On the solution of an initial-boundary value problem that combines neumann and integral condition for the wave equation

Numerical Methods for Partial Differential Equations

Volumn 21, Issue 1, 2005, Pages 24-40

  Dehghan, Mehdi ^a,b  

^a AMIRKABIR UNIVERSITY OF TECHNOLOGY   (Iran)

^b INSTITUTE FOR STUDIES IN THEORETICAL PHYSICS AND MATHEMATICS IPM   (Iran)

Author keywords

Existence; Finite difference schemes; Modified equivalent partial differential equation; Second order hyperbolic equation; Stability numerical differentiation; Thermoelasticity; Uniqueness; Viscoelasticity; Wave equation nonclassic boundary value problems

Indexed keywords

EID: 11144275388     PISSN: 0749159X     EISSN: None     Source Type: Journal    
DOI: 10.1002/num.20019     Document Type: Article

References (91)

1. 1 J. Chabrowski, On nonlocal problems for parabolic equations, Nagaya Math J 93 (1984), 109–131.
2. 2 K. L. Deckert and C. G. Maple, Solutions for diffusion equation with integral type boundary conditions, Proc Iowa Acad Sci 70 (1963), 345–361.
3. 3 K. Deng, Comparison principle for some nonlocal problems, Quart Appl Math 50 (1992), 517–522.
4. 4 A. Friedman, Monotonic decay of solutions of parabolic equation with nonlocal boundary conditions, Quart Appl Math 44 (1986), 401–407.
5. 5 A. Bouziani, Initial‐boundary value problem with a nonlocal condition for a viscosity equation, Int J Math Math Sci 30 (6) (2002), 327–338.
6. 6 M. Renardy, W. Hrusa, and J. A. Nohel, Mathematical problems in viscoelasticity, Longman Sci Tech, England, 1987.
7. 7 D. E. Carlson, Linear thermoelasticity, Encyclopedia of physics, Vol. 2, Springer, Berlin, 1972.
8. 8 W. A. Day, Existence of a property of solutions of the heat equation subject to linear thermoelasticity and other theories, Quart Appl Math 40 (1982), 319–330.
9. 9 W. A. Day, A decreasing property of solutions of a parabolic equation with applications to thermoelasticity and other theories, Quart Appl Math 41 (1983), 468–475.
10. 10 W. A. Day, Heat conduction within linear thermoelasticity, Springer‐Verlag, New York, 1985.
11. 11 W. A. Day, Parabolic equations and thermodynamics, Quart Appl Math 50 (1992), 523–533.
12. 12 J. R. Cannon, The solution of the heat equation subject to the specification of energy, Quart Appl Math 21 (1963), 155–160.
13. 13 J. R. Cannon, The one dimensional heat equation, Encyclopedia of mathematics and its applications, Vol. 23, Addison‐Welsey, Menlo Park, CA, 1984.
14. 14 J. R. Cannon and A. L. Matheson, A numerical procedure for diffusion subject to the specification of mass, Intern J Engng Sci 31 (3) (1993), 347–355.
15. 15 J. R. Cannon, S. Prez‐Esteva, and 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.
16. 16 J. R. Cannon and J. van der Hoek, 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. 17 J. R. Cannon and J. van der Hoek, Diffusion subject to specification of mass, J Math Anal Appl 115 (1986), 517–529.
18. 18 J. R. Cannon and J. van der Hoek, The one phase Stefan problem subject to the specification of energy, J Math Anal Appl 86 (1982), 281–289.
19. 19 J. R. Cannon and J. van der Hoek, The classical solution of the one‐dimensional two‐phase Stefan problem with energy specification, Annali Di Mat Pura ed Appl 130 (4) (1982), 385–398.
20. 20 J. R. Cannon, Y. Lin, and J. van der Hoek, A quasilinear parabolic equation with nonlocal boundary conditions, Rendicoti di Mathematica Roma Serie VII 9 (1989), 239–264.
21. 21 J. R. Cannon, Y. Lin, and S. Wang, An implicit finite difference scheme for the diffusion equation subject to the specification of mass, Intern J Engng Sci 28 (7) (1990), 573–578.
