Efficient techniques for the second-order parabolic equation subject to nonlocal specifications.

*(English)*Zbl 1063.65079Summary: Many physical phenomena are modeled by nonclassical parabolic boundary value problems with nonlocal boundary conditions. In place of the classical specification of boundary data, we impose a nonlocal boundary condition. Partial differential equations with nonlocal boundary conditions have received much attention in the last twenty years. Most of the papers were directed to the second-order parabolic equation, particularly to the heat conduction equation. One could generically classify these problems into two types; boundary value problems with nonlocal initial conditions, and boundary value problems with nonlocal boundary conditions.

We deal here with the second type of nonlocal boundary value problems that is the solution of nonlocal boundary value problems with standard initial condition. The main difficulty in the implicit treatment of the nonlocal boundary value problems is the nonclassical form of the resulting matrix of the system of linear algebraic equations.

In this paper, various approaches for the numerical solution of the one-dimensional heat equation subject to the specification of mass which have been considered in the literature, are reported. Several methods have been proposed for the numerical solution of this boundary value problem. Some remarks comparing our work with earlier work will be given throughout the paper.

Numerical examples are given at the end of this paper to compare the efficiency of the new techniques. Some specific applications in engineering models are introduced.

We deal here with the second type of nonlocal boundary value problems that is the solution of nonlocal boundary value problems with standard initial condition. The main difficulty in the implicit treatment of the nonlocal boundary value problems is the nonclassical form of the resulting matrix of the system of linear algebraic equations.

In this paper, various approaches for the numerical solution of the one-dimensional heat equation subject to the specification of mass which have been considered in the literature, are reported. Several methods have been proposed for the numerical solution of this boundary value problem. Some remarks comparing our work with earlier work will be given throughout the paper.

Numerical examples are given at the end of this paper to compare the efficiency of the new techniques. Some specific applications in engineering models are introduced.

##### MSC:

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

35K05 | Heat equation |

65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

##### Keywords:

Nonclassic boundary value problems; Explicit schemes; Implicit techniques; Three-level finite difference schemes; Runge-Kutta-Chebyshev scheme; Galerkin procedure; Boundary element method; Keller-box scheme; PadĂ© approximant; Second-order parabolic equation; Parallel algorithms; Product integration method; Comparison of methods; Heat equation; Numerical examples
PDF
BibTeX
XML
Cite

\textit{M. Dehghan}, Appl. Numer. Math. 52, No. 1, 39--62 (2005; Zbl 1063.65079)

Full Text:
DOI

##### References:

[1] | Allegretto, W.; Lin, Y.; Zhou, A., A box scheme for coupled systems resulting from micro-sensor thermistor problems, Dynam. contin. discr. impuls. syst, 5, 209-223, (1999) · Zbl 0979.78023 |

[2] | Ang, W.T., A method of solution for the one-dimensional heat equation subject to a nonlocal condition, SEA bull. math, 26, 2, 197-203, (2002) · Zbl 1032.35073 |

[3] | Berzins, M.; Fuzerland, R.M., Developing software for time-dependent problems using the method of lines and differential algebraic integrators, Appl. numer. math, 5, 1, 375-397, (1989) · Zbl 0679.65071 |

[4] | Boley, B.A.; Weiner, J.H., Theory of thermal stresses, (1960), Wiley New York · Zbl 0095.18407 |

[5] | Bouziani, A., Mixed problem with boundary integral conditions for a certain parabolic equation, J. appl. math. stoch. anal, 9, 323-330, (1996) · Zbl 0864.35049 |

[6] | Bouziani, A., On a class of parabolic equations with a nonlocal boundary condition, Acad. roy. belg. bull. cl. sci, 10, 61-77, (1999) · Zbl 1194.35200 |

[7] | Bouziani, A., Strong solution for a mixed problem with nonlocal condition for a certain pluriparabolic equations, Hiroshima math. J, 27, 373-390, (1997) · Zbl 0893.35061 |

[8] | Cannon, J.R., The solution of the heat equation subject to the specification of energy, Quart. appl. math, 21, 155-160, (1963) · Zbl 0173.38404 |

[9] | Cannon, J.R., The one-dimensional heat equation, Encyclopedia math. appl, vol. 23, (1984), Addison-Welsey Menlo Park, CA |

