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C 1 : Power Series Solutions and Special Functions

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C 1 : Power Series Solutions and Special Functions CONTENTS CHAPTER 1 POWER SERIES SOLUTIONS 03 1.1 INTRODUCTION 03 1.2 POWER SERIES SOLUTIONS 04 1.3 REGULAR SINGULAR POINTS – 05 FROBENIUS SERIES SOLUTIONS 1.4 GAUSS’S HYPER GEOMETRIC EQUATION 07 1.5 THE POINT AT INFINITY 09 CHAPTER 2 SPECIAL FUNCTIONS 11 2.1 LEGENDRE POLYNOMIALS 11 2.2 BESSEL FUNCTIONS – GAMMA FUNCTION 15 CHEPTER 3 SYSTEMS OF FIRST ORDER EQUATIONS 20 3.1 LINEAR SYSTEMS 20 3.2 HOMOGENEOUS LINEAR SYSTEMS WITH 21 CONSTANT COEFFICIENTS 3.3 NON LINEAR SYSTEM 24 CHAPTER 4 NON LIEAR EQUATIONS 26 4.1 AUTONOMOUS SYSTEM 26 4.2 CRITICAL POINTS & STABILITY 28 4.3 LIAPUNOV’S DIRECT METHOD 31 4.4 SIMPLE CRITICAL POINTS -NON LINEAR SYSTEM 34 CHAPTER 5 FUNDAMENTAL THEOREMS 38 5.1 THE METHOD OF SUCCESSIVE APPROXIMATIONS 38 5.2 PICARD’S THEOREM 39 Differential Equations 3 CHAPTER 6 FIRST ORDER PARTIAL DIFFERENTIAL EQUATIONS 46 6.1 INTRODUCTION – REVIEW 46 6.2 FORMATION OF FIRST ORDER PDE 48 6.3 CLASSIFICATION OF INTEGRALS 50 6.4 LINEAR EQUATIONS 54 6.5 PFAFFIAN DIFFERENTIAL EQUATIONS 56 6.6 CHARPIT’S METHOD 62 6.7 JACOBI’S METHOD 66 6.8 CAUCHY PROBLEM 70 6.9 GEOMETRY OF SOLUTIONS 74 CHAPTER 7 SECOND ORDER PARTIAL DIFFERENTIAL EQUATIONS 7.1 CLASSIFICATION 78 7.2 ONE DIMENSIONAL WAVE EQUATION 81 7.3 RIEMANN’S METHOD 87 7.4 LAPLACE EQUATION 89 7.5 HEAT CONDUCTION PROBLEM 95 - 98 Differential Equations 4 CHAPTER 1 POWER SERIES SOLUTIONS AND SPECIAL FUNCTIONS 1.1 Introduction An algebraic function is a polynomial, a rational function, or any function that satisfies a polynomial equation whose coefficients are polynomials. The elementary functions consists of algebraic functions, the elementary transcendental functions or non algebraic functions- the trigonometric functions and their inverses, exponential and logarithmic functions and all others that can be constructed from these by adding or multiplying or taking compositions. Any other function is called a special function. n Consider the power series an x . The series has a radius of convergence R, n0 0 R such that the series converges for | x | < R and diverges for | x | > R. lim a We have, R = n1 . n an x n For the geometric series 1+ x + x 2 + … , R = 1 and for the exponential series , R = n0 n! and the series n!x n converges only for x = 0. n0 Suppose = f(x) for | x | < R . Then f(x) has derivatives of all orders and the series can n1 n2 be differentiated term by term f '(x) nan x , f ''(x) n(n 1)an x and so on and n1 n1 f (0) each series converges for | x | < R. In fact, we get a n ,n . n n! n A function f(x) which can be expanded as a power series an (x x0 ) , valid in some n0 neighborhood of x 0, is said to be analytic at x 0. Polynomials, e x, sin x , cos x are analytic at all points, but 1/( 1+ x) is not at x = -1.. Differential Equations 5 1.2. Power series solutions It may be recalled that many differential equations can not be solved by the few analytical methods developed and these methods can be employed only if the differential equations are of a particular type. By applying the following method solutions can be obtained as a power series and hence known as power series method. Consider the equation y ' y. We may assume that this equation has a power series solution in the form y = that converges for | x | < R, for some R. ' 2 ' Then y a1 2a2 x 3a3 x ..... Since y y , by equating the coefficients of like powers of x, we get a1=a0, 2a2=a1,3a3=a2,… which reduces to a1=a0,a2=a1/ 2 = a0/2!,a3=a0/3!,…. 2 x Thus we obtain, y = a0( 1+ x/1! + x /2!