An application of the LTDRM to transient diffusion problems with nonlinear material properties and nonlinear boundary conditions
Transient diffusion problems with nonlinear material properties and a combination of Dirichlet, Neumann, convective and radiative boundary conditions are investigated. The nonlinearity of the thermal conductivity term \(K\) is as follows: \( K(u)= K_0(1+\beta u)\), where \(K_0,\;\beta\) are material dependent constants and \(u\) is a function of the unknown temperature. Two methods, with and without Kirchhoff transformation, are discussed. The nonlinear terms in both the governing equation and the boundary conditions are linearized first followed by the removal of the time variable from the linearized differential system with the Laplace transformation, and then the problem is solved by the dual reciprocity method. Two formulations of the Laplace transform dual reciprocity method (LTDRM) are derived: a formulation involving first-order spatial derivates (LPTDRM-1) and a formulation involving second-order spatial derivates (LPTDRM-2). Six examples are presented to show the efficiency and accuracy of the investigated method. These numerical solutions are compared with those obtained analytically or numerically.
scientific article; zbMATH DE number 1183167
scientific article