The Time Fractional Burger equation was solved in this study using the Mabel software and the Variational Iteration approach. where a number of instances of the Time Fractional Burger Equation were handled using this technique. Tables and images were used to present the collected numerical results. The difference between the exact and numerical solutions demonstrates the effectiveness of the Mabel program’s solution, as well as the accuracy and closeness of the results this method produced. It also demonstrates the Mabel program’s ability to quickly and effectively produce the numerical solution.
Dzherbashyan, M.M. and Nersesian, A.B. (1958) About Application of Some Integro-Differential Operators. Doklady Akademii Nauk (Proceedings of the Russian Academy of Sciences), 121, 210-213.
Dzherbashyan, M.M. and Nersesian, A.B. (1958) The Criterion of the Expansion of the Functions to Dirichlet Series. Izvestiya Akademii Nauk Armyanskoi SSR: Seriya Fiziko-Matemati-Cheskih Nauk, 11, 85-108.
Miller, K.S. and Ross, B. (1993) An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley & Sons, New York.
Oldham, K.B. and Spanier, J. (1974) The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order. Academic Press, New York.
Machado, J.T., Kiryakova, V. and Mainardi, F. (2011) Recent History of Fractional Calculus. Communications in Nonlinear Science and Numerical Simulation, 16, 1140-1153. https://doi.org/10.1016/j.cnsns.2010.05.027
Riemann, B. (1953) Versuch Einer Allgemeinen Auffassung der Integration und Differentiation. Gesammelte Mathematische Werke und Wissenschaftlicher Nachlass. Teubner, Leipzig, 1876, Dover, New York.
Grünwald, A.K. (1867) über “begrenzte” derivationen und deren anwendung. Zeitschrift für Mathematik und Physik, 12, 4441-4480.
Letnikov, A.V. (1868) Theory of Differentiation with an Arbitrary Index. Sbornik: Mathematics, 3, 1-66. (In Russian)
Weyl, H. (1917) Bemerkungen zum begriff des differentialquotien-ten gebrochener ordung vierteljahresschr. Naturforschende Gesellschaft in Zürich, 62, 296-302.
Riesz, M. (1949) L’intégrale de Riemann-Liouville et le probléme de Cauchy. Acta Mathematica, 81, 1-222. https://doi.org/10.1007/BF02395016
Riesz, M. (1939) L’intégrale de Riemann-Liouville et le probléme de Cauchy pour l’équation des ondes. Bulletin de la Société Mathématique de France, 67, 153-170. https://doi.org/10.24033/bsmf.1309
Caputo, M. (1967) Linear Models of Dissipation Whose q Is Almost Frequency Independent-II. Geophysical Journal of the Royal Astronomical Society, 13, 529-539. https://doi.org/10.1111/j.1365-246X.1967.tb02303.x
Figueiredo Camargo, R., Chiacchio, A.O. and Capelas de Oliveira, E. (2008) Differentiation to Fractional Orders and the Fractional Telegraph Equation. Journal of Mathematical Physics, 49, Article ID: 033505. https://doi.org/10.1063/1.2890375
Caponetto, R., Dongola, G., Fortuna, L. and Petras, I. (2010) Fractional Order Systems: Modeling and Control Applications. World Scientific, Singapore. https://doi.org/10.1142/7709
Davison, M. and Essex, C. (1998) Fractional Differential Equations and Initial Value Problems. The Mathematical Scientist, 23, 108-116.
Jumarie, G. (2005) On the Solution of the Stochastic Differential Equation of Exponential Growth Driven by Fractional Brownian Motion. Applied Mathematics Letters, 18, 817-826. https://doi.org/10.1016/j.aml.2004.09.012
Jumarie, G. (2012) An Approach to Differential Geometry of Fractional Order via Modified Riemann-Liouville Derivative. Acta Mathematica Sinica, 28, 1741-1768. https://doi.org/10.1007/s10114-012-0507-3
Jumarie, G. (2013) On the Derivative Chain-Rules in Fractional Calculus via Fractional Difference and Their Application to Systems Modelling. Central European Journal of Physics, 11, 617-633. https://doi.org/10.2478/s11534-013-0256-7
Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. (2006) Theory and Applications of Fractional Differential Equations. Vol. 204 of North-Holland Mathematics Studies. Elsevier, Amsterdam.
Monje, C.A., Chen, Y., Vinagre, B.M., Xue, D. and Feliu, V. (2010) Fractional-Order Systems and Controls: Fundamentals and Applications. Springer, London. https://doi.org/10.1007/978-1-84996-335-0
Podlubny, I. (1999) Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution, Vol. 198 of Mathematics in Science and Engineering. Academic Press, San Diego.
