Methods For Approximating And Stabilizing The Solution Of Nonlinear Riccati Matrix Delay Differential Equation

where ,A B and C are nn matrices such that , TBB TCC and ( ) . nn X t R This nonlinear Riccati matrix differential equation may also be viewed as a quadratic ordinary differential equation. The above equation may be generalized for delay differential equations with retarded arguments, in which the delay term occurs as a constant time delay in ()Xt but not in ()Xt (the derivative will disappear and the equation will become algebraic Riccati matrix equation after the initial condition is used). In this thesis we study the variational iteration method and use it to solve nonlinear Riccati matrix differential equation and nonlinear Riccati matrix delay differential equations. The solution approach requires, initially, the derivation of the variational iteration method for solving such types of equations and then proof of its convergence to the exact solution in two cases with and without delay. The Adomian decomposition method is then applied for solving nonlinear Riccati matrix differential equation in two cases with and without delay

An approximation algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method

In this paper we propose a collocation method for solving some well-known classes of Lane-Emden type equations which are nonlinear ordinary differential equations on the semi-infinite domain. They are categorized as singular initial value problems. The proposed approach is based on a Hermite function collocation (HFC) method. To illustrate the reliability of the method, some special cases of the equations are solved as test examples. The new method reduces the solution of a problem to the solution of a system of algebraic equations. Hermite functions have prefect properties that make them useful to achieve this goal. We compare the present work with some well-known results and show that the new method is efficient and applicable.Comment: 34 pages, 13 figures, Published in "Computer Physics Communications

Laplace Adomian Decomposition and Modify Laplace Adomian Decomposition Methods for Solving Linear Volterra Integro-Fractional Differential Equations with Constant Multi-Time Retarded Delay

     في هذا العمل نقدم تحويلات لابلاس مع طريقة أدوميان التحليلية المتسلسلة و كما اننا نعدل طريقة أدوميان التحليلية للمرة الاولى لحل معادلات فولتيرا التفاضلية-التكاملية الخطيه للرتب الكسرية كما في مفهوم كابوتو مع التأخير الحدي المتضاعف الثابت. هذه الطريقة تعتمد على مزيج ممتاز من طريقة تحويلات لابلاس، طريقة تحديد المتسلسلات، طريقة متعددات الحدود لادوميان مع التعديلات. أن التقنية المستخدمة تحول التأخير الحدي للمعادلات التفاضلية ذات التكاملات الكسرية الى معادلات جبرية متكررة عندما تكون نواة الفروق من نوع المنحل البسيط. و أخيراَ أعطيت أمثلة لتوضيح فعالية و ديقة الطرق المقترحة.In this work, we present Laplace transform with series Adomian decomposition and modify Adomian decomposition methods for the first time to solve linear Volterra integro-differential equations of the fractional order in Caputo sense with constant multi-time Retarded delay. This method is primarily based on the elegant mixture of Laplace transform method, series expansion method and Adomian polynomial with modifications. The proposed technique will transform the multi-term delay integro-fractional differential equations into some iterative algebraic equations, and it is capable of reducing computational analytical works where the kernel of difference and simple degenerate types. Analytical examples are presented to illustrate the efficiency and accuracy of the proposed methods

Optimal Perturbation Iteration Method for Bratu-Type Problems

In this paper, we introduce the new optimal perturbation iteration method based on the perturbation iteration algorithms for the approximate solutions of nonlinear differential equations of many types. The proposed method is illustrated by studying Bratu-type equations. Our results show that only a few terms are required to obtain an approximate solution which is more accurate and efficient than many other methods in the literature.Comment: 11 pages, 3 Figure