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As we known, reproducing kernel method (RKM) has been presented for solving differential equations for initial and boundary value problems. However, the direct application of the RKM presented in the previous works cannot produce good numerical results for Burgers’ equation. To solve this problem, this paper give a modified reproducing kernel method by piecewise technique. The exact solution is given by reproducing kernel functions in a series expansion form, the approximation solution is expressed by n-term summation of reproducing kernel functions. The three numerical experiments results show that the piecewise method is more easily implemented and effective. Some numerical results are also compared with the results obtained by other methods. [ABSTRACT FROM AUTHOR]

This paper research downhole temperature distribution in oil production and water injection using reproducing kernel Hilbert space method (RKHSM) for the first time. The aim of this paper is that using the highly accurate RKHSM can solve downhole temperature problems effectively. According to 2-D mathematical models of downhole temperature distribution, the analytical solution was given in a series expansion form and the approximate solution was expressed by n -term summation of reproducing kernel functions which initial and boundary conditions were selected properly. Numerical results of downhole temperature distribution with multiple pay zones, in which different radial distance and different injection–production conditions (such as injection rate, injection temperature, injection time, oil layer thickness), were carried out by mathematical 7.0, and numerical results correspond to general knowledge and show that use RKHSM to research downhole temperature distribution is feasible and effective. [ABSTRACT FROM AUTHOR]