Liu, J.G. and Zhang, J. (2023) A New Approximate Method to the Time Fractional Damped Burger Equation. AIMS Mathematics, 8, 13317-13324. https://doi.org/10.3934/math.2023674
Burgers, J.M. (1948) A Mathematical Model Illustrating the Theory of Turbulence. Advances in Applied Mechanics. Academic Press, New York. https://doi.org/10.1016/S0065-2156(08)70100-5
Hussain, M., Haq, S., Ghafoor, A. and Ali, I. (2020) Numerical Solutions of Time-Fractional Coupled Viscous Burgers’ Equations Using Meshfree Spectral Method. Computational and Applied Mathematics, 39, Article No. 6. https://doi.org/10.1007/s40314-019-0985-3
Debnath, L. (2011) Nonlinear Partial Differential Equations for Scientists and Engineers. Springer Science & Business Media, Berlin. https://doi.org/10.1007/978-0-8176-8265-1
Garra, R. (2011) Fractional-Calculus Model for Temperature and Pressure Waves in Fluid-Saturated Porous Rocks. Physical Review E, 84, Article ID: 036605. https://doi.org/10.1103/PhysRevE.84.036605
Aghdam, Y.E., Mesgrani, H., Javidi, M. and Nikan, O. (2020) A Computational Approach for the Space-Time Fractional Advection—Diffusion Equation Arising in Contaminant Transport through Porous Media. Engineering with Computers, 37, 3615-3627. https://doi.org/10.1007/s00366-020-01021-y
Kumar, D. and Singh, J. (2020) Fractional Calculus in Medical and Health Science. CRC Press, Boca Raton. https://doi.org/10.1201/9780429340567
Khavari, M., Priyadarshi, A., Morton, J., Porfyrakis, K., Pericleous, K., Eskin, D. and Tzanakis, I. (2023) Cavitation-Induced Shock Wave Behaviour in Different Liquids. Ultrasonics Sonochemistry, 94, Article ID: 106328. https://doi.org/10.1016/j.ultsonch.2023.106328
Rezaei, I. and Vaghefi, M. (2023) Numerical Solution of the Three-Dimensional Burger’s Equation by Using the DQ-FD Combined Method in the Determination of the 3D Velocity of the Flow. Applied Water Science, 13, Article No. 5. https://doi.org/10.1007/s13201-022-01822-0
Momani, S. (2006) Non-Perturbative Analytical Solutions of the Space- and Time-Fractional Burgers Equations. Chaos Solitons Fractals, 28, 930-937. https://doi.org/10.1016/j.chaos.2005.09.002
Tian, L. and Yin, J. (2007) Shock-Peakon and Shock-Compacton Solutions for K(p,q) Equation by Variational Iteration Method. Journal of Computational and Applied Mathematics, 207, 46-52. https://doi.org/10.1016/j.cam.2006.07.026
Shahzad, F., Jamshed, W., Sajid, T., Shamshuddin, M.D., Safdar, R., Salawu, S.O. and Krawczuk, M. (2022) Electromagnetic Control and Dynamics of Generalized Burgers’ Nanoliquid Flow Containing Motile Microorganisms with Cattaneo-Christov Relations: Galerkin Finite Element Mechanism. Applied Sciences, 12, Article No. 8636. https://doi.org/10.3390/app12178636
Aksan, E.N. (2006) Quadratic B-Spline Finite Element Method for Numerical Solution of the Burgers’ Equation. Applied Mathematics and Computation, 174, 884-896. https://doi.org/10.1016/j.amc.2005.05.020
Kutluay, S. and Esen, A. (2004) A Lumped Galerkin Method for Solving the Burgers Equation. International Journal of Computer Mathematics, 81, 1433-1444. https://doi.org/10.1080/00207160412331286833
Abbasbandy, S. and Darvishi, M.T. (2005) A Numerical Solution of Burgers’ Equation by Modified Adomian Method. Applied Mathematics and Computation, 163, 1265-1272. https://doi.org/10.1016/j.amc.2004.04.061
Tomar, S., Singh, M., Vajravelu, K. and Ramos, H. (2023) Simplifying the Variational Iteration Method: A New Approach to Obtain the Lagrange Multiplier. Mathematics and Computers in Simulation, 204, 640-644. https://doi.org/10.1016/j.matcom.2022.09.003
Inokuti, M., Sekine, H. and Mura, T. (1978) General Use of the Lagrange Multiplier in Nonlinear Mathematical Physics. In: Nemat-Nasser, S., Ed., Variational Methods in the Mechanics of Solids, Pergamon Press, New York, 156-162. https://doi.org/10.1016/B978-0-08-024728-1.50027-6
He, J.H. (2006) Non-Perturbative Methods for Strongly Nonlinear Problems. Dissertation, Deverlag im Internet GmbH, Berlin.
He, J.H., Wan, Y.Q. and Guo, Q. (2004) An Iteration Formulation for Normalized Diode Characteristics. International Journal of Circuit Theory and Applications, 32, 629-632. https://doi.org/10.1002/cta.300
He, J.H. (2000) Variational Iteration Method for Autonomous Ordinary Differential Systems. Applied Mathematics and Computation, 114, 115-123. https://doi.org/10.1016/S0096-3003(99)00104-6
He, J.H. (1997) Variational Iteration Method for Delay Differential Equations. Communications in Nonlinear Science and Numerical Simulation, 2, 235-236. https://doi.org/10.1016/S1007-5704(97)90008-3
He, J.H. (1998) Approximate Analytical Solution for Seepage Flow with Fractional Derivatives in Porous Media. Computer Methods in Applied Mechanics and Engineering, 167, 57-68. https://doi.org/10.1016/S0045-7825(98)00108-X
Das, S. (2009) Analytical Solution of a Fractional Diffusion Equation by Variational, Iteration Method. Computers & Mathematics with Applications, 57, 483-487. https://doi.org/10.1016/j.camwa.2008.09.045
Yulita Molliq, R., Noorani, M.S.M. and Hashim, I. (2009) Variational Iteration Method for Fractional Heat- and Wave-Like Equations. Nonlinear Analysis: Real World Applications, 10, 1854-1869. https://doi.org/10.1016/j.nonrwa.2008.02.026
Inc, M. (2008) The Approximate and Exact Solutions of the Space-and-Time-Fractional Burgers Equations with Initial Conditions by Variational Iteration Method. Journal of Mathematical Analysis and Applications, 345, 476-484. https://doi.org/10.1016/j.jmaa.2008.04.007
He, J.H. (2003) Generalized Variational Principles in Fluids. Science and Culture Publishing House of China, Hong Kong. (In Chinese)