Numerical solution of singularly perturbed 2D parabolic initial-boundary-value problems based on reproducing kernel theory: error and stability analysis. (English) Zbl 1533.65203
Summary: The main aim of this article is to propose two computational approaches on the basis of the reproducing kernel Hilbert space method for solving singularly perturbed 2D parabolic initial-boundary-value problems. For each approach, the solution in reproducing kernel Hilbert space is constructed with series form, and the approximate solution \(U_{m}\) is given as an \(m\)-term summation. Furthermore, convergence of the proposed approaches is presented which provides the theoretical basis of these approaches. Finally, some numerical experiments are considered to demonstrate the efficiency and applicability of proposed approaches.
{© 2020 Wiley Periodicals LLC.}
{© 2020 Wiley Periodicals LLC.}
MSC:
| 65M99 | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 65K10 | Numerical optimization and variational techniques |
| 35B25 | Singular perturbations in context of PDEs |
| 76S05 | Flows in porous media; filtration; seepage |
| 35A20 | Analyticity in context of PDEs |
| 35C05 | Solutions to PDEs in closed form |
| 68W30 | Symbolic computation and algebraic computation |
| 46E22 | Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) |
| 47B32 | Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces) |
Keywords:
approximate solution; convection-diffusion; convergence analysis; error analysis; reaction-diffusion; reproducing kernel; singularly perturbed problemsReferences:
| [1] | N.Aronszajn, Theory of reproducing kernels, Harvard University, Cambridge, MA, 1951. · Zbl 0037.20701 |
| [2] | O. A.Arqub and M.Al‐Smadi, Numerical algorithm for solving two‐point, second‐order periodic boundary value problems for mixed integro‐differential equations, Appl. Math. Comput.243 (2014), 911-922. · Zbl 1337.65083 |
| [3] | E.Babolian, S.Javadi, and E.Moradi, RKM for solving Bratu‐type differential equations of fractional order, Math. Methods Appl. Sci.39 (2015), 1548-1557. · Zbl 1350.65080 |
| [4] | 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 (2016), 300-311. · Zbl 1416.65527 |
| [5] | M.Cui and Z. H.Deng, Numerical functional method in reproducing kernel space, The Publication of Harbin Institute of Technology, China, 1988. |
| [6] | M.Cui and Y.Lin, Nonlinear numerical analysis in the reproducing kernel space, Nova Science, New York, 2009. · Zbl 1165.65300 |
| [7] | M.Cui and Y.Yan, The representation of the solution of a kind of operator equation Au = f, Numer. Math. A J. Chin. Univ.1 (1995), 82-86. · Zbl 0823.47011 |
| [8] | M.Fardi and M.Ghasemi, Solving nonlocal initial‐boundary value problems for parabolic and hyperbolic integro‐differential equations in reproducing kernel hilbert space, Numer. Methods Partial Differential Equations33 (2016), 174-198. · Zbl 1357.65317 |
| [9] | M.Fardi, R. K.Ghaziani, and M.Ghasemi, The reproducing kernel method for some variational problems depending on indefinite integrals, Math. Model. Anal.21 (2016), 412-429. · Zbl 1488.49015 |
| [10] | M.Foroutan, A.Ebadian, and R.Asadi, Reproducing kernel method in Hilbert spaces for solving the linear and nonlinear four‐point boundary value problems, Int. J. Comput. Math.95(10) (2017), 1-15. |
| [11] | F.Geng and S.Qian, Piecewise reproducing kernel method for singularly perturbed delay initial value problems, Appl. Math. Lett.37 (2014), 67-71. · Zbl 1321.34086 |
| [12] | M.Ghasemi, M.Fardi, and R. K.Ghaziani, Numerical solution of nonlinear delay differential equations of fractional order in reproducing kernel Hilbert space, Appl. Math. Comput.268 (2015), 815-831. · Zbl 1410.34185 |
| [13] | W.Jiang, M.Cui, and Y.Lin, Anti‐periodic solutions for Rayleigh‐type equations via the reproducing kernel Hilbert space method, Commun. Nonlinear Sci. Numer. Simul.15 (2010), 1754-1758. · Zbl 1222.65085 |
| [14] | O. A.Ladyženskaja, V. A.Solonnikov, and N. N.Uraĺceva, Linear and quasilinear equations of parabolic type, American Mathematical Society, Providence, 1968. · Zbl 0174.15403 |
| [15] | J.Mercer, Functions of positive and negative type and their connection with the theory of integral equations, Philos. Trans. R. Soc. Lond.209 (1909), 415-446. · JFM 40.0408.02 |
| [16] | M.Mohammadi and R.Mokhtari, Solving the generalized regularized long wave equation on the basis of a reproducing kernel space, J. Comput. Appl. Math.235 (2011), 4003-4014. · Zbl 1220.65143 |
| [17] | M.Mohammadi and R.Mokhtari, A reproducing kernel method for solving a class of nonlinear systems of PDEs, Math. Model. Anal.19 (2014), 180-198. · Zbl 1488.65548 |
| [18] | A. H.Nayfeh, Introduction to perturbation techniques, John Wiley & Sons, New York, 1981. · Zbl 0449.34001 |
| [19] | H.G.Roos, Stynes, M., and Tobiska, L., Robust numerical methods for singularly perturbed differential equations. Springer‐Verlag, Berlin, 2008. · Zbl 1155.65087 |
| [20] | S.Vahdati, M.Fardi, and M.Ghasemi, Option pricing using a computational method based on reproducing kernel, J. Comput. Appl. Math.328 (2018), 252-266. · Zbl 1405.91701 |
| [21] | Y.Wang et al., A new method for solving a class of mixed boundary value problems with singular coefficient, Appl. Math. Comput.217 (2010), 2768-2772. · Zbl 1202.65095 |
| [22] | M.Xu and Y.Lin, Simplified reproducing kernel method for fractional differential equations with delay, Appl. Math. Lett.52 (2016), 156-161. · Zbl 1330.65102 |
| [23] | S.Zaremba, Sur le calcul numerique des founctions demandness dans le problems de dirichlet et le problems hydrodynamique, Bull. Int. I Acad. Sci. Crac. (1908), 125-195. |
| [24] | Y.Zhou, M.Cui, and Y.Lin, Numerical algorithm for parabolic problems with non‐classical conditions, J. Comput. Appl. Math.230 (2009), 770-780. · Zbl 1190.65136 |
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