توصيفگر ها :
مسأله براچيستوچرون , حساب تغييرات , روش هاي عددي , موجك هاي لژاندر
چكيده فارسي :
در اين پايان‌نامه، يك روش عددي براي حل مسائل غيرخطي حساب تغييرات ارائه شده است. روش ارائه شده مبتني بر موجك‌هاي لژاندر است. ابتدا نحوه ساخت موجك‌هاي لژاندر و ماتريس‌هاي عملياتي انتگرال و حاصلضرب متناظر با موجك‌هاي لژاندر را بررسي مي‌كنيم، سپس با استفاده از ماتريس‌هاي عملياتي انتگرال و حاصلضرب، مسأله كنترل بهينه مورد مطالعه به يك مسأله بهينه‌سازي پارامتري تبديل مي‌شود كه حل آن به مراتب ساده تر از حل مسأله اصلي است. براي بررسي و ارزيابي دقت و كارايي روش ارائه شده، مثال‌هاي گوناگوني ارائه شده است.
چكيده انگليسي :
In this thesis, an efficient numerical method based on the Legendre wavelets is presented. There has been a considerable renewal of interest in the classical problems of the calculus of variations both from the point of view of mathematics and of applications in physics, engineering and applied mathematics. Finding the brachistochrone, or path of quickest decent, is a historically interesting problem that is discussed in virtually all textbooks dealing with the calculus of variations in 1696, the brachistochrone problem was posed as a challenge to mathematicians by John Bernoulli. The solution of the brachistochrone problem is often cited as the origin of the calculus of variations. The classical brachistochrone problem deals with a mass moving along a smooth path in a uniform gravitational field. A mechanical analogy is the motion of a bead sliding down a frictionless wire. The solution to this problem was obtained by various methods such as the gradient method, successive sweep algorithm, the classical Chebyshev method, and multistage Monte Carlo method. Orthogonal functions (OFs) have received considerable attention in dealing with various problems of dynamic systems. The main characteristics of this technique is that it reduces these problems to those of solving a system of algebraic equations, thus, greatly simplifying the problem. The approach is based on converting the underlying differential equations into integral equations through integration, approximating various signals moved in the equation by truncated orthogonal series and using the operational matrix of integration to eliminate the integral operations. The form of this matrix depends on the particular choice of the orthogonal functions. Special attention has been given to applications of Walsh functions, block-pulse functions, Laguerre series, shifted Legendre polynomials, and shifted Chebyshev polynomials. Among these orthogonal functions, the shifted Legendre, which is obtained from Legendre polynomials by shifting the defining interval [-1, 1] to [0, L], is computationally more effective. This is because the operational matrix of integration is tridiagonal, the weight function of orthogonality is unity, the convergence rate is rapid. There are three classes of sets of OFs which are widely used. The first includes sets of piecewise constant basis functions such as Walsh functions and block-pulse functions. The second consists of sets of orthogonal polynomials such as Laguerre, Legendre, and Chebyshev polynomials. The third is the widely used sets of sine-cosine functions in Fourier series. While Orthogonal polynomials and sine-cosine functions together form a class of continuous basis functions, PCBFs have inherent discontinuous or jumps. The inherent features of a set of OFs largely determine their merit for application a given situation. If a continuous function is approximated by PCBFs, the resulting approximation is piecewise constant. On the other hand, if a discontinuous function is approximated by continuous basis functions, the discontinuities are not properly modeled. Wavelet analysis possesses several useful properties, such as orthogonality, compact support, exact representation of polynomials to a certain degree, and ability to represent functions at different levels of resolution. In this thesis, we introduce a new numerical method to solve the nonlinear problems of the calculus of variations. The method consists of reducing the calculus of variations problems to a set of algebraic equations by expanding the candidate function as a Legendre wavelets with unknown coefficients. we demonstrate the accuracy of the proposed numerical scheme by comparing our numerical finding with the exact solutions.
