WINSEM2018-19_MAT2002_ETH_SJT502_VL2018195005967_Reference Material II_Module 5.4_Frobenius_Method.p

WINSEM2018-19_MAT2002_ETH_SJT502_VL2018195005967_Reference Material II_Module 5.4_Frobenius_Method.p

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Frobenius Method Dr. T. Phaneendra Professor of Mathematics School of Advanced Sciences VIT University, Vellore - 632 014 (TN) E-mail:[email protected] February 13, 2018
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Table of contents 1 Regular and Singular Points 2 Frobenius Method 3 Bessel Functions Dr. T. Phaneendra (VIT) System of Differential Equations February 13, 2018 2 / 25
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Regular and Singular Points Singular Point Recall that a point x = x 0 is called the regular or ordinary point of the second order homogeneous differential equation y 00 + p ( x ) y 0 + q ( x ) y = 0 if p ( x ) and q ( x ) are analytic at x = x 0 . If x = x 0 is not regular point then it is called a singular point A singular point is further classified as regular singular point irregular singular point Dr. T. Phaneendra (VIT) System of Differential Equations February 13, 2018 3 / 25
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Regular and Singular Points Regular Singular Point The point x = x 0 is a regular singular point of the equation y 00 + p ( x ) y 0 + q ( x ) y = 0 if x = x 0 is not an ordinary point but both ( x - x 0 ) p ( x ) and ( x - x 0 ) 2 q ( x ) are analytic at x = x 0 . Example 1.1 Consider 2 x 2 y 00 + 7 x ( x + 1 ) y 0 - 3 y = 0 . Here, the point x = 0 is a regular singular point Consider 8 x 2 y 00 + 10 xy 0 + ( x - 1 ) y = 0 . Here, the point x = 0 is a regular singular point Dr. T. Phaneendra (VIT) System of Differential Equations February 13, 2018 4 / 25
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Regular and Singular Points Exercise 1.1 Determine whether the point x = 0 is a regular singular point for the follow- ing, with justification: x 3 y 00 + 2 x 2 y 0 + y = 0. y 00 - xy 0 + 2 y = 0. Dr. T. Phaneendra (VIT) System of Differential Equations February 13, 2018 5 / 25
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Frobenius Method Method of Frobenius If x = 0 is a regular singular point of y 00 + p ( x ) y 0 + q ( x ) y = 0 , (2.1) then this equation has at least one solution of the form y = x r X n = 0 a n x n = X n = 0 a n x n + r , (2.2) where r and a n ( n = 0 , 1 , 2 , · · · ) are constants. This series is known as the Frobenius series, which is convergent in the interval 0 < x < R for some real R . Then y 0 = X n = 0 ( n + r ) a n x n + r - 1 , and y 00 = X n = 0 ( n + r )( n + r - 1 ) a n x n + r - 2 (2.3) Insert (2.2) and (2.3) into (2.1). Dr. T. Phaneendra (VIT) System of Differential Equations February 13, 2018 6 / 25
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Frobenius Method Computing the Coefficients a n and r in (2.2) Collect the coefficients of like powers of x and equate to zero With the coefficient of general power x n + r , the resulting relation will give a recurrence formula Equating to zero the coefficient of the lowest power of x , we obtain a quadratic equation in r , called the indicial equation, of the problem Find the two roots r 1 and r 2 of the indicial equation. Suppose that r 1 and r 2 do not differ by an integer.
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