解非线性比例方程的迭代方法
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摘要
论文研究了微分变换法(DTM)和拉普拉斯分解法(LDA)，针对这两种方法做了比较。见年来，DTM和LDA形成和发展成为强有力的数学方法。微分变换法是在1986年由中国科学家周提出，拉普拉斯分解法是在2001年由阿拉伯研究者Khuri提出，两种方法使用非常广泛。论文针对特定类型的非线性比例微分方程，评价了这两种方法的效果。
论文共分五章。第一章概述了微分变换法的基本性质，介绍了Laplace分解算法(SLDA)。第二章给出了标准Laplace分解方法(MLDA)，该方法使收敛速度加快；同时，将SLDA、MLDA和DTM用于求解带比例延迟的泛函微分方程。第三章给出了高阶非线性比例积分-微分方程的解。第四章研究了SLDA、MLDA和DTM在求解中立型泛函微分方程的情况。最后一章给出了变分迭代法(VIM)和修正Laplace分解法的联系；同时，将SLDA、MLDA和DTM应用于带比例延迟的偏微分方程。附录中对变分方法作了简单介绍。
This thesis presents a comparative study between two powerful methods that weredeveloped recently. The first method is the diferential transform method (DTM), whichdeveloped by the Chinese scientist Zhou in1986and the second one, is the Laplace de-composition algorithm (LDA), which introduced by the Arabian researcher Khuri in2001.Both methods are used heavily in the literature. The efectiveness of these two method-s when used for certain categories of nonlinear pantograph diferential equation is alsoevaluated in this thesis.
This thesis is divided into five chapters. The first Chapter is devoted to clarify thefundamental properties of the diferential transform method (DTM) and it introduces thestandard Laplace decomposition algorithm (SLDA). The second chapter introduces a newform of SLDA called modified Laplace decomposition algorithm (MLDA), which ledto an increase in the speed of convergence. In this chapter, SLDA, MLDA and DTM areadapted and applied to solve Functional diferential equations with proportional delays. Inthe third chapter, the solution of the higher-order nonlinear pantograph integro-diferentialequation is presented. The fourth chapter is devoted to apply SLDA, MLDA and DTM tosolve neutral functional–diferential equations. Finally, a relation between the variationaliteration method (VIM) and the modified Laplace decomposition algorithm is presented.The application of SLDA, MLDA and DTM for solving partial diferential equation withproportional delays is presented in the fifth Chapter. In addition, a brief introduction aboutthe variational iteration method (VIM) is given in the appendix.
The conclusion that is pointed out in this study is that the diferential transformmethod is easier to use, time and efort saving more than Laplace decomposition algorith-m. However, the rate of convergence of Laplace decomposition algorithm is higher thanthat of the diferential transform method.
论文共分五章。第一章概述了微分变换法的基本性质，介绍了Laplace分解算法(SLDA)。第二章给出了标准Laplace分解方法(MLDA)，该方法使收敛速度加快；同时，将SLDA、MLDA和DTM用于求解带比例延迟的泛函微分方程。第三章给出了高阶非线性比例积分-微分方程的解。第四章研究了SLDA、MLDA和DTM在求解中立型泛函微分方程的情况。最后一章给出了变分迭代法(VIM)和修正Laplace分解法的联系；同时，将SLDA、MLDA和DTM应用于带比例延迟的偏微分方程。附录中对变分方法作了简单介绍。
This thesis presents a comparative study between two powerful methods that weredeveloped recently. The first method is the diferential transform method (DTM), whichdeveloped by the Chinese scientist Zhou in1986and the second one, is the Laplace de-composition algorithm (LDA), which introduced by the Arabian researcher Khuri in2001.Both methods are used heavily in the literature. The efectiveness of these two method-s when used for certain categories of nonlinear pantograph diferential equation is alsoevaluated in this thesis.
This thesis is divided into five chapters. The first Chapter is devoted to clarify thefundamental properties of the diferential transform method (DTM) and it introduces thestandard Laplace decomposition algorithm (SLDA). The second chapter introduces a newform of SLDA called modified Laplace decomposition algorithm (MLDA), which ledto an increase in the speed of convergence. In this chapter, SLDA, MLDA and DTM areadapted and applied to solve Functional diferential equations with proportional delays. Inthe third chapter, the solution of the higher-order nonlinear pantograph integro-diferentialequation is presented. The fourth chapter is devoted to apply SLDA, MLDA and DTM tosolve neutral functional–diferential equations. Finally, a relation between the variationaliteration method (VIM) and the modified Laplace decomposition algorithm is presented.The application of SLDA, MLDA and DTM for solving partial diferential equation withproportional delays is presented in the fifth Chapter. In addition, a brief introduction aboutthe variational iteration method (VIM) is given in the appendix.
The conclusion that is pointed out in this study is that the diferential transformmethod is easier to use, time and efort saving more than Laplace decomposition algorith-m. However, the rate of convergence of Laplace decomposition algorithm is higher thanthat of the diferential transform method.
引文
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[15] X. Lv, Y. Gao. The Rkhsm for Solving Neutral Functional–diferential Equationswith Proportional Delays. Mathematical Methods in the Applied Sciences.2012,36(6):642–649
[16] G. Bocharov, F. Rihan. Numerical Modelling in Biosciences Using Delay Dif-ferential Equations. Journal of Computational and Applied Mathematics.2000,125(1):183–199
[17] C. Baker, G. Bocharov, F. Rihan. A Report on the Use of Delay Diferential E-quations in Numerical Modelling in the Biosciences. Numerical Analysis Report.1999,343
[18] G. Hutchinson. Circular Causal Systems in Ecology. Annals of the New YorkAcademy of Sciences.1948,50(4):221–246
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