22. 22 J. R. Cannon and Y. Lin, A Galerkin procedure for diffusion equations with boundary integral conditions, Intern J Engng Sci 28 (7) (1990), 579–587.
23. 23 J. R. Cannon and H. M. Yin, On a class of non‐classical parabolic problems, Diff Eqs 79 (1989), 266–288.
24. 24 A. A. Samarskii, Some problems in modern theory of differential equations, Differentsialnie Uravnenia 16 (1980), 1221–1228.
25. 25 J. H. Cushman, Diffusion in fractal porous media, Water Resource Res 27 (4) (1991), 643–644.
26. 26 J. H. Cushman and T. R. Ginn, Nonlocal dispersion in media with continuously evolving scales of heterogeneity, Transport Porous Media 13 (1993), 123–138.
27. 27 J. H. Cushman, H. Xu, and F. Deng, Nonlocal reactive transport with physical and chemical heterogeneity: localization error, Water Resource Res 31 (1995), 2219–2237.
28. 28 S. A. Beilin, Existence of solutions for one‐dimensional wave equations with non‐local conditions, Electron J Differential Eq 76 (2001), 1–8.
29. 29 A. Bouziani, Mixed problem for certain nonclassical equations containing a small parameter, Acad Roy Belg Bull CI, Sci 6 (5) (1994), 389–400.
30. 30 S. Mesloub and A. Bouziani, On a class of singular hyperbolic equation with a weighted integral condition, Intern J Math Math Sci 22 (1999), 511–520.
31. 31 L. S. Pulkina, A nonlocal problem with integral conditions for hyperbolic equations, Electr J Differential Eq 45 (1999), 1–6.
32. 32 L. S. Pulkina, On solvability in L 2 of nonlocal problem with integral conditions for hyperbolic equations, Differets Uravn VN 2 (2000), 1–6.
33. 33 A. K. Gushin and V. P. Mikhailov, On solvability of nonlocal problems for second‐order elliptic equations, Matem Sbornik 185 (1994), 121–160.
34. 34 B. A. Boley and J. H. Weiner, Theory of thermal stresses, Wiley, New York, 1960.
35. 35 Y. S. Choi and K. Y. Chan, A parabolic equation with nonlocal boundary conditions arising from electrochemistry, Nonlinear Anal Theory Methods Appl 18 (4) (1992), 317–331.
36. 36 G. Dagan, The significance of heterogeneity of evolving scales to transport in porous formations, Water Resource Res 13 (1994), 3327–3336.
37. 37 A. M. Nakhushev, On certain approximate methods for boundary‐value problems for differential equations and its applications in ground waters dynamics, Differentsialnie Uravnenia 18 (1982), 72–81.
38. 38 V. V. Shelukhin, A non‐local in time model for radionuclides propagation in Stokes fluids, dynamics of fluids with free boundaries, Siberian Russian Acad Sci Inst Hydrodynam 107 (1993), 180–193.
39. 39 W. Allegretto, J. R. Cannon, and Y. Lin, A parabolic integro‐differential equation arising from thermoelastic contact, Dynam Discr Contin Impuls Sys 3 (2) (1997), 217–234.
40. 40 B. Jumarhon and S. McKee, On a heat equation with nonlinear and nonlocal boundary conditions, J Math Anal Appl 190 (1995), 806–820.
41. 41 B. Jumarhon and M. Pidcock, On a nonlinear Volterra integro‐differential equation with a weakly singular kernel, Zamm Z Angew Math Mech 76 (6) (1996), 357–360.
42. 42 B. Jumarhon and S. McKee, Product integration methods for solving a system of nonlinear Volterra integral equations, J Comput Appl Math 67 (6) (1996), 24–41.
43. 43 G. Fairweather and R. D. Saylor, The reformulation and numerical solution of certain nonclassical initial‐boundary value problems, SIAM J Sci Stat Comput 12 (1) (1991), 127–144.
44. 44 S. Wang and Y. Lin, A finite difference solution to an inverse problem determining a control function in a parabolic partial differential equation, Inverse Problems 5 (1989), 631–640.
45. 45 V. Capasso and K. Kunisch, A reaction‐diffusion system arising in modeling man‐environment diseases, Quart Appl Math 46 (1988), 431–449.
