On using sinc collocation approach for solving a parabolic PDE with nonlocal boundary conditions
-
1411
Downloads
-
3083
Views
Authors
Mohamed El-Gamel
- Department of Mathematical Sciences, Faculty of Engineering, Mansoura University, Egypt.
Mahmoud Abd El-Hady
- Department of Mathematical Sciences, Faculty of Engineering, Mansoura University, Egypt.
Abstract
This work suggests a simple method based on a
sinc approximation at sinc nodes
for solving parabolic partial differential equations
with nonlocal boundary conditions. Sinc approximation are typified by errors of the form O\(\left(e^{-k/h}\right)\),
where \(k > 0\) is a constant and \(h\) is a step size. Some numerical examples are utilized to
reveal the efficaciousness and precision of this method. The suggested method is flexible, easy to programme and efficient.
Share and Cite
ISRP Style
Mohamed El-Gamel, Mahmoud Abd El-Hady, On using sinc collocation approach for solving a parabolic PDE with nonlocal boundary conditions, Journal of Nonlinear Sciences and Applications, 14 (2021), no. 1, 29--38
AMA Style
El-Gamel Mohamed, Abd El-Hady Mahmoud, On using sinc collocation approach for solving a parabolic PDE with nonlocal boundary conditions. J. Nonlinear Sci. Appl. (2021); 14(1):29--38
Chicago/Turabian Style
El-Gamel, Mohamed, Abd El-Hady, Mahmoud. "On using sinc collocation approach for solving a parabolic PDE with nonlocal boundary conditions." Journal of Nonlinear Sciences and Applications, 14, no. 1 (2021): 29--38
Keywords
- Sinc function
- nonlocal
- collocation
- numerical solutions
MSC
References
-
[1]
A. Abdrabou, M. El-Gamel, On the sinc-Galerkin method for triharmonic boundary-value problems, Comput. Math. Appl., 76 (2018), 520--533
-
[2]
W. Allegretto, Y. P. Lin, A. Zhou, A box scheme for coupled systems resulting from microsensor thermistor problems, Dynam. Contin. Discrete Impuls. Systems, 5 (1999), 209--223
-
[3]
M. Bastani, D. k. Khojasteh, Numerical studies of a non-local parabolic partial differential equations by spectral collocation method with preconditioning, Comput. Math. Modeling, 24 (2013), 81--89
-
[4]
B. Bialecki, Sinc-collocation methods for two-point boundary value problems, IMA J. Numer. Anal., 11 (1991), 357--375
-
[5]
A. Bouziani, Mixed problem with boundary integral conditions for a certain parabolic equation, J. Appl. Math. Stochastic Anal., 9 (1996), 323--330
-
[6]
A. Bouziani, Strong solution for a mixed problem with nonlocal condition for a certain pluriparabolic equations, Hiroshima Math. J., 27 (1997), 373--390
-
[7]
A. Bouziani, On a class of parabolic equations with a nonlocal boundary condition, Acad. Roy. Belg. Bull. Cl. Sci. (6), 10 (1999), 61--77
-
[8]
A. Bouziani, N. Merazga, S. Benamira, Galerkin method applied to a parabolic evolution problem with nonlocal boundary conditions, Nonlinear Anal., 69 (2008), 1515--1524
-
[9]
J. R. Cannon, The solution of the heat equation subject to the specification of energy, Quart. Appl. Math., 21 (1963), 155--160
-
[10]
J. R. Cannon, Y. P. Lin, S. M. Wang, An implicit finite difference scheme for the diffusion equation subject to mass specification, Int. J. Engrg. Sci., 28 (1990), 573--578
-
[11]
J. R. Cannon, A. L. Matheson, A numerical procedure for diffusion subject to the specification of mass, Int. J. Engrg. Sci., 31 (1993), 347--355
-
[12]
J. R. Cannon, S. Perez Esteva, J. van der Hoek, A Galerkin procedure for the diffusion equation subject to the specification of mass, SIAM J. Numer. Anal., 24 (1987), 499--515
-
[13]
J. H. Cushman, T. R. Ginn, Nonlocal dispersion in porous media with continuously evolving scales of heterogeneity, Transp. Porous Media, 13 (1993), 123--138
-
[14]
J. H. Cushman, H. X. Xu, F. W. Deng, Nonlocal reactive transport with physical and chemical heterogeneity: localization error, Water Resources Res., 31 (1995), 2219--2237
-
[15]
G. Dagan, The significance of heterogeneity of evolving scales to transport in porous formations, Water Resources Res., 13 (1994), 3327--3336
-
[16]
W. A. Day, A decreasing property of solutions of parabolic equations with applications to thermoelasticity, Quart. Appl. Math., 40 (1982), 468--475
-
[17]
W. A. Day, Extensions of a property of the heat equation to linear thermoelasticity and other theories, Quart. Appl. Math., 41 (1982/83), 319--330