+…..) = a0 e , where a0 is left undetermined and hence arbitrary. n an x n0 Now let us consider the general second order homogeneous equation, y '' P(x)y ' Q(x)y 0. (*). If both P(x) and Q(x) are analytic at x =x0, we say x0 is an ordinary point of the equation n We may assume the solution of the equation (*) as a power series y = an (x x0 ) valid for n0 |x-x0| < R, for some R. The various coefficients can be found in terms of a0 and a1, which is left undetermined. Consider y '' y 0. Here P(x)=0 and Q(x) = 1, which are analytic at x = 0. Assume y = . Then the equation gives the recurrence relation (n+1)(n+2) an+2+an=0 , n for n=0,1,2,….. .Substituting n =0,1,2,..successively and reducing we get a2n+1 = (-1) a1/ 2 4 3 5 n x x x x (2n+1)! and a2n=(-1) a0/(2n)!. Hence y = a (1 ...) a (x ....) 0 2! 4! 1 3! 5! = a0 cos x + a1 sin x. Consider the Legendre’s equation 1 x2 y '' 2xy' p( p 1)y 0 , where p is a constant. 2x p( p 1) Here P(x) = and Q(x) = , which are analytic at x = 0. 1 x 2 1 x 2 Differential Equations 6 Let y = . Then the equation gives the recurrence relation (n+1)(n+2)an+2-(n-1)an- p( p 1) ( p 1)( p 2) 2nan+p(p+1) an= 0. Put n = 0,1,2,..which gives a a ,a a , 2 2! 0 3 3! 1 p( p 2)( p 1)( p 3) ( p 1)( p 3)( p 2)( p 4) a a ,a a ,..... 4 4! 0 5 5! 1 p( p 1) 2 ( p 2) p( p 1)( p 3) 4 Thus y = a0 1 x x ...... 2! 4! ( p 1)( p 2) 3 ( p 3)( p 1)( p 2)( p 4) 5 +a1 x x x ...... 3! 5! The radius of convergence for each of the series in the brackets is R = 1. The series in the first bracket terminates for p = 0,2,4,6,.. and the series in the second bracket terminates for p = n an x 1,3,5,….. The resulting polynomials aren0 called Legendre polynomials whose properties will be discussed later. Ex. The equation y '' ( p 1 2 1 4 x2 )y 0 , where p is a constant, has a power series solution y = at x = 0. Show that the coefficients are related by the three term recurrence relation (n 1)(n 2)an2 ( p 1 2)an 1 4an2 0. If the dependent variable y x2 is replaced by y = w e 4 , show that the equation is transformed to w'' xw' pw 0 and its power series solution at x = 0, involves only a two term recurrence relation. 1.3. Regular singular points x = x0 is a singular point of (*) if either P(x) or Q(x) is not analytic at x0. In this case the power series solution may not exist in a neighborhood of x0. But the solutions near a singular point is important in a physical context, and most of the cases they exist. Origin is a singular '' 2 ' 2 -2 point of y y y 0 and for x > 0, y = c1 x + c2 x , is its general solution. x x 2 2 '' ' 2 A singular point x0 of (*) is called regular 1singular x y if 2bothxy ( px(-px0)P(x)1)y &0 (x-x0) Q(x) are analytic at x0. Consider the Legendre’s equation , for which x =1 , -1 are singular points but they are regular singular. For the Bessel equation of order p, Differential Equations 7 x2 y '' xy' (x2 p 2 )y 0 , where p is a non negative constant, x = 0 is a regular singular point. 2 If x = x0 is regular singular point of (*), then by definition ( x-x0)P(x) & (x-x0) Q(x) are n analytic at x0 and hence we may take ( x-x0)P(x) = pn (x x0 ) and n0 2 n (x-x0) Q(x) = qn (x x0 ) . A solution of the equation (*) as a Frobenius series n0 m n y = x x0 an (x x0 ) , where m is a real number and a0 is assumed non zero, can be n0 expected. On substituting, y = in (*), and equating the coefficients, we get n1 the recursion formula an [(m n)(m n 1) (m n) p0 q0 ] ak [(m k) pnk qnk ] 0 k 0 lim lim 2 ( ** ). Here p0 (x x0 )P(x) and q0 (x x0 ) Q(x) . x x0 x x0 For n=0, ( ** )gives m(m 1) mp0 q0 0 ***, called the indicial equation, which determines the values of m. Substituting the values of m and taking n=1,2,3,.. in ( ** ) an’s can be determined in terms of a0 and the respective solutions can be obtained . Eg. Consider the equation 2x2 y '' x(2x 1)y ' y 0 . x = 0 is a regular singular point of m n the equation. Let us assume that the solution at x = 0, is y = x an x . n0 1 1 we get the indicial equation, m(m 1) m 0 --(1). m = 1 , -1/2 . For m = 1 , -1/2 2 2 respectively we get the solutions on determining the an’s successively from the recurrence 2 4 1 relation ( ** ) as y a (x x 2 x 3 ...) and y a x 1 2 (1 x x 2 ...) which 1 0 5 35 2 0 2 are independent also and thereby the general solution is y = c1 y1 + c2 y2, where c1& c2 are arbitrary constants.
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