A Two Dimensional Mathematical Model of Heat Propagation Equation and its Applications
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A TWO DIMENSIONAL MATHEMATICAL MODEL OF HEAT PROPAGATION EQUATION AND ITS APPLICATIONS Nassima Talbi,1 Anis Ben Dhahbi,2 Salah Boulaaras,3 Hadj Baltache,4 and Mohammad Alnegga5 We propose a two-dimensional mathematical model based on Galerkin’s spatial method combined with a theta time scheme applied to the heat equations. This model has been applied to a hypothetical example in which the obtained results are compared with the real experimental data. This comparison allow us to predict the soil temperature at different depths as well as at different time periods according to certain conditions imposed on the weather. Keywords: Mathematical model, heat propagation equations, soil temperature, finite elements, finite differences time scheme.
1. Introduction Partial differential equations on surfaces have become a domain of extensive research within the last few decades. This mathematical issue has a wide variety of applications in several fields, such as fluid mechanics and cell biology. The applications cover the determination of the shape of rotating drops, as can be seen in [3] and [14], the dynamics of surfactant at the interface between two fluids, as in [2], phase separation on biological membranes [7] and modeling of cell motility [8], where the considered surfaces can be defined as interfaces of two phases (e.g., liquid and gas), films (liquid soap bubbles), or fluid membranes (e.g., biological tissues). Therefore, partial differential equations on the surfaces are making a challenge in relation with the general theme of numerical analysis. The first results based on the numerical analysis of partial differential equations on surfaces were introduced in [8]. Thus, Dziuk [11] was one of the pioneers in the field. He developed an intrinsic approach, based on the finite-element method, for the Laplace–Beltrami equation on arbitrary surfaces. With this approach, he realized the first techniques of a priori error estimates in the case of piecewise affine approximations. Since the beginning of the 2000s, his approach has been frequently used, especially for more complex partial differential equations involving curvature terms as shown in [14] and [9]. The issue has been then generalized by using isoparametric approximations of higher polynomial degree (see, e.g., [3]). The first variant based on a current function formulation was proposed in [18] and then used in [3] for the flows of incompressible two-phase fluids. Then the second variant based on discontinuous Galerkin’s method was proposed in [15] for the partial differential equations of convectiondiffusion type on surfaces. In addition to these works, it is necessary to mention a mixed approach based on Koiter’s model introduced in [4] and [14]. Note that [12] can be equally reviewed for more details on the theoretical and numerical analysis of this mixed approach. Since the solution of the cited model is vectorial, the mixed approach 1
Department of Physics, Faculty of Exact Sciences, University Mustapha Stambouli of Mascara, Mascara 29000, Algeria. Univers
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