[10] | Cannon, J.R.; Matheson, A.L., A numerical procedure for diffusion subject to the specification of mass, Internat. J. engrg. sci, 31, 3, 347-355, (1993) · Zbl 0773.65069 |

[11] | Cannon, J.R.; Prez-Esteva, S.; van der Hoek, J., A Galerkin procedure for the diffusion equation subject to the specification of mass, SIAM J. numer. anal, 24, 499-515, (1987) · Zbl 0677.65108 |

[12] | Cannon, J.R.; van der Hoek, J., Implicit finite difference scheme for the diffusion of mass in porous media, (), 527-539 |

[13] | Cannon, J.R.; van der Hoek, J., Diffusion subject to specification of mass, J. math. anal. appl, 115, 517-529, (1986) · Zbl 0602.35048 |

[14] | Cannon, J.R.; van der Hoek, J., The one phase Stefan problem subject to the specification of energy, J. math. anal. appl, 86, 281-289, (1982) · Zbl 0508.35074 |

[15] | Cannon, J.R.; van der Hoek, J., The classical solution of the one-dimensional two-phase Stefan problem with energy specification, Ann. mat. pura appl, 130, 4, 385-398, (1982) · Zbl 0493.35080 |

[16] | Cannon, J.R.; Lin, Y.; van der Hoek, J., A quasilinear parabolic equation with nonlocal boundary conditions, Rend. math. roma ser. VII, 9, 239-264, (1989) · Zbl 0726.35065 |

[17] | Cannon, J.R.; Lin, Y.; Wang, S., An implicit finite difference scheme for the diffusion equation subject to the specification of mass, Internat. J. engrg. sci, 28, 7, 573-578, (1990) · Zbl 0721.65046 |

[18] | Cannon, J.R.; Lin, Y., A Galerkin procedure for diffusion equations with boundary integral conditions, Internat. J. engrg. sci, 28, 7, 579-587, (1990) · Zbl 0721.65054 |

[19] | Cannon, J.R.; Yin, H.M., On a class of non-classical parabolic problems, Differential equations, 79, 266-288, (1989) · Zbl 0702.35120 |

[20] | Capasso, V.; Kunisch, K., A reaction – diffusion system arising in modeling man-environment diseases, Quart. appl. math, 46, 431-449, (1988) · Zbl 0704.35069 |

[21] | Carlson, D.E.; Truesdell, C., Linear thermoelasticity, Encyclopedia of physics, vol. 2, (1972), Springer Berlin, pp. 297-345 |

[22] | Chabrowski, J., On nonlocal problems for parabolic equations, Nagaya math. J, 93, 109-131, (1984) · Zbl 0506.35048 |

[23] | Choi, Y.S.; Chan, K.Y., A parabolic equation with nonlocal boundary conditions arising from electrochemistry, Nonlinear anal. theory methods appl, 18, 4, 317-331, (1992) · Zbl 0757.35031 |

[24] | Crandall, S.H., An optimum implicit recurrence formula for the heat conduction equation, Quart. appl. math, 13, 318-320, (1955) · Zbl 0066.10501 |

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

[26] | Cushman, J.H.; Xu, H.; Deng, F., Nonlocal reactive transport with physical and chemical heterogeneity: localization error, Water resour. res, 31, 2219-2237, (1995) |

[27] | Cushman, J.H., Diffusion in fractal porous media, Water resour. res, 27, 4, 643-644, (1991) |

[28] | Dagan, G., The significance of heterogeneity of evolving scales to transport in porous formations, Water resour. res, 13, 3327-3336, (1994) |

[29] | Day, W.A., Existence of a property of solutions of the heat equation subject to linear thermoelasticity and other theories, Quart. appl. math, 40, 319-330, (1982) · Zbl 0502.73007 |

[30] | Day, W.A., Parabolic equations and thermodynamics, Quart. appl. math, 50, 523-533, (1992) · Zbl 0794.35069 |

[31] | Day, W.A., A decreasing property of solutions of a parabolic equation with applications to thermoelasticity and other theories, Quart. appl. math, 41, 468-475, (1983) · Zbl 0514.35038 |

[32] | Day, W.A., Heat conduction within linear thermoelasticity, (1985), Springer New York · Zbl 0577.73009 |

[33] | Deckert, K.L.; Maple, C.G., Solutions for diffusion equations with integral type boundary conditions, Proc. iowa acad. sci, 70, 345-361, (1963) · Zbl 0173.12803 |