-
[18]
G. Ekolin, Finite difference methods for a nonlocal boundary value problem for the heat equation, BIT, 31 (1991), 245--261
-
[19]
M. El-Gamel, A numerical scheme for solving nonhomogeneous time-dependent problems, Z. Angew. Math. Phys., 57 (2006), 369--383
-
[20]
M. El-Gamel, Numerical solution of Troesch's problem by sinc-collocation method, Appl. Math., 4 (2013), 707--712
-
[21]
M. El-Gamel, A note on solving the fourth-order parabolic equation by the sinc-Galerkin method, Calcolo, 52 (2015), 327--342
-
[22]
M. El-Gamel, Error analysis of sinc-Galerkin method for time-dependent partial differential equations, Numer. Algorithms, 77 (2018), 517--533
-
[23]
M. El-Gamel, A. Abdrabou, Sinc-Galerkin solution to eighth-order boundary value problems, SeMA J., 76 (2019), 249--270
-
[24]
M. El-Gamel, A. I. Zayed, A comparison between the wavelet-Galerkin and the Sinc-Galerkin methods in solving nonhomogeneous heat equations, in: Inverse Problem, Image Analysis, and Medical Imaging, 2002 (2002), 97--116
-
[25]
R. Ewing, R. Lazarov, Y. P. Lin, Finite volume element approximations of nonlocal reactive flows in porous media, Numer. Methods Partial Differential Equations, 16 (2000), 285--311
-
[26]
G. Fairweather, R. D. Saylor, The reformulation and numerical solution of certain nonclassical initial-boundary value problems, SIAM J. Sci. Statist. Comput., 12 (1991), 127--144
-
[27]
A. Golbabai, M. Javidi, A numerical solution for nonclassical parabolic problem based on Chebyshev spectral collocation method, Appl. Math. Comput., 190 (2007), 179--185
-
[28]
V.-M. Hokkanen, G. Morosanu, Functional Methods in Differential Equations, Chapman & Hall/CRC, Boca Raton (2002)
-
[29]
N. Ionkin, Solution of a boundary value problem in heat conduction with a non-classical boundary condition, Differential Equations, 13 (1977), 204--211
-
[30]
J. Lund, K. L. Bowers, Sinc Methods for Quadrature and Differential Equations, SIAM, Philadelphia (1992)
-
[31]
A. M. Nakhushev, On certain approximate method for boundary-value problems for differential equations and its applications in ground waters dynamics, Differ. Uravn., 18 (1982), 72--81
-
[32]
M. K. Ng, Fast iterative methods for symmetric sinc-Galerkin system, IMA J. Numer. Anal., 19 (1999), 357--373
-
[33]
A. K. Pani, A finite element method for a diffusion equation with constrained energy and nonlinear boundary conditions, J. Austral. Math. Soc. Ser. B, 35 (1993), 87--102
-
[34]
M. Renardy, W. J. Hrusa, J. A. Nohel, Mathematical Problems in Viscoelasticity, John Wiley & Sons, New York (1987)
-
[35]
A. Samarskii, Some problems in differential equations theory, Differential Equations, 16 (1980), 1221--1228
-
[36]
R. C. Smith, G. A. Bogar, K. L. Bowers, J. Lund, The Sinc-Galerkin method for fourth-order differential equations, SIAM J. Numer. Anal., 28 (1991), 760--788
-
[37]
F. Stenger, A "sinc-Galerkin" method of solution of boundary value problems, Math. Comput., 33 (1979), 85--109
-
[38]
F. Stenger, Numerical Methods Based on Sinc and Analytic Functions, Springer-Verlag, New York (1993)
-
[39]
F. Stenger, Summary of sinc numerical methods, J. Comput. Appl. Math., 121 (2000), 379--420
-
[40]
M. Tadi, M. Radenkovic, A numerical method for 1-D parabolic equation with nonlocal boundary conditions, Int. J. Comput. Math., 2014 (2014), 9 pages
-
[41]
A. S. Vasudeva Murthy, J. G. Verwer, Solving parabolic integro-differential equations by an explicit integration method, J. Comput. Appl. Math., 39 (1992), 121--132
-
[42]
V. Vodakhova, A boundary-value problem with nonlocal condition for certain pseudo-parabolic water-transfer equation, Differentsialnie Uravnenia, 18 (1982), 280--285
-
[43]
G. Y. Yin, Sinc-Collocation method with orthogonalization for singular problem-like poisson, Math. Comp., 62 (1994), 21--40
-
[44]
S. A. Yousefi, M. Behroozifar, M. Dehghan, The operational matrices of Bernstein polynomials for solving the parabolic equation subject to specification of the mass, J. Comput. Appl. Math., 235 (2011), 5272--5283
-
[45]
R. Zolfaghari, A. Shidfar, Solving a parabolic PDE with nonlocal boundary conditions using the Sinc method, Numer. Algorithms, 62 (2013), 411--427
