The paper considers a scalar linear differential difference equation （LDDE） of mixed type x（t） = （a0 ＋ a1t）X（t） ＋ （b0 ＋ b1t）x（t - 1） ＋ （d0 ＋ d1tx（t ＋ 1） ＋ f（t）, t ∈ R, （＊） where f（t） = ∑...The paper considers a scalar linear differential difference equation （LDDE） of mixed type x（t） = （a0 ＋ a1t）X（t） ＋ （b0 ＋ b1t）x（t - 1） ＋ （d0 ＋ d1tx（t ＋ 1） ＋ f（t）, t ∈ R, （＊） where f（t） = ∑n=0^F fn^tn. This equation is investigated with the use of the method of polynomial quasisolutions based on the representation of an unknown function in the form of polynomial x（t） = ∑n=0^N xn^tn. As a result of substitution of this function into equation （＊）, there appears a residual △（t） = 0（t^N）, for which an exact analytical representation has been obtained. In turn, this allows one to find the unknown coefficients xn and consequently the polynomial quasisolution x（t）. Several examples are considered.展开更多
Based on Wu＇s elimination method and ＂divide-and-conquer＂ strategy, the undetermined coefficient algorithm to construct polynomial form conservation laws for nonlinear differential-difference equations （DDEs） is ...Based on Wu＇s elimination method and ＂divide-and-conquer＂ strategy, the undetermined coefficient algorithm to construct polynomial form conservation laws for nonlinear differential-difference equations （DDEs） is improved. Furthermore, a Maple package named CLawDDEs, which can entirely automatically derive polynomial form conservation laws of nonlinear DDEs is presented. The effective- ness of CLawDDEs is demonstrated by application to different kinds of examples.展开更多
文摘The paper considers a scalar linear differential difference equation （LDDE） of mixed type x（t） = （a0 ＋ a1t）X（t） ＋ （b0 ＋ b1t）x（t - 1） ＋ （d0 ＋ d1tx（t ＋ 1） ＋ f（t）, t ∈ R, （＊） where f（t） = ∑n=0^F fn^tn. This equation is investigated with the use of the method of polynomial quasisolutions based on the representation of an unknown function in the form of polynomial x（t） = ∑n=0^N xn^tn. As a result of substitution of this function into equation （＊）, there appears a residual △（t） = 0（t^N）, for which an exact analytical representation has been obtained. In turn, this allows one to find the unknown coefficients xn and consequently the polynomial quasisolution x（t）. Several examples are considered.
基金supported by the National Natural Science Foundation of China under Grant Nos.10771072 and 11071274
文摘Based on Wu＇s elimination method and ＂divide-and-conquer＂ strategy, the undetermined coefficient algorithm to construct polynomial form conservation laws for nonlinear differential-difference equations （DDEs） is improved. Furthermore, a Maple package named CLawDDEs, which can entirely automatically derive polynomial form conservation laws of nonlinear DDEs is presented. The effective- ness of CLawDDEs is demonstrated by application to different kinds of examples.
