Implementing reproducing kernel method to solve singularly perturbed convection-diffusion parabolic problems
Abstract
In the present paper, reproducing kernel method (RKM) is introduced, which is employed to solve singularly perturbed convection-diffusion parabolic problems (SPCDPPs). It is noteworthy to mention that regarding very serve singularities, there are regular boundary layers in SPCDPPs. On the other hand, getting a reliable approximate solution could be difficult due to the layer behavior of SPCDPPs. The strategy developed in our method is dividing the problem region into two regions, so that one of them would contain a boundary layer behavior. For more illustrations of the method, certain linear and nonlinear SPCDPP are solved.
Keyword : reproducing kernel method, singularly perturbed parabolic problems, error analysis, boundary layer behavior
How to Cite
Abbasbandy, S., Sahihi, H., & Allahviranloo, T. (2021). Implementing reproducing kernel method to solve singularly perturbed convection-diffusion parabolic problems. Mathematical Modelling and Analysis, 26(1), 116-134. https://doi.org/10.3846/mma.2021.12057
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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A. Akgül. Solutions of the linear and nonlinear differential equations within the generalized fractional derivatives. Chaos. Interdiscipl. J. Nonlinear. Sci., 29(2):023108, 2019. https://doi.org/10.1063/1.5084035
A. Akgül, A. Cordero and J.R. Torregrosa. Solutions of fractional gas dynamics equation by a new technique. Math. Meth. in the Appl. Sci, 43:1349–1358, 2020. https://doi.org/10.1002/mma.5950
A. Akgül and M. Modanli. Crank-Nicholson difference method and reproducing kernel function for third order fractional differential equations in the sense of Atangana–Baleanu Caputo derivative. Chaos. Solit. Fract., 127:10–16, 2019. https://doi.org/10.1016/j.chaos.2019.06.011
B. Azarnavid, F. Parvaneh and S. Abbasbandy. Picard–reproducing kernel Hilbert space method for solving generalized singular nonlinear Lane-Emden type equations. Math. Model. Anal., 20:754–767, 2015. https://doi.org/10.3846/13926292.2015.1111953
E. Babolian and D. Hamedzadeh. A splitting iterative method for solving second kind integral equations in reproducing kernel spaces. J. Comput. Appl. Math., 326:204–216, 2017. https://doi.org/10.1016/j.cam.2017.05.025
E. Babolian, S. Javadi and E. Moradi. Error analysis of reproducing kernel Hilbert space method for solving functional integral equations. J. Comput. Appl. Math., 300:300–311, 2016. https://doi.org/10.1016/j.cam.2016.01.008
D. Baleanu, A. Fernandez and A. Akgül. On a fractional operator combining proportional and classical differintegrals. Mathematics, 8(3):360, 2020. https://doi.org/10.3390/math8030360
C. Clavero, J.C. Jorge and F. Lisbona. A uniformly convergent scheme on a nonuniform mesh for convection–diffusion parabolic problems. J. Comput. Appl. Math., 154:415–429, 2003. https://doi.org/10.1016/S0377-0427(02)00861-0
M.G. Cui and Y. Lin. Nonlinear Numerical Analysis in the Reproducing Kernel Space. Nova Science-Hauppauge-New York-United States, 2009.
