22] Kulish, V. V.; Lage, J. L., Application of fractional calculus to fluid mechanics, Journal of Fluids Engineering, 124, 3, 803-806 (2002) · doi:10.1115/1.1478062
[23] Bapna, I. B.; Mathur, N., Application of fractional calculus in statistics, International Journal of Contemporary Mathematical Sciences, 7, 17-20, 849-856 (2012) · Zbl 1248.62028
[24] Kurt, A.; Şenol, M.; Tasbozan, O.; Chand, M., Two reliable methods for the solution of fractional coupled burgers’ equation arising as a model of polydispersive sedimentation, Applied Mathematics and Nonlinear Sciences, 4, 2, 523-534 (2019) · Zbl 1524.35699 · doi:10.2478/AMNS.2019.2.00049
[25] Agarwal, P.; Qi, F.; Chand, M.; Singh, G., Some fractional differential equations involving generalized hypergeometric functions, Journal of Applied Analysis, 25, 1, 37-44 (2019) · Zbl 1423.26010 · doi:10.1515/jaa-2019-0004
[26] Bonyah, E.; Atangana, A.; Chand, M., Analysis of 3D IS-LM macroeconomic system model within the scope of fractional calculus, Chaos, Solitons & Fractals: X, 2, article 100007 (2019) · doi:10.1016/j.csfx.2019.100007
[27] Chand, M.; Agarwal, P.; Hammouch, Z., Certain sequences involving product of k-Bessel function, International Journal of Applied and Computational Mathematics, 4, 4, 1-9 (2018) · Zbl 1400.33006 · doi:10.1007/s40819-018-0532-8
[28] Xu, L.; He, J.-H.; Liu, Y., Electrospun nanoporous spheres with Chinese drug, International Journal of Nonlinear Sciences and Numerical Simulation, 8, 2, 199-202 (2007) · doi:10.1515/IJNSNS.2007.8.2.199
[29] Wang, H.; Fu, H. M.; Zhang, H. F.; Hu, Z. Q., A practical thermodynamic method to calculate the best glass-forming composition for bulk metallic glasses, International Journal of Nonlinear Sciences and Numerical Simulation, 8, 2, 171-178 (2007) · doi:10.1515/IJNSNS.2007.8.2.171
[30] Bo, T.-L.; Xie, L.; Zheng, X. J., Numerical approach to wind ripple in desert, International Journal of Nonlinear Sciences and Numerical Simulation, 8, 2, 223-228 (2007) · doi:10.1515/IJNSNS.2007.8.2.223
[31] Sun, F. Z.; Gao, M.; Lei, S. H.; Zhao, Y. B.; Wang, K.; Shi, Y. T.; Wang, N. H., The fractal dimension of the fractal model of dropwise condensation and its experimental study, International Journal of Nonlinear Sciences and Numerical Simulation, 8, 2, 211-222 (2007) · doi:10.1515/IJNSNS.2007.8.2.211
[32] Akyuz-Dascioglu, A., Chebyshev polynomial solutions of systems of linear integral equations, Applied Mathematics and Computation, 151, 1, 221-232 (2004) · Zbl 1049.65149 · doi:10.1016/S0096-3003(03)00334-5
[33] Zedan, H. A.; Alaidarous, E., Haar wavelet method for the system of integral equations, Abstract and Applied Analysis, 2014 (2014) · Zbl 1470.65223 · doi:10.1155/2014/418909
[34] Almasieh, H.; Roodaki, M., Triangular functions method for the solution of Fredholm integral equations system, Ain Shams Engineering Journal, 3, 4, 411-416 (2012) · Zbl 1238.65134 · doi:10.1016/j.asej.2012.04.006
[35] Sahin, N.; Yuzbasi, S.; Gulsu, M., A collocation approach for solving systems of linear Volterra integral equations with variable coefficients, Computers & Mathematics with Applications, 62, 2, 755-769 (2011) · Zbl 1228.65248 · doi:10.1016/j.camwa.2011.05.057
[36] Sahu, P. K.; Ray, S. S., Legendre wavelets operational method for the numerical solutions of nonlinear Volterra integro-differential equations system, Applied Mathematics and Computation, 256, 715-723 (2015) · Zbl 1338.65208 · doi:10.1016/j.amc.2015.01.063
[37] Maleknejad, K.; Aghazadeh, N.; Rabbani, M., Numerical solution of second kind Fredholm integral equations system by using a Taylor-series expansion method, Applied Mathematics and Computation, 175, 2, 1229-1234 (2006) · Zbl 1093.65124 · doi:10.1016/j.amc.2005.08.039
[38] Roul, P.; Meyer, P., Numerical solutions of systems of nonlinear integro-differential equations by homotopy-perturbation method, Applied Mathematical Modelling, 35, 9, 4234-4242 (2011) · Zbl 1225.65081 · doi:10.1016/j.apm.2011.02.043
[39] Bushnaq, S.; Maayah, B.; Momani, S.; Alsaedi, A., A reproducing kernel Hilbert space method for solving systems of fractional integrodifferential equations, Abstract and Applied Analysis, 2014 (2014) · Zbl 1470.65123 · doi:10.1155/2014/103016
[40] Biazar, J., Solution of systems of integral-differential equations by Adomian decomposition method, Applied Mathematics and Computation, 168, 2, 1232-1238 (2005) · Zbl 1082.65594 · doi:10.1016/j.amc.2004.10.015
[41] Wang, Y.; Zhu, L., Solving nonlinear Volterra integro-differential equations of fractional order by using Euler wavelet method, Advances in Difference Equations, 2017, 1 (2017) · Zbl 1422.45001 · doi:10.1186/s13662-017-1085-6
[42] Saberi-Nadjafi, J.; Tamamgar, M., The variational iteration method: a highly promising method for solving the system of integro-differential equations, Computers & Mathematics with Applications, 56, 2, 346-351 (2008) · Zbl 1155.65399 · doi:10.1016/j.camwa.2007.12.014
[43] Ma, X.; Huang, C., Spectral collocation method for linear fractional integro-differential equations, Applied Mathematical Modelling, 38, 4, 1434-1448 (2014) · Zbl 1427.65421 · doi:10.1016/j.apm.2013.08.013
[44] T, O.; OA, T.; JU, A.; ZO, O., Numerical studies for solving fractional integro-differential equations by using least squares method and bernstein polynomials, Fluid Mechanics Open Access, 3, 3, 1-7 (2016) · doi:10.4172/2476-2296.1000142
[45] Abbasbandy, S.; Hashemi, M. S.; Hashim, I., On convergence of homotopy analysis method and its application to fractional integro-differential equations, Quaestiones Mathematicae, 36, 1, 93-105 (2013) · Zbl 1274.65229 · doi:10.2989/16073606.2013.780336
[46] Arikoglu, A.; Ozkol, I., Solutions of integral and integro-differential equation systems by using differential transform method, Computers & Mathematics with Applications, 56, 9, 2411-2417 (2008) · Zbl 1165.45300 · doi:10.1016/j.camwa.2008.05.017
[47] Yi, M.; Wang, L.; Huang, J., Legendre wavelets method for the numerical solution of fractional integro- differential equations with weakly singular kernel, Applied Mathematical Modelling, 40, 4, 3422-3437 (2016) · Zbl 1452.65144 · doi:10.1016/j.apm.2015.10.009
[48] He, J.-H.; Li, Z.-B.; Wang, Q.-l., A new fractional derivative and its application to explanation of polar bear hairs, Journal of King Saud University-Science, 28, 2, 190-192 (2016) · doi:10.1016/j.jksus.2015.03.004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.