# On using sinc collocation approach for solving a parabolic PDE with nonlocal boundary conditions

Volume 14, Issue 1, pp 29--38
Publication Date: June 13, 2020 Submission Date: November 28, 2018 Revision Date: March 29, 2020 Accteptance Date: April 08, 2020
• 94 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

•  65L60
•  45J05

### 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