R.E. Ewing and H. Wang. A summary of numerical methods for time dependent advection–dominated partial differential equations. J. Comp. Appl. Math., 128:423–445, 2001. https://doi.org/10.1016/B978-0-444-50616-0.50018-X
M. Fardi and M. Ghasemi. Solving nonlocal initial–boundary value problems for parabolic and hyperbolic integro–differential equations in reproducing kernel Hilbert space. Wiley. Online. Library., 33:174–198, 2017. https://doi.org/10.1002/num.22079
F.Z. Geng and S.P. Qian. Reproducing kernel method for singularly perturbed turning point problems having twin boundary layers. Appl. Math. Lett., 26:998– 1004, 2013. https://doi.org/10.1016/j.aml.2013.05.006
F.Z. Geng and S.P. Qian. Piecewise reproducing kernel method for singularly perturbed delay initial value problems. Appl. Math. Lett., 37:67–71, 2014. https://doi.org/10.1016/j.aml.2014.05.014
F.Z. Geng and S.P. Qian. Modified reproducing kernel method for singularly perturbed boundary value problems with a delay. Appl. Math. Model., 39:5592– 5597, 2015. https://doi.org/10.1016/j.apm.2015.01.021
S. Gowrisankar and S. Natesan. Robust numerical scheme for singularly perturbed convection–diffusion parabolic initial–boundary–value problems on equidistributed grids. Comput. Phys. Commun., 185:2008–2019, 2014. https://doi.org/10.1016/j.cpc.2014.04.004
W. Hundsdorfer and J. Verwer. Numerical solution of time–dependent advection– diffusion-reaction equations, volume 33 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 2003. https://doi.org/10.1007/978-3-662-09017-6
R. Ketabchi, R.Mokhtari and E.Babolian. Some error estimates for solving Volterra integral equations by using the reproducing kernel method. J. Comput. Appl. Math., 273:245–250, 2015. https://doi.org/10.1016/j.cam.2014.06.016
H.O. Kreiss and J. Lorenz. Initial–boundary value problems and the NavierStokes equations. Classics in Appl. Math. SIAM., 47, 2004. https://doi.org/10.1137/1.9780898719130
Z.Y. Li, Y.L. Wang, F.G. Tan, X.H. Wan, H. Yu and J.S. Duan. Solving a class of linear nonlocal boundary value problems using the reproducing kernel. Appl. Math. Comput., 265:1098–1105, 2015. https://doi.org/10.1016/j.amc.2015.05.117
T. Linß. Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems. Lecture Notes in Mathematics-Springer-Verlag-Berlin-Heidelberg, 1985.
P.A. Markowich, C. Ringhofer and S. Schmeiser. Semiconductor Equations. Springer-Vienna, 1990. https://doi.org/10.1007/978-3-7091-6961-2
H.G. Roos, M. Stynes and L. Tobiska. Robust Numerical Methods for Singularly Perturbed Differential Equations, Convection–Diffusion–Reaction and Flow Problems. Springer-Verlag Berlin Heidelberg, 2008.
H. Sahihi, T. Allahviranloo and S. Abbasbandy. Solving system of second–order BVPs using a new algorithm based on reproducing kernel Hilbert space. Appl. Num. Math., 151:27–39, 2020. https://doi.org/10.1016/j.apnum.2019.12.008
M.G. Sakar. Iterative reproducing kernel Hilbert spaces method for Riccati differential equations. J. Comput. Appl. Math., 309:163–174, 2017. https://doi.org/10.1016/j.cam.2016.06.029
M.G. Sakar, O. Saldr and F. Erdogan. An iterative approximation for timefractional Cahn-Allen equation with reproducing kernel method. Comput. Appl. Math., 37:5951–5964, 2018. https://doi.org/10.1007/s40314-018-0672-9
M.G. Sakar, O. Saldr and F. Erdogan. A hybrid method for singularly perturbed convection-diffusion equation. Int. J. Appl. Comput. Math., 5(135), 2019. https://doi.org/10.1007/s40819-019-0714-z
O. Saldr, M.G. Sakar and F. Erdogan. Numerical solution of time-fractional Kawahara equation using reproducing kernel method with error estimate. Comput. Appl. Math., 38(198), 2019. https://doi.org/10.1007/s40314-019-0979-1
R. Salmon. Lectures on Geophysical Fluid Dynamics. Oxford University PressNew York, 1998. https://doi.org/10.1093/oso/9780195108088.001.0001
M. Stynes and E. O’Riordan. Uniformly convergent difference schemes for singularly perturbed parabolic diffusion–convection problems without turning points. Numer. Math., 55:521–544, 1989. https://doi.org/10.1007/BF01398914
Y. Wang, T. Chaolu and Z. Chen. Using reproducing kernel for solving a class of singular weakly nonlinear boundary value problems. Int. J. Comput. Mathe., 87:367–380, 2010. https://doi.org/10.1080/00207160802047640
Y. Wang, T. Chaolu and P. Jing. New algorithm for second–order boundary value problems of integro–differential equation. J. Comp. Appl. Math., 229:1–6, 2009. https://doi.org/10.1016/j.cam.2008.10.007
Y. Wang, L. Su, X. Cao and X. Li. Using reproducing kernel for solving a class of singularly perturbed problems. Comput. Math. Appl., 61:421–430, 2011. https://doi.org/10.1016/j.camwa.2010.11.019