توصيفگر ها :
موجك‌هاي مونتس , حساب تغييرات , ماتريس عملياتي انتگرال , ضرايب لاگرانژ , شرايط تقاطعي
چكيده فارسي :
در اين پايان نامه، يك روش عددي براي حل مسائل حساب تغييرات پيشنهاد شده است. روش ارائه شده، مبتني بر موجك هاي مونتس است. با استفاده از خواص و ويژگي هاي موجك هاي مونتس و ماتريس عملياتي انتگرال متناظر با آنها، مسأله حساب تغييرات مورد نظر به يك مسأله بهينه سازي پارامتري تبديل مي شود، كه با استفاده از روش ضرايب لاگرانژ حل شده است. با ارائه مثال هاي گوناگون از مسائل حساب تغييرات، دقت و كارايي روش پيشنهادي مورد بررسي و ارزيابي قرار گرفته است.
چكيده انگليسي :
The problem of determining a function which optimizes a certain functional is called variational problem. The variational problems have been analyzed extensively by engineers, mathematicians, and scientists. Such types of problems appear in science, engineering, and several fields of real life such as economics, biology, solid mechanics, etc. Moreover, the variational problems have drawn great attention in various practical applications such as heat conduction model. The functions that extremize functional can be determined by using the Euler–Lagrange equation, but that equation cannot always be solved. Therefore, various direct techniques based on orthogonal functions and polynomial series have been used to solve the variational problems. It is necessary to find the extremum of a certain functional in many problems of applied sciences and engineering such as geometry, economics, mechanics, analysis, and so on. The variational problems with different boundary conditions have gained tremendous attention because of the important role of this subject in mathematics, engineering, and applied sciences. Some applications of such types of variational problems appear in the heat conduction problem, brachistochrone problem, Ramsey growth model, shortest path problem, ordinary differential equation, systems of boundary value problems, etc. Due to a lot of applications of the variational problems with different boundary conditions in several areas, the focus of the researchers is on the numerical solutions of the variational problems with different boundary conditions. So, motivated by the above discussions, in this study, we consider the variational problems with moving or fixed boundary conditions. In this thesis, an efficient approximation approach based on Müntz wavelets is introduced for solving this kind of the variational problems. This approach is a common approach, but implementation with Müntz wavelets basis functions is new. The properties of Müntz wavelets and the associated operational matrix of integration are used to convert the variational problem under study into a parameter optimization problem. The method consists of reducing the given variational problems into an algebraic system by expressing the higher derivative term in the form of Müntz wavelets with the unknown wavelet coefficients. The properties of Müntz wavelets along with the operational matrix of integration and the Lagrange multipliers are applied to calculate the unknown wavelet coefficients and find the approximate wavelet solution in the given problems. In this thesis, an approximation method with an integral operational matrix based on the Müntz wavelets basis is presented to solve the variational problems of moving or fixed boundary conditions and a computational algorithm is given for the suggested approach. First, the integral operational matrix is created through the Müntz wavelets. Then, by using this integral operational matrix with Lagrange multipliers, the present approach reduces the variational problem into the system of algebraic equations. This approach is examined by some illustrative examples, and the acquired results prove that the suggested approach can solve the variational problems effectively with higher accuracy. The proposed approach yields better and comparable results with some other existing schemes given in the literature. The approximate wavelet solutions derived by the suggested approach are very identical to the corresponding exact solution.
