روش انتگرال گيري ضربي براي حل معادلات ديفرانسيل معمولي كسري

چكيده انگليسي :

A Product Integration Method For Solving Fractional Ordinary Differential Equations Saeed Raisi Zargari s raisi@math iut ac ir October 2020 Master of Science Thesis in Farsi Departement of Mathematical Sciences Isfahan University of Technology Isfahan 84156 8311 IranSupervisor Dr Reza Mokhtari mokhtari@iut ac irAdvisor Dr Mehdi Tatari mtatari@iut ac ir2000 MSC 65L20 26A33Keywords Ordinary nonlinear fractional differential equations Fractional calculus Product integration Fixed point iteration Legendre polynomials Abstract In this thesis we have examined two categories of fractional calculus problems These two categories are linear andnonlinear fractional ordinary differential equations For this purpose first of all some necessary definitions and the orems such as the introduction of the weight function and continuous inner product and orthogonality and orthogonalpolynomials Gaussian quadrature and an introduction to three special functions gamma beta and Mittag Leffler that play the most important role in the theory of fractional calculus and fractional differential equations are given Then fractional integrals and derivatives of Grunwald Letnikov Riemann Liouville and Caputo and their propertiesare considered After that the relationship between these approaches and two important inequalities i e Gronwalland Cauchy Schwartz inequalities are represented and then fixed point iterative method and contraction mappingtheorem are mentioned Forward and backward Euler methods and their stability analyses as well as the regions ofstability of these two methods are investigated Then the existence and uniqueness of the solution of the fractionaldifferential equation are expressed After that we proceed to the numerical method of product integration to solve fractional differential equations Weconvert the fractional differential equation to a second type Volterra integral equation Many numerical methodsare created by considering this conversion Then with two division steps we divide the domain of the differentialequation into subintervals Then we replace the nonlinear function f with the Fourier expansion In this expansion we need a kind of orthogonal polynomial as the basis of the Fourier expansion Here we use Legendre orthogonalpolynomials Since the Legendre polynomials are defined in the interval 1 1 we move it to the interval 0 1 using a linear transformation and the obtained polynomials are called the shifted Legendre polynomials By selectingthese polynomials we rewrite the Fourier expansion and its coefficients Here we calculate the expansion coefficientsusing a quadrature formula and we obtain the knot points and quadrature weights using the two Gauss Legendre andGauss Radau quadratures For each we consider two two point and three point modes and a total of four methodsare obtained Finally we substitute the nonlinear function f for the Fourier expansion f in the piecewise Volterraintegral equation type II The resulting discrete formula is a set of nonlinear implicit equations that is best solved bythe fixed point method Because in terms of time and cost the program is more economical than other methods Thenwe get the error of the mentioned method and its order After that we consider the convergence condition of the fixedpoint iteration method Then we examine the stability of the stated method for four methods and each method threevalues of the parameter of the order of the equation and draw the regions of stability Finally we obtain the numericalresults of the method for several examples of this type of equation with tables and graphs We show that the methodunder investigation is numerically very efficient and effective

رئيسي زرگري، سعيد

روش انتگرال گيري ضربي براي حل معادلات ديفرانسيل معمولي كسري

معادلات ديفرانسيل معمولي كسري غير خطي , روش انتگرال گيري ضربي