Geodesic
On using sinc collocation approach for solving a parabolic PDE with nonlocal boundary conditions
El-Gamel, Mohamed 1 ; Abd El-Hady, Mahmoud 1
1 Department of Mathematical Sciences, Faculty of Engineering, Mansoura University, Egypt
Journal of nonlinear sciences and its applications, Tome 14 (2021) no. 1, p. 29-38.

Voir la notice de l'article provenant de la source International Scientific Research Publications

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.
DOI : 10.22436/jnsa.014.01.04
Classification : 65L60, 45J05
Keywords: Sinc function, nonlocal, collocation, numerical solutions

El-Gamel, Mohamed  1 ; Abd El-Hady, Mahmoud  1

1 Department of Mathematical Sciences, Faculty of Engineering, Mansoura University, Egypt 
@article{JNSA_2021_14_1_a3,
     author = {El-Gamel, Mohamed  and Abd El-Hady, Mahmoud },
     title = {On using  sinc collocation approach for solving  a parabolic {PDE} with nonlocal boundary 	conditions},
     journal = {Journal of nonlinear sciences and its applications},
     pages = {29-38},
     publisher = {mathdoc},
     volume = {14},
     number = {1},
     year = {2021},
     doi = {10.22436/jnsa.014.01.04},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.22436/jnsa.014.01.04/}
}
TY  - JOUR
AU  - El-Gamel, Mohamed 
AU  - Abd El-Hady, Mahmoud 
TI  - On using  sinc collocation approach for solving  a parabolic PDE with nonlocal boundary 	conditions
JO  - Journal of nonlinear sciences and its applications
PY  - 2021
SP  - 29
EP  - 38
VL  - 14
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.22436/jnsa.014.01.04/
DO  - 10.22436/jnsa.014.01.04
LA  - en
ID  - JNSA_2021_14_1_a3
ER  - 
%0 Journal Article
%A El-Gamel, Mohamed 
%A Abd El-Hady, Mahmoud 
%T On using  sinc collocation approach for solving  a parabolic PDE with nonlocal boundary 	conditions
%J Journal of nonlinear sciences and its applications
%D 2021
%P 29-38
%V 14
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.22436/jnsa.014.01.04/
%R 10.22436/jnsa.014.01.04
%G en
%F JNSA_2021_14_1_a3
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 its applications, Tome 14 (2021) no. 1, p. 29-38. doi : 10.22436/jnsa.014.01.04. http://geodesic.mathdoc.fr/articles/10.22436/jnsa.014.01.04/

[1] Abdrabou, A.; El-Gamel, M. On the sinc-Galerkin method for triharmonic boundary-value problems, Comput. Math. Appl., Volume 76 (2018), pp. 520-533 | DOI | Zbl

[2] Allegretto, W.; Lin, Y. P.; Zhou, A. A box scheme for coupled systems resulting from microsensor thermistor problems, Dynam. Contin. Discrete Impuls. Systems, Volume 5 (1999), pp. 209-223 | Zbl

[3] Bastani, M.; Khojasteh, D. k. Numerical studies of a non-local parabolic partial differential equations by spectral collocation method with preconditioning, Comput. Math. Modeling, Volume 24 (2013), pp. 81-89

[4] Bialecki, B. Sinc-collocation methods for two-point boundary value problems, IMA J. Numer. Anal., Volume 11 (1991), pp. 357-375 | DOI | Zbl

[5] Bouziani, A. Mixed problem with boundary integral conditions for a certain parabolic equation, J. Appl. Math. Stochastic Anal., Volume 9 (1996), pp. 323-330 | DOI | Zbl

[6] Bouziani, A. Strong solution for a mixed problem with nonlocal condition for a certain pluriparabolic equations, Hiroshima Math. J., Volume 27 (1997), pp. 373-390 | Zbl

