(4) To conclude, the solution of the above system allows obtaining the coefficients. (3) As a consequence, a linear algebraic system for the unknowns of ( 3) is obtained. (2) We equate each coefficient of the above-mentioned polynomial to zero. (1) Equation ( 3) is substituted into ( 1), and then we regroup the resulting polynomial equation in terms of powers of. The method of solution for differential equations can be summarized as follows. Where are unknown functions to be determined by series method. PSM assumes that the solution of a differential equation can be written in the following form: Where is a general differential operator, is a boundary operator, is a known analytical function, and is the domain boundary for. It can be considered that a nonlinear differential equation can be expressed as Finally, a brief conclusion is given in Section 6. Besides, a discussion on the results is presented in Section 5. Section 4 presents three case studies: one singular nonlinear index-one system, one singular linear index-two system, and one singular nonlinear index-two system. Section 3 provides a brief explanation of application of PSM to solve SPDAEs. In Section 2, we introduce the basic idea of power series method. More generally, we will see that the combination of PSM and Padé posttreatment could be effective to improve the PSM’s truncated series solutions in convergence rate what is more, sometimes it ends up giving the exact solution of the system, such as what will happen in our third case study. These systems turn out to be difficult even for numerical methods. This study shows that power series method (PSM) is able to address the above difficulties to obtain power series solutions for singular partial differential-algebraic equations (SPDAEs), that is, PDAEs with singular points. Also, a few exact solutions to nonlinear differential equations have been reported occasionally. In recent years, several methods focused on approximating nonlinear and linear problems, as an alternative to classical methods, have been reported, such as those based on variational approaches, tanh method, exp-function, Adomian’s decomposition method, parameter expansion, homotopy perturbation method, homotopy analysis method, homotopy asymptotic method, series method, and perturbation method, among many others. A further difficulty to be considered that arises and affects also other kinds of systems of differential equations, as well as differential equations, is the presence of singularities, which are related to points at which some terms of the differential equations become infinite or undefined. A fact that justifies the search for other methods of solution to these equations is that the solutions of higher index PDAEs (index greater than one) become very complicated, even for numerical methods, and many application problems lead to PDAEs with different indices. The differentiation index is defined as the minimum number of times that all or part of the PDAEs must be differentiated with respect to time, in order to obtain the time derivative of the solution, as a continuous function of the solution and its space derivatives. Despite the importance of this topic, it may be considered relatively new and little known.Īlthough the case of constant-coefficient linear PDAEs has been investigated by means of numerical methods, for instance, in, perhaps the more relevant aspect of PDAEs, both linear and nonlinear, is the concept of index. These systems arise, for example, in nanoelectronics, electrical networks, and mechanical systems, among many others. The importance of research on partial differential-algebraic equations (PDAEs) is that they are used in the mathematical modeling of many phenomena, both practical and theoretical. What is more, we will see that, in some cases, Padé posttreatment, besides enlarging the domain of convergence, may be employed in order to get the exact solution from the truncated series solutions of PSM. We will solve three examples to show that PSM method can be used to search for analytical solutions of SPDAEs. This paper proposes power series method (PSM) in order to find solutions for singular partial differential-algebraic equations (SPDAEs).
0 Comments
Leave a Reply. |