[34] | Dehghan, M., Numerical solution of a nonlocal boundary value problem with Neumann’s boundary conditions, Comm. numer. methods engrg, 19, 1-12, (2003) · Zbl 1014.65072 |

[35] | Dehghan, M., On the numerical solution of the diffusion equation with a nonlocal boundary condition, Math. problems engrg, 2, 81-92, (2003) · Zbl 1068.65115 |

[36] | M. Dehghan, On the solution of an initial-boundary value problem which combine Neumann and integral condition for wave equation, Numer. Methods Partial Differential Equations, in press |

[37] | Deng, K., Comparison principle for some nonlocal problems, Quart. appl. math, 50, 517-522, (1992) · Zbl 0777.35006 |

[38] | Diaz, C.; Fairweather, G.; Keast, P., Fortran package for solving certain almost block diagonal linear systems by modified alternate row and column elimination, ACM trans. math. software, 9, 358-375, (1983) · Zbl 0516.65013 |

[39] | Dixon, J.A., On the order of the error in discretization methods for weakly singular second kind Volterra integral equations with non-smooth solutions, Bit, 25, 624-634, (1985) · Zbl 0584.65091 |

[40] | Ekolin, G., Finite difference methods for a nonlocal boundary value problem for the heat equation, Bit, 31, 2, 245-255, (1991) · Zbl 0738.65074 |

[41] | Ewing, R.E.; Lazarov, R.D.; Lin, Y., Finite volume element approximations of nonlocal in time one-dimensional flows in porous media, Computing, 64, 157-182, (2000) · Zbl 0969.76052 |

[42] | Ewing, R.E.; Lazarov, R.D.; Lin, Y., Finite volume element approximations of nonlocal reactive flows in porous media, Numer. methods partial differential equations, 16, 285-311, (2000) · Zbl 0961.76050 |

[43] | Fairweather, G.; Saylor, R.D., The reformulation and numerical solution of certain nonclassical initial-boundary value problems, SIAM J. sci. statist. comput, 12, 1, 127-144, (1991) · Zbl 0722.65062 |

[44] | Fairweather, G.; Lopez-Marcos, J.C., Galerkin methods for a semilinear parabolic problem with nonlocal boundary conditions, Adv. comput. math, 6, 243-262, (1996) · Zbl 0868.65068 |

[45] | G. Fairweather, J.C. Lopez-Marcos, A. Boutayeb, Orthogonal spline collocation for a quasilinear nonlocal parabolic problem, in preparation |

[46] | Friedman, A., Monotonic decay of solutions of parabolic equation with nonlocal boudary conditions, Quart. appl. math, 44, 401-407, (1986) · Zbl 0631.35041 |

[47] | Gumel, A.B., On the numerical solution of the diffusion equation subject to the specification of mass, J. austral. math. soc. ser. B, 40, 475-483, (1999) · Zbl 0962.65078 |

[48] | Hadizadeh, M.; Maleknejad, K., On the decomposition method to the heat equation with non-linear and nonlocal boundary conditions, Kybernetes, 27, 4, 426-434, (1998) · Zbl 0934.65112 |

[49] | Ionkin, N.I., Solution of a boundary value problem in heat conduction with a non-classical boundary condition, Differential equations, 13, 204-211, (1977) · Zbl 0403.35043 |

[50] | Ionkin, N.I., Stability of a problem in heat transfer theory with a non-classical boundary condition, Differential equations, 15, 911-914, (1980) · Zbl 0431.35045 |

[51] | Jumarhon, B.; Mckee, S., On a heat equation with nonlinear and nonlocal boundary conditions, J. math. anal. appl, 190, 806-820, (1995) · Zbl 0822.35076 |

[52] | Jumarhon, B.; Pidcock, M., On a nonlinear Volterra integro-differential equation with a weakly singular kernel, Z. angew. math. mech, 76, 6, 357-360, (1996) · Zbl 0881.45007 |

[53] | Jumarhon, B.; Mckee, S., Product integration methods for solving a system of nonlinear Volterra integral equations, J. comput. appl. math, 67, 6, 24-41, (1996) |

[54] | Kamynin, L.I., A boundary value problem in the theory of heat conduction with a non-classical boundary condition, USSR comput. math. math. phys, 4, 33-59, (1964) · Zbl 0206.39801 |

