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, short- est 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. There are two main approaches to solve problems of the fractional calculus of variations or optimal control problems: indirect methods and direct methods. One involves solving fractional Euler-Lagrange equations or fractional Pontryagin-type conditions, which is the indirect approach; the other involves addressing directly the problem, without involving necessary optimality conditions, which is the direct approach. The emphasis in the literature has been put on direct methods.Several classes of wavelet analysis possess some useful properties, such as orthogonality, compact support,exact representation of polynomials to certain degree, and ability to represent functions at different levels of resolution.Moreover, wavelets establish a connection with fast numerical algorithm.In this thesis, a numerical method based on Bernoulli wavelets is presented for solving linear and nonlinear variational problems. First, the construction of Bernoulli wavelets is described, and then the operational matrices of integration and multiplication corresponding to Bernoulli wavelets are derived. Using the properties of Bernoulli wavelets and the aforementioned operational matrices, the variational problem under study is transformed into a parametric optimization problem, which is significantly simpler to solve than the original problem. Finally, various examples are provided to examine and eva‎luate the accuracy and efficiency of the proposed method.We derive the operational matrices of integration and product for Bernoulli wavelets and apply them to solve linear and nonlinear problems in calculus of variations.The operational matrices of integration and product with the Bernoulli wavletes contain many zero elements, making them sparse and thereby reducing computer memory requirements. Additionally, a small number of Bernoulli wavelets suffice to achieve satisfactory results, as demonstrated by the numerical examples provided.Various types of linear and nonlinear variational problems are investigated to show the efficiency and accuracy of the proposed numerical method.

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محمود منجگاني , رسول عاشقي حسين آبادي