On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation. (English) Zbl 1059.65072

Dehghan, Mehdi

Numer. Methods Partial Differ. Equations 21, No. 1, 24-40 (2005).

Summary: Numerical solution of hyperbolic partial differential equation with an integral condition continues to be a major research area with widespread applications in modern physics and technology. Many physical phenomena are modeled by nonclassical hyperbolic 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 specifications have received much attention in last 20 years. However, most of the articles were directed to the second-order parabolic equation, particularly to heat conduction equation.
We deal here with a new type of nonlocal boundary value problems that is the solution of hyperbolic partial differential equations with nonlocal boundary specifications. These nonlocal conditions arise mainly when the data on the boundary can not be measured directly. Several finite difference methods have been proposed for the numerical solution of this one-dimensional nonclassic boundary value problem. These computational techniques are compared using the largest error terms in the resulting modified equivalent partial differential equation. Numerical results supporting theoretical expectations are given. Restrictions on using higher order computational techniques for the studied problem are discussed. Suitable references on various physical applications and the theoretical aspects of solutions are introduced at the end of the article.

MSC:

65M06	Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12	Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L05	Wave equation

Keywords:

Wave equation-nonclassic boundary value problems; finite difference schemes; stability; numerical differentiation; second-order hyperbolic equation; thermoelasticity; viscoelasticity; numerical results

Software:

COLROW

References:

