A new numerical method for heat equation subject to integral specifications
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Authors
H. M. Jaradat
- Department of Mathematics, Al al-Bayt University, Jordan.
- Department of Mathematics and Applied Sciences, Dhofar University, Salalah, Oman.
M. M. M. Jaradat
- Department of Mathematics, Statistics and Physics, Qatar University, Doha, Qatar.
F. Awawdeh
- Department of Mathematics, Hashemite University, Jordan.
- Department of Mathematics and Applied Sciences, Dhofar University, Salalah, Oman.
Z. Mustafa
- Department of Mathematics, Statistics and Physics, Qatar University, Doha, Qatar.
O. Alsayyed
- Department of Mathematics, Hashemite University., Jordan.
Abstract
We develop a numerical technique for solving the one-dimensional heat equation that combine classical
and integral boundary conditions. The combined Laplace transform, high-precision quadrature schemes,
and Stehfest inversion algorithm are proposed for numerical solving of the problem. A Laplace transform
method is introduced for solving considered equation, definite integrals are approximated by high-precision
quadrature schemes. To invert the equation numerically back into the time domain, we apply the Stehfest
inversion algorithm. The accuracy and computational efficiency of the proposed method are verified by
numerical examples.
Share and Cite
ISRP Style
H. M. Jaradat, M. M. M. Jaradat, F. Awawdeh, Z. Mustafa, O. Alsayyed, A new numerical method for heat equation subject to integral specifications, Journal of Nonlinear Sciences and Applications, 9 (2016), no. 5, 2117--2125
AMA Style
Jaradat H. M., Jaradat M. M. M., Awawdeh F., Mustafa Z., Alsayyed O., A new numerical method for heat equation subject to integral specifications. J. Nonlinear Sci. Appl. (2016); 9(5):2117--2125
Chicago/Turabian Style
Jaradat, H. M., Jaradat, M. M. M., Awawdeh, F., Mustafa, Z., Alsayyed, O.. "A new numerical method for heat equation subject to integral specifications." Journal of Nonlinear Sciences and Applications, 9, no. 5 (2016): 2117--2125
Keywords
- Heat equation
- nonlocal boundary value problems
- Laplace inversion
- high-precision quadrature schemes
- Stehfest inversion algorithm.
MSC
References
-
[1]
W. T. Ang, A numerical method for the wave equation subject to a non-local conservation condition, Appl. Numer. Math., 56 (2006), 1054-1060.
-
[2]
D. H. Bailey, X. S. Li, K. Jeyabalan, A comparison of three high-precision quadrature programs, Exp. Math., 14 (2005), 317-329.
-
[3]
J. G. Batten, Second-order correct boundary conditions for the numerical solution of the mixed boundary problem for parabolic equations, Math. Comp., 17 (1963), 405-413.
-
[4]
A. Bouziani, N. E. Benouar , Problème mixte avec conditions intégrales pour uneclasse d'équations paraboliques , C. R. Acad. Sci., Paris, 321 (1995), 1177-1182.
-
[5]
J. R. Cannon , The solution of the heat equation subject to the specification of energy, Quart. Appl. Math., 21 (1963), 155-160.
-
[6]
J. R. Cannon, Y. Lin, Determination of parameter p(t) in some quasilinear parabolic differential equations, Inverse Probl., 4 (1988), 35-45.
-
[7]
J. R. Cannon, Y. Lin , Determination of parameter p(t) in Holder classes for some semilinear parabolic equations, Inverse Probl., 4 (1988), 595-606.
-
[8]
J. R. Cannon, Y. Lin , A Galerkin procedure for diffusion equations with boundary integral conditions, Int. J. Eng. Sci., 28 (1990), 579-587.
-
[9]
J. R. Cannon, S. Prez-Esteva, J. A. van der Hoek, A. Galerkin, procedure for the diffusion equation subject to the specification of mass, SIAM J. Numer. Anal., 24 (1987), 499-515.
-
[10]
A. M. Cohen, Numerical methods for Laplace transform inversion, Numer. Methods Algorithms 5. Springer, Dordrecht (2007)
-
[11]
B. Davies, B. Martin , Numerical inversion of the Laplace transform, a survey and comparison of methods, J. Comput. Phys., 33 (1979), 1-32.
-
[12]
M. Dehghan, Numerical solution of a parabolic equation with non-local boundary specifications, Appl. Math. Comput., 145 (2003), 185-194.