[55] | Lapidus, L.; Pinder, G.F., Numerical solution of partial differential equations in science and engineering, (1982), Wiley New York · Zbl 0584.65056 |

[56] | Lees, M., A priori estimates for the solutions of difference approximations to parabolic partial differential equations, Duke J. math, 27, 297-311, (1960) · Zbl 0092.32803 |

[57] | Y. Lin, Parabolic Partial Differential Equations Subject to Nonlocal Boundary Conditions, Ph.D. Dissertation, Department of Pure and Applied Mathematics, Washington State University, 1988 |

[58] | Lin, Y.; Xu, S.; Yin, H.M., Finite difference approximations for a class of nonlocal parabolic equations, Internat. J. math. math. sci, 20, 1, 147-164, (1997) |

[59] | Liu, Y., Numerical solution of the heat equation with nonlocal boundary conditions, J. comput. appl. math, 110, 1, 115-127, (1999) · Zbl 0936.65096 |

[60] | Makarov, V.L.; Kulyev, D.T., Solution of a boundary value problem for a quasi-linear parabolic equation with nonclassical boundary conditions, Differential equations, 21, 296-305, (1985) · Zbl 0573.35048 |

[61] | Mesloub, S.; Bouziani, A., On a class of singular hyperbolic equation with a weighted integral condition, Internat. J. math. math. sci, 22, 511-520, (1999) · Zbl 0961.35080 |

[62] | Murthy, A.S.V.; Verwer, J.G., Solving parabolic integro-differential equations by an explicit integration method, J. comput. appl. math, 39, 121-132, (1992) · Zbl 0746.65102 |

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

[64] | Pani, A.K., A finite element method for a diffusion equation with constrained energy and nonlinear boundary conditions, J. Australian math. soc. ser. B, 35, 87-102, (1993) · Zbl 0797.65073 |

[65] | Pulkina, L.S., A nonlocal problem with integral conditions for hyperbolic equations, Electronic J. differential equations, 45, 1-6, (1999) · Zbl 0935.35027 |

[66] | Renardy, M.; Hrusa, W.; Nohel, J.A., Mathematical problems in viscoelasticity, (1987), Longman London · Zbl 0719.73013 |

[67] | Samarskii, A.A., Some problems in differential equations theory, Differential equations, 16, 1221-1228, (1980) · Zbl 0519.35069 |

[68] | Shelukhin, V.V., A nonlocal in time model for radionuclides propagation in Stokes fluids, dynamics of fluids with free boundaries, Siberian Russian acad. sci. inst. hydrodynam, 107, 180-193, (1993) · Zbl 0831.76086 |

[69] | Shi, P., Weak solution to an evolution problem with a nonlocal constraint, SIAM J. math. anal, 24, 1, 46-58, (1993) · Zbl 0810.35033 |

[70] | Strickwerda, J.C., Finite difference schemes and partial differential equations, (1989), Chapman and Hall New York |

[71] | Sun, Z.Z., A second-order accurate finite difference scheme for a class of nonlocal parabolic equations with natural boundary conditions, J. comput. appl. math, 76, 137-146, (1996) · Zbl 0873.65129 |

[72] | Vodakhova, V.A., A boundary-value problem with nonlocal condition for certain pseudo-parabolic water-transfer equation, Differentsialnie uravnenia, 18, 280-285, (1982) · Zbl 0486.35045 |

[73] | Wang, S.; Lin, Y., A numerical method for the diffusion equation with nonlocal boundary specifications, Internat. J. engrg. sci, 28, 6, 543-546, (1990) · Zbl 0718.76096 |

[74] | Wang, S.; Lin, Y., A finite difference solution to an inverse problem determining a control function in a parabolic partial differential equations, Inverse problems, 5, 631-640, (1989) · Zbl 0683.65106 |

[75] | Wang, S., The numerical method for the conduction subject to moving boundary energy specification, Numer. heat transfer, 130, 35-38, (1990) |

[76] | Warming, R.F.; Hyett, B.J., The modified equation approach to the stability and accuracy analysis of finite-difference methods, J. comput. phys, 14, 2, 159-179, (1974) · Zbl 0291.65023 |

[77] | Yurchuk, N.I., Mixed problem with an integral condition for certain parabolic equations, Differential equations, 22, 1457-1463, (1986) · Zbl 0654.35041 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.