46. 46 P. Shi, Weak solution to an evolution problem with a nonlocal constraint, SIAM J Math Anal 24 (1) (1993), 46–58.
47. 47 N. Borovykh, Stability in the numerical solution of the heat equation with non‐local boundary conditions, Appl Numer Math 42 (2002), 17–27.
48. 48 I. S. Gordeziani and G. A. Avalishvili, On the constructing of solutions of the nonlocal initial boundary problems for one‐dimensional medium oscillation equations, Matem Modelirovanie 12 (1) (2000), 94–103.
49. 49 N. I. Kavalloris and D. S. Tzanetis, Behaviour of critical solutions of a nonlocal hyperbolic problem in ohmic heating of foods, Appl Math E‐Notes 2 (2002), 59–65.
50. 50 L. A. Muravei and A. V. Philinovskii, Non‐local boundary value problems for hyperbolic equation, Matem Zametki 54 (1993), 98–116.
51. 51 W. T. Ang, A method of solution for the one‐dimensional heat equation subject to a nonlocal condition, SEA Bull Math 26 (2) (2002), 197–203.
52. 52 G. W. Batten, Jr., Second‐order correct boundary conditions for the numerical solution of the mixed boundary problem for parabolic equations, Math Comput 17 (1963), 405–413.
53. 53 A. Bouziani, Mixed problem with boundary integral conditions for a certain parabolic equation, J Appl Math Stochastic Anal 9 (1996), 323–330.
54. 54 A. Bouziani, On a class of parabolic equations with a nonlocal boundary condition, Acad Roy Belg Bull Cl Sci 10 (1999), 61–77.
55. 55 M. Dehghan, Numerical solution of a parabolic equation with non‐local boundary specifications, Appl Math Comput 145 (2003), 185–194.
56. 56 M. Dehghan, The use of Adomian decomposition method for solving the one‐dimensional parabolic equation with non‐local boundary specifications, Int J Comput Math 81 (2004), 25–34.
57. 57 G. Ekolin, Finite difference methods for a non‐local boundary value problem for the heat equation, BIT 31 (2) (1991), 245–255.
58. 58 R. E. Ewing, R. D. Lazarov, and Y. Lin, Finite volume element approximations of nonlocal in time one‐dimensional flows in porous media, Computing 64 (2000), 157–182.
59. 59 R. E. Ewing, R. D. Lazarov, and Y. Lin, Finite volume element approximations of nonlocal reactive flows in porous media, Numer Methods Partial Differential Eq 16 (2000), 285–311.
60. 60 G. Fairweather and J. C. Lopez‐Marcos, Galerkin methods for a semilinear parabolic problem with nonlocal boundary conditions, Adv Comput Math 6 (1996), 243–262.
61. 61 G. Fairweather, J. C. Lopez‐Marcos, and A. Boutayeb, Orthogonal spline collocation for a quasilinear nonlocal parabolic problem, to appear.
62. 62 A. B. Gumel, On the numerical solution of the diffusion equation subject to the specification of mass, J Austral Math Soc Ser B 40 (1999), 475–483.
63. 63 L. I. Kamynin, A boundary value problem in the theory of heat conduction with a non‐classical boundary condition, USSR Comp Math Math Physics 4 (1964), 33–59.
64. 64 N. I. Ionkin, Solution of a boundary value problem in heat conduction with a non‐classical boundary condition, Diff Eqs 13 (1977), 204–211.
65. 65 N. I. Ionkin, Stability of a problem in heat transfer theory with a non‐classical boundary condition, Diff Eqs 15 (1980), 911–914.
66. 66 Y. Lin, Parabolic partial differential equations subject to non‐local boundary conditions, Ph.D. dissertation, Department of Pure and Applied Mathematics, Washington State University, 1988.
67. 67 Y. Lin, Analytical and numerical solutions for a class of nonlocal nonlinear parabolic differential equations, SIAM J Math Anal 25 (6) (1994), 1577–1594.
68. 68 Y. Lin, A nonlocal parabolic system in linear thermoelasticity, Dynam Discr Contin Impuls Sys 2 (3) (1996), 267–283.
