Analytical and approximate solution of two-dimensional convection-diffusion problems
In this work, we have used reduced differential transform method (RDTM) to compute an approximate solution of the Two-Dimensional Convection-Diffusion equations (TDCDE). This method provides the solution quickly in the form of a convergent series. Also, by using RDTM the approximate solution of two-dimensional convection-diffusion equation is obtained. Further, we have computed exact solution of non-homogeneous CDE by using the same method. To the best of my knowledge, the research work carried out in the present paper has not been done, and is new. Examples are provided to support our work.
[1] Baskonus, H.M. (2019). Complex Soliton So-lutions to the Gilson-Pickering Model. Ax- ioms, 8(1), 18.
[2] Ilhan, O.A., Esen, A., Bulut, H., & Baskonus, H.M. (2019). Singular Solitons in the Pseudo- parabolic Model Arising in Nonlinear Surface Waves. Results in Physics, 12, 17121715.
[3] Cattani, C., Sulaiman, T.A., Baskonus, H.M., & Bulut, H. (2018). Solitons in an inhomo- geneous Murnaghan’s rod. European Physical Journal Plus, 133(228), 1-12.
[4] Baskonus, H.M., Sulaiman, T.A. & Bulut, H.(2018). Dark, bright and other optical solitons to the decoupled nonlinear Schrdinger equa- tion arising in dual-core optical fibers. Optical and Quantum Electronics, 50(4), 1-12.
[5] Ciancio, A., Baskonus, H.M., Sulaiman, T.A., & Bulut, H. (2018). New Structural Dynamics of Isolated Waves Via the Coupled Nonlinear Maccari’s System with Complex Structure. Indian Journal of Physics, 92(10), 12811290.
[6] Ilhan, O.A., Sulaiman, T.A., Bulut, H., & Baskonus, H.M. (2018). On the New Wave Solutions to a Nonlinear Model Arising in Plasma Physics. European Physical Journal Plus, 133(27), 1-6.
[7] Yel, G., Baskonus, H.M., & Bulut, H.(2017). Novel archetypes of new coupled Kon- noOono equation by using sineGordon expan- sion method. Optical and Quantum Electron- ics, 49(285), 1-10.
[8] Baskonus, H.M. (2017). New Complex and Hyperbolic Function Solutions to the Gen- eralized Double Combined Sinh-Cosh-Gordon Equation. AIP Conference Proceedings. 1798, 1-9 (020018).
[9] Baskonus, H.M. (2016). New acoustic wave behaviors to the DaveyStewartson equa- tion with power-law nonlinearity arising in 丑uid Dynamics. Nonlinear Dynamics, 86(1), 177183.
[10] Karaa, S., & Zhang, J. (2004). Higher order ADI method for solving unsteady convection- difusion problems. Journal of Computational Physics, 198, 1-9.
[11] Tian, Z. (2011). A rational high-order com- pact ADI method for unsteady convection- difusion equations. Computer Physics Com- munications, 182, 649-662.
[12] Rui, H. (2003). An alternative direction it- erative method with second-order upwind scheme for convection-difusion equations. In- ternational Journal of Computer Mathemat- ics, 80(4), 527-533.
[13] Mekuria, G.T., & Rao, J.A. (2016). Adaptive finite element method for steady convection- difusion equation. American Journal of Com- putational Mathematics, 6(3), 275-285.
[14] Li, L., Jiang, Z., & Yin, Z. (2018). Fourth- order compact finite diference method for solving two-dimensional convection-difusion equation. Advances in Di erence Equations, 2018, 1-24.
[15] Momani, S. (2008). A Decomposition Method for Solving Unsteady Convection- Difusion Problems. Turkish Journal of Mathematics, 32, 51-60.
[16] Saqib, M., Hasnain, S., & Mashat, D.S.(2017). Computational solutions of two di- mensional convection-difusion equation using Crank Nicolson and time e伍cient ADI. Amer- ican Journal of Computational Mathematics, 7(3), 208-227.
[17] Ismail, H.N.A., Elbarbary, E. M. E., & Salem, G. S. E. (2004). Restrictive Tay- lor’s approximation for solving convection- difusion equation. Applied Mathematics and Computation, 147, 355-363.
[18] Castillo, M., & Power, H. (2008). The Neumann series as a fundamental solution of the two-dimensional convection-difusion equation with variable velocity. Journal of Engineering Mathematics, 62, 189-202.
[19] Noye, B.J., & Tan, H.H. (1989). Finite diference methods for solving the two- dimensional advection-difusion equation. In- ternational Journal for Numerical Methods in Fluids, 9(1), 75-98.
[20] Sun, H., & Li, L. (2014). A CCD-ADI method for unsteady convection-difusion equations. Computer Physics Communica- tions, 185, 790-797.
[21] Kalita, J.C., Dalal, D.C., & Dass, A.K.(2002). A class of higher order compact schemes for the unsteady two-dimensional convection-difusion equation with variable convection coe伍cients. International Journal for Numerical Methods in Fluids, 38, 1111- 1131.
[22] Shu, C.W. (2017). Bound-preserving high or- der finite volume schemes for conservation laws and convection-difusion equations. Fi- nite Volumes for Complex Applications VIII- Methods and Theoretical Aspects, Springer Proceedings, 3-14.
[23] Ammi, M.R.S., & Jamiai, I. (2017). Finite diference and Legendre spectral method for a time-fractional difusion-convection equation for image restoration. Discrete & Continuous Dynamical Systems, 11(1), 103-117.
[24] Koley, U., Risebro, N.H., Schwab, C., et al.(2017). A multilevel monte carlo finite dif- ference method for random scalar degener- ate convection-difusion equations. Journal of Hyperbolic Di erential Equations, 14(3), 415- 454.
[25] Keskin, Y. (2010). Solving partial di erential equations by the reduced di erential transform method. PhD Thesis. Selcuk University.
[26] Ziqan, A.M., Armiti, S., & Suwan, L. (2016). Solving three-dimensional volterra integral equation by the reduced diferential trans- form method. International Journal of Ap- plied Mathematical Research, 5(2), 103-106.