[7] Bouziani, A. On a class of parabolic equations with a nonlocal boundary condition, Acad. Roy. Belg. Bull. Cl. Sci. (6), Volume 10 (1999), pp. 61-77 | Zbl

[8] Bouziani, A.; Merazga, N.; Benamira, S. Galerkin method applied to a parabolic evolution problem with nonlocal boundary conditions, Nonlinear Anal., Volume 69 (2008), pp. 1515-1524 | Zbl | DOI

[9] Cannon, J. R. The solution of the heat equation subject to the specification of energy, Quart. Appl. Math., Volume 21 (1963), pp. 155-160 | Zbl

[10] Cannon, J. R.; Lin, Y. P.; Wang, S. M. An implicit finite difference scheme for the diffusion equation subject to mass specification, Int. J. Engrg. Sci., Volume 28 (1990), pp. 573-578 | Zbl | DOI

[11] Cannon, J. R.; Matheson, A. L. A numerical procedure for diffusion subject to the specification of mass, Int. J. Engrg. Sci., Volume 31 (1993), pp. 347-355 | Zbl | DOI

[12] Cannon, J. R.; Esteva, S. Perez; Hoek, J. van der A Galerkin procedure for the diffusion equation subject to the specification of mass, SIAM J. Numer. Anal., Volume 24 (1987), pp. 499-515 | DOI | Zbl

[13] Cushman, J. H.; Ginn, T. R. Nonlocal dispersion in porous media with continuously evolving scales of heterogeneity, Transp. Porous Media, Volume 13 (1993), pp. 123-138

[14] Cushman, J. H.; Xu, H. X.; Deng, F. W. Nonlocal reactive transport with physical and chemical heterogeneity: localization error, Water Resources Res., Volume 31 (1995), pp. 2219-2237

[15] Dagan, G. The significance of heterogeneity of evolving scales to transport in porous formations, Water Resources Res., Volume 13 (1994), pp. 3327-3336

[16] Day, W. A. A decreasing property of solutions of parabolic equations with applications to thermoelasticity, Quart. Appl. Math., Volume 40 (1982), pp. 468-475 | Zbl | DOI

[17] Day, W. A. Extensions of a property of the heat equation to linear thermoelasticity and other theories, Quart. Appl. Math., Volume 41 (1982/83), pp. 319-330 | Zbl | DOI

[18] Ekolin, G. Finite difference methods for a nonlocal boundary value problem for the heat equation, BIT, Volume 31 (1991), pp. 245-261 | Zbl | DOI

[19] El-Gamel, M. A numerical scheme for solving nonhomogeneous time-dependent problems, Z. Angew. Math. Phys., Volume 57 (2006), pp. 369-383 | Zbl | DOI

[20] El-Gamel, M. Numerical solution of Troesch's problem by sinc-collocation method, Appl. Math., Volume 4 (2013), pp. 707-712

[21] El-Gamel, M. A note on solving the fourth-order parabolic equation by the sinc-Galerkin method, Calcolo, Volume 52 (2015), pp. 327-342 | DOI | Zbl

[22] El-Gamel, M. Error analysis of sinc-Galerkin method for time-dependent partial differential equations, Numer. Algorithms, Volume 77 (2018), pp. 517-533 | Zbl

[23] El-Gamel, M.; Abdrabou, A. Sinc-Galerkin solution to eighth-order boundary value problems, SeMA J., Volume 76 (2019), pp. 249-270 | DOI

[24] El-Gamel, M.; Zayed, A. I. A comparison between the wavelet-Galerkin and the Sinc-Galerkin methods in solving nonhomogeneous heat equations, in: Inverse Problem, Image Analysis, and Medical Imaging, Volume 2002 (2002), pp. 97-116 | Zbl

[25] Ewing, R.; Lazarov, R.; Lin, Y. P. Finite volume element approximations of nonlocal reactive flows in porous media, Numer. Methods Partial Differential Equations, Volume 16 (2000), pp. 285-311 | DOI | Zbl