-
[13]
M. Dehghan , Numerical solution of one-dimensional parabolic inverse problem , Appl. Math. Comput., 136 (2003), 333-344.
-
[14]
M. Dehghan , The use of Adomian decomposition method for solving the one dimentional parabolic equation with non-local boundary spesification, Int. J. Comput. Math., 81 (2004), 25-34.
-
[15]
M. Dehghan, Efficient techniques for the second-order parabolic equation subject to nonlocal specifications, Appl. Numer. Math., 52 (2005), 39-62.
-
[16]
M. Dehghan , On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation, Numer. Methods Partial Differ. Equations, 21 (2005), 24-40.
-
[17]
M. Dehghan, Parameter determination in a partial differential equation from the overspecified data, Math. Comput. Model., 41 (2005), 197-213.
-
[18]
M. Dehghan, Solution of partial integro-differential equation arising from viscoelasticity, Int. J. Comput. Math., 83 (2006), 123-129.
-
[19]
M. Dehghan, A. Saadatmandi , A tau method for the one-dimensional parabolic inverse problem subject to temperature overspecification, Comput. Math. Appl., 52 (2006), 933-940.
-
[20]
D. P. Gaver, Observing stochastic processes and approximate transform inversion, Oper. Res, 14 (1966), 444-459.
-
[21]
A. B. Gumel, On the numerical solution of the diffusion equation subject to the specification of mass, J. Aust. Math. Soc. Ser. B, 40 (1999), 475-483.
-
[22]
H. Hassanzadeh, M. Pooladi-Darvish, Comparison of different numerical Laplace inversion methods for engineering applications, Appl. Math. Comput., 189 (2007), 1966-1981.
-
[23]
N. I. Ionkin, E. I. Moiceev , Solutions of boundary value problem in heat conductions theory with nonlocal boundary conditions, Differ. Equations, 13 (1977), 294-304.
-
[24]
L. I. Kamynin, A boundary value problem in the theory of the heat conduction with nonclassical boundary condition, USSR Comput. Math. Math. Phys., 4 (1964), 33-59.
-
[25]
K. L. Kuhlman, L. Kristopher , Review of inverse Laplace transform algorithms for Laplace-space numerical approaches, Numer. Algorithms, 63 (2013), 339-355.
-
[26]
Y. Lin , An inverse problem for a class of quasilinear parabolic equation, SIAM. J. Math. Anal., 22 (1991), 146-156.
-
[27]
A. Murli, M. Rizzardi , Algorithm 682 Talbots Method for the Laplace Inversion Problem, ACM Trans. Softw., 16 (1990), 158-168.
-
[28]
R. Piessens , New quadrature formulas for the numerical inversion of the Laplace transform, BIT Numer. Math., 9 (1969), 351-361.
-
[29]
R. Piessens , Gaussian quadrature formulas for the numerical inversion of the Laplace transform, J. Eng. Math., 5 (1971), 1-9.
-
[30]
M. Ramezani, M. Dehghan, M. Razzaghi, Combined finite difference and spectral methods for the numerical solution of hyperbolic equation with an integral condition, Numer. Methods Partial Differ. Equations, 24 (2007), 1-8.
-
[31]
T. Sakurai , Numerical inversion of the Laplace transform of functions with discontinuities, Adv. Appl. Prob., 36 (2004), 616-642.
-
[32]
A. A. Samarskii, Some problems in differential equations theory , Differ. Equations, 16 (1980), 1925-1935.
-
[33]
P. Shi, M. Shilor, On design of contact patterns in one-dimensional thermoelasticity , in Theoretical aspects of industrial design (Wright-Patterson Air Force Base), SIAM Proccedings, Philadelphia, (1992), 76-82.
-
[34]
H. Stehfest , Algorithm 368: Numerical inversion of the Laplace transform, Comm. ACM, 13 (1970), 47-49.
-
[35]
H. Takahasi, M. Mori , Error estimation in the numerical integration of analytic functions, Rep. Comput. Centre Univ. Tokyo, 3 (1970), 41-108.
-
[36]
H. Takahasi, M. Mori , Quadrature formulas obtained by variable transformation, Numer. Math., 21 (1973), 206- 219.
-
[37]
H. Takahasi, M. Mori , Double exponential formulas for numerical integration, Pubi. Res. Inst. Math. Sci., 9 (1974), 721-741.
-
[38]
A. Talbot, The accurate inversion of Laplace transforms, IMA J. Appl. Math., 23 (1979), 97-120.