Parallel Solution of Higher Order Differential Equations

The paper focuses on a mathematical approach which uses Modern Taylor Series Method (MTSM) for solving differential equations in a parallel way. Even though this method is not much preferred in the literature, some experimental calculations have shown and verified that the accuracy and stability of the MTSM exceeds the currently used algorithms for solving differential equations. Further, the MTSM has properties suitable for parallel processing, i.e. many independent calculations. The MTSM allows these calculations to be performed independently on several processors using basic mathematical operations. Hardware representation of these operations and their principle are discussed in this paper. Generally, the MTSM can only solve systems of ordinary differential equations (ODEs) that are formed as initial value problems (IVPs). Therefore, this paper also presents methods for solving higher order differential equations, PDEs and their transformations to the corresponding systems of ODEs (IVPs). Effectiveness of hardware implementation of the MTSM is also discussed in this paper, e.g. implementation on FPGA. In many cases, the MTSM obtains results faster than the commonly used Runge-Kutta methods.