Abstract
In this article, the nonlinear heat equilibrium problems are solved by the local multiquadric (MQ) radial basis function (RBF) collocation method. The system of nonlinear algebraic equations is solved by iteration based on the residual norm-based algorithm, in which the direction of evolution is determined by a linear equation. In addition, the role of the collocation point and source point is clearly defined such that in our proposed method the field value of any interested point can be expressed. Six numerical examples are shown to check the performance of the proposed method. As the number of supporting points (mp) increases, the accuracy of numerical solution increases. Among all examples, mp = 50 can perform well. In addition, the selection of shape parameter, c, affects the accuracy. However, as c < 2 the maximum relative absolute error percentage is less than 1%.
Highlights
The nonlinear heat equilibrium problems can be written as: ∇ · (k(T)∇T) = 0 (1)where k(T) is the temperature-dependent thermal conductivity, and T is the temperature field
The nonlinear heat equilibrium problems are solved by the local multiquadric radial basis functions (MQ RBF)
Even this localized MQ RBF method is compared with other localized RBF such as localized Gaussian RBF, the current approach is more accurate since multiquadric RBF is recognized as the most accurate RBF
Summary
The nonlinear heat equilibrium problems can be written as:. where k(T) is the temperature-dependent thermal conductivity, and T is the temperature field. In comparison with the global radial basis function collocation method, the current approach does not encounter a full matrix form for the Jacobian matrix, and this merit makes it possible for us to increase the number of collocation points. Even this localized MQ RBF method is compared with other localized RBF such as localized Gaussian RBF, the current approach is more accurate since multiquadric RBF is recognized as the most accurate RBF.
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