[26] Fairweather, G.; Saylor, R. D. The reformulation and numerical solution of certain nonclassical initial-boundary value problems, SIAM J. Sci. Statist. Comput., Volume 12 (1991), pp. 127-144 | DOI | Zbl

[27] Golbabai, A.; Javidi, M. A numerical solution for nonclassical parabolic problem based on Chebyshev spectral collocation method, Appl. Math. Comput., Volume 190 (2007), pp. 179-185

[28] Hokkanen, V.-M.; Morosanu, G. Functional Methods in Differential Equations, Chapman & Hall/CRC, Boca Raton, 2002 | Zbl

[29] Ionkin, N. Solution of a boundary value problem in heat conduction with a non-classical boundary condition, Differential Equations, Volume 13 (1977), pp. 204-211

[30] Lund, J.; Bowers, K. L. Sinc Methods for Quadrature and Differential Equations, SIAM, Philadelphia, 1992 | Zbl

[31] Nakhushev, A. M. On certain approximate method for boundary-value problems for differential equations and its applications in ground waters dynamics, Differ. Uravn., Volume 18 (1982), pp. 72-81

[32] Ng, M. K. Fast iterative methods for symmetric sinc-Galerkin system, IMA J. Numer. Anal., Volume 19 (1999), pp. 357-373

[33] Pani, A. K. A finite element method for a diffusion equation with constrained energy and nonlinear boundary conditions, J. Austral. Math. Soc. Ser. B, Volume 35 (1993), pp. 87-102 | DOI | Zbl

[34] Renardy, M.; Hrusa, W. J.; Nohel, J. A. Mathematical Problems in Viscoelasticity, John Wiley & Sons, New York, 1987 | Zbl

[35] Samarskii, A. Some problems in differential equations theory, Differential Equations, Volume 16 (1980), pp. 1221-1228

[36] Smith, R. C.; Bogar, G. A.; Bowers, K. L.; Lund, J. The Sinc-Galerkin method for fourth-order differential equations, SIAM J. Numer. Anal., Volume 28 (1991), pp. 760-788 | DOI | Zbl

[37] Stenger, F. A "sinc-Galerkin" method of solution of boundary value problems, Math. Comput., Volume 33 (1979), pp. 85-109 | DOI | Zbl

[38] Stenger, F. Numerical Methods Based on Sinc and Analytic Functions, Springer-Verlag, New York, 1993 | DOI | Zbl

[39] Stenger, F. Summary of sinc numerical methods, J. Comput. Appl. Math., Volume 121 (2000), pp. 379-420 | Zbl | DOI

[40] Tadi, M.; Radenkovic, M. A numerical method for 1-D parabolic equation with nonlocal boundary conditions, Int. J. Comput. Math., Volume 2014 (2014), pp. 1-9 | DOI

[41] Murthy, A. S. Vasudeva; Verwer, J. G. Solving parabolic integro-differential equations by an explicit integration method, J. Comput. Appl. Math., Volume 39 (1992), pp. 121-132 | DOI | Zbl

[42] Vodakhova, V. A boundary-value problem with nonlocal condition for certain pseudo-parabolic water-transfer equation, Differentsialnie Uravnenia, Volume 18 (1982), pp. 280-285

[43] Yin, G. Y. Sinc-Collocation method with orthogonalization for singular problem-like poisson, Math. Comp., Volume 62 (1994), pp. 21-40

[44] Yousefi, S. A.; Behroozifar, M.; Dehghan, M. The operational matrices of Bernstein polynomials for solving the parabolic equation subject to specification of the mass, J. Comput. Appl. Math., Volume 235 (2011), pp. 5272-5283 | Zbl | DOI

[45] Zolfaghari, R.; Shidfar, A. Solving a parabolic PDE with nonlocal boundary conditions using the Sinc method, Numer. Algorithms, Volume 62 (2013), pp. 411-427 | DOI | Zbl

Cité par Sources :