##
**Application of the Adomian decomposition method for two-dimensional parabolic equation subject to nonstandard boundary specifications.**
*(English)*
Zbl 1054.65105

Summary: We are concerned with the solution of a nonlocal boundary value problem. An approach is presented for solving the two-dimensional parabolic partial differential equation subject to integral boundary specifications. The main objective is to propose an alternative method of solution, one not based on finite difference methods or finite element schemes or spectral techniques.

The aim of the present paper is to investigate the application of the Adomian decomposition method for solving the two-dimensional linear parabolic partial differential equation with nonlocal boundary specifications replacing the classical boundary conditions. The Adomian decomposition method is used by many researchers to investigate several scientific applications and requires less work if compare with the traditional techniques. The introduction of this idea as will be discussed, not only provides the solution in a series form but it also guarantees considerable saving of the calculations volume.

The solutions will be handle more easily, quickly and elegantly without linearizing the problem by implementing the decomposition method rather than the standard methods for the exact solutions. In this approach the solution is found in the form of a convergent power series with easily computed components. The Adomian decomposition scheme is easy to program in applied problems and provides immediate and convergent solutions without any need for linearization or discretization. To give a clear overview of the methodology, we have selected illustrative example.

The aim of the present paper is to investigate the application of the Adomian decomposition method for solving the two-dimensional linear parabolic partial differential equation with nonlocal boundary specifications replacing the classical boundary conditions. The Adomian decomposition method is used by many researchers to investigate several scientific applications and requires less work if compare with the traditional techniques. The introduction of this idea as will be discussed, not only provides the solution in a series form but it also guarantees considerable saving of the calculations volume.

The solutions will be handle more easily, quickly and elegantly without linearizing the problem by implementing the decomposition method rather than the standard methods for the exact solutions. In this approach the solution is found in the form of a convergent power series with easily computed components. The Adomian decomposition scheme is easy to program in applied problems and provides immediate and convergent solutions without any need for linearization or discretization. To give a clear overview of the methodology, we have selected illustrative example.

### MSC:

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

35K05 | Heat equation |

### Keywords:

Adomian decomposition method; two-dimensional parabolic equation; nonlocal boundary conditions; analytic solutions; method of lines; specification of energy; boundary element techniques; closed form solutions; parallel algorithms; numerical examples; diffusion equation
PDF
BibTeX
XML
Cite

\textit{M. Dehghan}, Appl. Math. Comput. 157, No. 2, 549--560 (2004; Zbl 1054.65105)

Full Text:
DOI

### References:

[1] | Adomian, G., Solving frontier problems of physics: the decomposition method, (1994), Kluwer Boston, MA · Zbl 0802.65122 |

[2] | Adomian, G., A review of the decomposition method in applied mathematics, J. math. anal. appl, 135, 501-544, (1988) · Zbl 0671.34053 |

[3] | Adomian, G., Modification of the decomposition approach to the heat equation, J. math. anal. appl, 124, 290-291, (1987) · Zbl 0693.35068 |

[4] | Adomian, G., A new approach to the heat equation,an application of the decomposition method, J. math. anal. appl, 113, 202-209, (1986) · Zbl 0606.35037 |

[5] | Adomian, G., Solving frontier problems modelled by nonlinear partial differential equations, Comput. math. appl, 22, 91-94, (1991) · Zbl 0767.35016 |

[6] | 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 |

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

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

[9] | Cannon, J.R.; Esteva, S.P.; 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 |

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

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

[12] | Cherruault, Y., Convergence of Adomian’s method, Math. comput. modell, 14, 83-86, (1990) · Zbl 0728.65056 |

[13] | Datta, B.K., A new approach to the wave equation–an application of the decomposition method, J. math. anal. appl, 142, 6-12, (1989) · Zbl 0685.35059 |

[14] | 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 |

[15] | 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 |

[16] | Dehghan, M., Numerical solution of a parabolic equation with non-local boundary specifications, Appl. math. comput, 145, 185-194, (2003) · Zbl 1032.65104 |

[17] | Dehghan, M., Saulyev’s techniques for solving a parabolic equation with a non linear boundary specification, Int. J. comput. math, 80, 2, 257-265, (2003) · Zbl 1018.65104 |

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

[19] | Friedman, A., Monotonic decay of solutions of parabolic equation with nonlocal boundary conditions, Quart. appl. math, 44, 468-475, (1986) |

[20] | Kawohl, B., Remark on a paper by D.A. day on a maximum principle under nonlocal boundary conditions, Quart. appl. math, 45, 751-752, (1987) · Zbl 0617.35064 |

[21] | Kaya, D., An application of the decomposition method to the second-order wave equation, Int. J. comput. math, 75, 235-245, (2000) · Zbl 0964.65113 |

[22] | Lesnic, D., A computational algebraic investigation of the decomposition method for time-dependent problems, Appl. math. comput, 119, 197-206, (2001) · Zbl 1023.65107 |

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

[24] | 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 |

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

[26] | 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 |

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

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

[29] | Wazwaz, A.M., A reliable modification of Adomian decomposition method, Appl. math. comput, 102, 77-86, (1999) · Zbl 0928.65083 |

[30] | Yee, E., Application of the decomposition method to the solution of the reaction – convection – diffusion equation, Appl. math. comput, 56, 1-27, (1993) · Zbl 0773.76055 |

[31] | Dehghan, M., Locally explicit schemes for three-dimensional diffusion with a non-local boundary condition, Appl. math. comput, 138, 489-501, (2003) · Zbl 1027.65112 |

[32] | Dehghan, M., Numerical solution of a non-local boundary value problem with Neumann’s boundary conditions, Communications in numerical methods in engineering, 19, 1-12, (2003) · Zbl 1014.65072 |

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.