69. 69 Y. Lin, S. Xu, and H. M. Yin, Finite difference approximations for a class of nonlocal parabolic equations, Intern J Math Math Sci 20 (1) (1997), 147–164.
70. 70 Y. Lin and R. J. Tait, Finite difference approximations for a class of nonlocal parabolic boundary value problems, J Comput Appl Math 47 (3) (1993), 335–350.
71. 71 Y. Lin and R. J. Tait, On a class of non‐local parabolic boundary value problems, Int J Engng Sci 32 (3) (1994), 395–407.
72. 72 Y. Liu, Numerical solution of the heat equation with nonlocal boundary conditions, J Comput Appl Math 110 (1) (1999), 115–127.
73. 73 V. L. Makarov and D. T. Kulyev, Solution of a boundary value problem for a quasi‐linear parabolic equation with nonclassical boundary conditions, Diff Eqs 21 (1985), 296–305.
74. 74 M. Hadizadeh and K. Maleknejad, On the decomposition method to the heat equation with non‐linear and non‐local boundary conditions, Kybernetes 27 (4) (1998), 426–434.
75. 75 V. A. Vodakhova, A boundary‐value problem with Nakhgushev non‐local condition for certain pseudo‐parabolic water‐transfer equations, Differentsialnie Uravnenia 18 (1982), 280–285.
76. 76 A. K. Pani, A finite element method for a diffusion equation with constrained energy and nonlinear boundary conditions, J Australian Math Soc Series B 35 (1993), 87–102.
77. 77 B. P. Paneiah, On certain nonlocal boundary value problems for linear differential operators, Matem Zametki 35 (3) (1984), 425–434.
78. 78 Z. Z. Sun, A second‐order accurate finite difference scheme for a class of nonlocal parabolic equations with natural boundary conditions, J Comput Appl Math 76 (1996), 137–146.
79. 79 S. Wang and Y. Lin, A numerical method for the diffusion equation with nonlocal boundary specifications, Intern J Engng Sci 28 (6) (1990), 543–546.
80. 80 S. Wang, The numerical method for the conduction subject to moving boundary energy specification, Numerical Heat Transfer 130 (1990), 35–38.
81. 81 N. I. Yurchuk, Mixed problem with an integral condition for certain parabolic equations, Diff Eqs 22 (1986), 1457–1463.
82. 82 A. Bouziani, Strong solution for a mixed problem with nonlocal condition for certain pluriparabolic equations, Hiroshima Math J 27 (3) (1997), 373–390.
83. 83 A. L. Araujo and F. A. Oliveira, Semi‐discretization method for the heat equation with non‐local boundary conditions, Commun Numer Methods Engng 10 (1994), 751–758.
84. 84 M. Berzins and R. M. Fuzerland, Developing software for time‐dependent problems using the method of lines and differential algebraic integrators, Appl Numer Math 5 (1) (1989), 375–397.
85. 85 C. Diaz, G. Fairweather, and P. Keast, Fortran package for solving certain almost block diagonal linear systems by modified alternate row and column elimination, ACM Trans Math Software 9 (1983), 358–375.
86. 86 J. A. Dixon, On the order of the error in discretization methods for weakly singular second kind Volterra integral equations with non‐smooth solutions, BIT 25 (1985), 624–634.
87. 87 A. S. V. Murthy and J. G. Verwer, Solving parabolic integro‐differential equations by an explicit integration method, J Comput Appl Math 39 (1992), 121–132.
88. 88 R. F. Warming and B. J. Hyett, The modified equation approach to the stability and accuracy analysis of finite‐difference methods, J Comput Physics 14 (2) (1974), 159–179.
89. 89 L. Lapidus and G. F. Pinder, Numerical solution of partial differential equations in science and engineering, Wiley, New York, 1982.
90. 90 J. C. Strickwerda, Finite difference schemes and partial differential equations, Chapman and Hall, New York, 1989.
91. 91 E. H. Twizell, An explicit difference method for the wave equation with extended stability range, BIT 19 (1979), 378–383.