Selection Of Bases In Operational Calculus And Applications

Nonlinear problems always play an important role in not only the scientific research but also the engineering practice.In 2011,Gabriel Bengochea and Luis Verde-Star constructed an operational calculus for a modified shift operator on an abstract space of formal Laurent series.It is a general model for many operational calculi for a lot of operators.In this method,the construction of a linear algebraic setting makes it easy to solve many types of equations,no matter linear problems or nonlinear problems.Some other kinds of problems can also be solved by this method,such as the difference equations and the fractional differential equations.In the construct of the algebraic setting,one important thing is to find a modified left shift operator and a corresponding base.There are many problems dealt with this operational calculus simply.However,the modified left shift operator and the base are given directly.So far,there has no result on how to select the modified left shift operator and the base that cooperates with the operator in this method.In this article,a theorem is proposed to guide us the way in selecting bases that cooperates with the modified left shift operator in some problems.We take two examples to show that we can find the right base with this theorem.One is to select the base in solving Emden-fowler equation using this operational calculus;the other is to select the base in dealing with differential equations with variable coefficients.Another contribution of this article is the solution of differential-difference equation,which is given by operational calculus and Pade approximate(called POCM).This result is exactly the same with that giving by Adomian decomposition method and Pade approximate(called PADM).However,integrals are not involved in the new approach.So POCM is more efficient.