[1]	Chabrowski, Nagaya Math J 93 pp 109– (1984) · Zbl 0506.35048
[2]	Deckert, Proc Iowa Acad Sci 70 pp 345– (1963)
[3]	Deng, Quart Appl Math 50 pp 517– (1992)
[4]	Friedman, Quart Appl Math 44 pp 401– (1986)
[5]	Bouziani, Int J Math Math Sci 30 pp 327– (2002)
[6]	and Mathematical problems in viscoelasticity, Longman Sci Tech, England, 1987.
[7]	Linear thermoelasticity, Encyclopedia of physics, Vol. 2, Springer, Berlin, 1972.
[8]	Day, Quart Appl Math 40 pp 319– (1982)
[9]	Day, Quart Appl Math 41 pp 468– (1983)
[10]	Heat conduction within linear thermoelasticity, Springer-Verlag, New York, 1985. · Zbl 0577.73009
[11]	Day, Quart Appl Math 50 pp 523– (1992)
[12]	Cannon, Quart Appl Math 21 pp 155– (1963)
[13]	The one dimensional heat equation, Encyclopedia of mathematics and its applications, Vol. 23, Addison-Welsey, Menlo Park, CA, 1984.
[14]	Cannon, Intern J Engng Sci 31 pp 347– (1993)
[15]	Cannon, SIAM J Numer Anal 24 pp 499– (1987)
[16]	and Implicit finite difference scheme for the diffusion of mass in porous media, Numerical solutions of partial differential equations, North Holland, Amsterdam, 1982, pp. 527-539.
[17]	Cannon, J Math Anal Appl 115 pp 517– (1986)
[18]	Cannon, J Math Anal Appl 86 pp 281– (1982)
[19]	Cannon, Annali Di Mat Pura ed Appl 130 pp 385– (1982)
[20]	Cannon, Rendicoti di Mathematica Roma Serie VII 9 pp 239– (1989)
[21]	Cannon, Intern J Engng Sci 28 pp 573– (1990)
[22]	Cannon, Intern J Engng Sci 28 pp 579– (1990)
[23]	Cannon, Diff Eqs 79 pp 266– (1989)
[24]	Samarskii, Differentsialnie Uravnenia 16 pp 1221– (1980)
[25]	Cushman, Water Resource Res 27 pp 643– (1991)
[26]	Cushman, Transport Porous Media 13 pp 123– (1993)
[27]	Cushman, Water Resource Res 31 pp 2219– (1995)
[28]	Beilin, Electron J Differential Eq 76 pp 1– (2001)
[29]	Bouziani, Acad Roy Belg Bull CI, Sci 6 pp 389– (1994)
[30]	Mesloub, Intern J Math Math Sci 22 pp 511– (1999)
[31]	Pulkina, Electr J Differential Eq 45 pp 1– (1999)
[32]	Pulkina, Differets Uravn VN 2 pp 1– (2000)
[33]	Gushin, Matem Sbornik 185 pp 121– (1994)
[34]	and Theory of thermal stresses, Wiley, New York, 1960.
[35]	Choi, Nonlinear Anal Theory Methods Appl 18 pp 317– (1992)
[36]	Dagan, Water Resource Res 13 pp 3327– (1994)
[37]	Nakhushev, Differentsialnie Uravnenia 18 pp 72– (1982)
[38]	Shelukhin, Siberian Russian Acad Sci Inst Hydrodynam 107 pp 180– (1993)
[39]	Allegretto, Dynam Discr Contin Impuls Sys 3 pp 217– (1997)
[40]	Jumarhon, J Math Anal Appl 190 pp 806– (1995)
[41]	Jumarhon, Zamm Z Angew Math Mech 76 pp 357– (1996)
[42]	Jumarhon, J Comput Appl Math 67 pp 24– (1996)
[43]	Fairweather, SIAM J Sci Stat Comput 12 pp 127– (1991)
[44]	Wang, Inverse Problems 5 pp 631– (1989)
[45]	Capasso, Quart Appl Math 46 pp 431– (1988)
[46]	Shi, SIAM J Math Anal 24 pp 46– (1993)
[47]	Borovykh, Appl Numer Math 42 pp 17– (2002)
[48]	Gordeziani, Matem Modelirovanie 12 pp 94– (2000)
[49]	Kavalloris, Appl Math E-Notes 2 pp 59– (2002)
[50]	Muravei, Matem Zametki 54 pp 98– (1993)
[51]	Ang, SEA Bull Math 26 pp 197– (2002)
[52]	Batten, Math Comput 17 pp 405– (1963)
[53]	Bouziani, J Appl Math Stochastic Anal 9 pp 323– (1996)
[54]	Bouziani, Acad Roy Belg Bull Cl Sci 10 pp 61– (1999)
[55]	Dehghan, Appl Math Comput 145 pp 185– (2003)
[56]	Dehghan, Int J Comput Math 81 pp 25– (2004)
[57]	Ekolin, BIT 31 pp 245– (1991)
[58]	Ewing, Computing 64 pp 157– (2000)
[59]	Ewing, Numer Methods Partial Differential Eq 16 pp 285– (2000)
[60]	Fairweather, Adv Comput Math 6 pp 243– (1996)
[61]	and Orthogonal spline collocation for a quasilinear nonlocal parabolic problem, to appear.
[62]	Gumel, J Austral Math Soc Ser B 40 pp 475– (1999)
[63]	Kamynin, USSR Comp Math Math Physics 4 pp 33– (1964)
[64]	Ionkin, Diff Eqs 13 pp 204– (1977)
[65]	Ionkin, Diff Eqs 15 pp 911– (1980)
[66]	Parabolic partial differential equations subject to non-local boundary conditions, Ph.D. dissertation, Department of Pure and Applied Mathematics, Washington State University, 1988.
[67]	Lin, SIAM J Math Anal 25 pp 1577– (1994)
[68]	Lin, Dynam Discr Contin Impuls Sys 2 pp 267– (1996)
[69]	Lin, Intern J Math Math Sci 20 pp 147– (1997)
[70]	Lin, J Comput Appl Math 47 pp 335– (1993)
[71]	Lin, Int J Engng Sci 32 pp 395– (1994)
[72]	Liu, J Comput Appl Math 110 pp 115– (1999)
[73]	Makarov, Diff Eqs 21 pp 296– (1985)
[74]	Hadizadeh, Kybernetes 27 pp 426– (1998)
[75]	Vodakhova, Differentsialnie Uravnenia 18 pp 280– (1982)
[76]	Pani, J Australian Math Soc Series B 35 pp 87– (1993)
[77]	Paneiah, Matem Zametki 35 pp 425– (1984)
[78]	Sun, J Comput Appl Math 76 pp 137– (1996)
[79]	Wang, Intern J Engng Sci 28 pp 543– (1990)
[80]	Wang, Numerical Heat Transfer 130 pp 35– (1990)
[81]	Yurchuk, Diff Eqs 22 pp 1457– (1986)
[82]	Bouziani, Hiroshima Math J 27 pp 373– (1997)
[83]	Araujo, Commun Numer Methods Engng 10 pp 751– (1994)
[84]	Berzins, Appl Numer Math 5 pp 375– (1989)
[85]	Diaz, ACM Trans Math Software 9 pp 358– (1983)
[86]	Dixon, BIT 25 pp 624– (1985)
[87]	Murthy, J Comput Appl Math 39 pp 121– (1992)
[88]	Warming, J Comput Physics 14 pp 159– (1974)
[89]	and Numerical solution of partial differential equations in science and engineering, Wiley, New York, 1982. · Zbl 0584.65056
[90]	Finite difference schemes and partial differential equations, Chapman and Hall, New York, 1989.
[91]	Twizell, BIT 19 pp 378– (1979)