Section 1.1 Frobenius Method In this section, we consider a method to find a general solution to a second order ODE about a singular point, written in either of the two equivalent forms below: \begin{equation} x^2 y'' + xb(x)y' + c(x) y = 0\label{frobenius-standard-form1}\tag{1.1.1} \end{equation} also Singular point). 343.8 593.8 312.5 937.5 625 562.5 625 593.8 459.5 443.8 437.5 625 593.8 812.5 593.8 /Name/F8 \end{equation*}, In the following, the zeros $\lambda _ { i }$ of the indicial polynomial will be ordered by requiring, \begin{equation*} \operatorname { Re } \lambda _ { 1 } \geq \ldots \geq \operatorname { Re } \lambda _ { \nu }. 1062.5 1062.5 826.4 288.2 1062.5 708.3 708.3 944.5 944.5 0 0 590.3 590.3 708.3 531.3 << >> /BaseFont/XKICMY+CMSY10 /LastChar 196 This article was adapted from an original article by Franz Rothe (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. 544 516.8 380.8 386.2 380.8 544 516.8 707.2 516.8 516.8 435.2 489.6 979.2 489.6 489.6 www.springer.com << 0 0 0 0 0 0 0 0 0 0 0 0 675.9 937.5 875 787 750 879.6 812.5 875 812.5 875 0 0 812.5 /FirstChar 33 In particular, this can happen if the coe cients P(x) and Q(x) in the ODE y00+ P(x)y0+ Q(x)y = 0 fail to be de ned at a point x 0. However, the method of Frobenius can be extended to the case where , , and are functions that can be represented by power series in on some interval that contains zero, and . One gets $L _ { 0 } ( u ^ { \lambda } ) = \pi ( \lambda ) z ^ { \lambda }$ with the indicial polynomial, \begin{equation} \tag{a5} \pi ( \lambda ) = \sum _ { n = 0 } ^ { N } ( \lambda + n ) ( \lambda + n - 1 ) \ldots ( \lambda + 1 ) a ^ { n _0} = \end{equation}, \begin{equation*} = a _ { 0 } ^ { N } \prod _ { i = 1 } ^ { \nu } ( \lambda - \lambda _ { i } ) ^ { n _ { i } }. Ferdinand Georg Frobenius (26 October 1849 – 3 August 1917) was a German mathematician, best known for his contributions to the theory of elliptic functions, differential equations, number theory, and to group theory.He is known for the famous determinantal identities, known as Frobenius–Stickelberger formulae, governing elliptic functions, and for developing the theory of biquadratic forms. Since the general situation is rather complex, two special cases are given first. /Filter[/FlateDecode] 875 531.3 531.3 875 849.5 799.8 812.5 862.3 738.4 707.2 884.3 879.6 419 581 880.8 0 is y(x) … cxe1=x, which could not be captured by a Frobenius expansion. /FirstChar 33 /FirstChar 33 820.5 796.1 695.6 816.7 847.5 605.6 544.6 625.8 612.8 987.8 713.3 668.3 724.7 666.7 Frobenius Method If is an ordinary point of the ordinary differential equation, expand in a Taylor series about. /FirstChar 33 << The solution of the … 826.4 295.1 531.3] Because for $i = 1 , \dots , \nu$ and $l = 0 , \dots , n _ { i } - 1$, all leading terms are different, the method of Frobenius does indeed yield a fundamental system of $N$ linearly independent solutions of the differential equation (a3). \begin{equation*} u ( z , \lambda ) = z ^ { \lambda } \sum _ { k = 0 } ^ { \infty } c _ { k } ( \lambda ) z ^ { k }, \end{equation*}, the coefficients have to be calculated from the requirement (a7). 444.4 611.1 777.8 777.8 777.8 777.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 << /Type/Font /Subtype/Type1 /Widths[351.8 611.1 1000 611.1 1000 935.2 351.8 481.5 481.5 611.1 935.2 351.8 416.7 endobj with $\lambda = \lambda _ { 2 }$ in the second function, are two linearly independent solutions of the differential equation (a9). /LastChar 196 This is usually the method we use for complicated ordinary differential equations. 334 405.1 509.3 291.7 856.5 584.5 470.7 491.4 434.1 441.3 461.2 353.6 557.3 473.4 \end{equation*}, Here, $p _ { i } ( \lambda )$ are polynomials of degree at most $N$ determined by setting, \begin{equation*} p _ { i } ( z ) z ^ { \lambda } = \sum _ { n = 0 } ^ { N } a ^ { n _ { i } } z ^ { n } ( \frac { \partial } { \partial z } ) ^ { n } z ^ { \lambda }. 21 0 obj /Widths[295.1 531.3 885.4 531.3 885.4 826.4 295.1 413.2 413.2 531.3 826.4 295.1 354.2 We classify a point x n; y2(x) =xr2. { l ! } /FirstChar 33 805.5 896.3 870.4 935.2 870.4 935.2 0 0 870.4 736.1 703.7 703.7 1055.5 1055.5 351.8 /Subtype/Type1 The method looks simpler in the most common case of a differential operator, \begin{equation} \tag{a9} L = a ^ { [ 2 ] } ( z ) z ^ { 2 } \left( \frac { d } { d z } \right) ^ { 2 } + a ^ { [ 1 ] } ( z ) z \left( \frac { d } { d z } \right) + a ^ { [ 0 ] } ( z ). 4 Named after the German mathematician Ferdinand Georg Frobenius (1849 – 1917). 300 325 500 500 500 500 500 814.8 450 525 700 700 500 863.4 963.4 750 250 500] 896.3 896.3 740.7 351.8 611.1 351.8 611.1 351.8 351.8 611.1 675.9 546.3 675.9 546.3 295.1 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 295.1 295.1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 753.7 1000 935.2 831.5 Method of Frobenius: The Exceptional Cases Now, we have to take a look at what happens when r 1 − r 2 is an integer. Case (d) Complex conjugate roots If c 1 = λ+iμ and c 2 = λ−iμ with μ = 0, then in the intervals −d < x < 0 and 0 < x < d the two linearly independent solutions of the differential equation are 462.4 761.6 734 693.4 707.2 747.8 666.2 639 768.3 734 353.2 503 761.2 611.8 897.2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 642.9 885.4 806.2 736.8 /BaseFont/LQKHRU+CMSY8 500 1000 500 500 500 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 /FontDescriptor 35 0 R /LastChar 196 /Widths[609.7 458.2 577.1 808.9 505 354.2 641.4 979.2 979.2 979.2 979.2 272 272 489.6 ���ů�f4[rI�[��l�rC\�7 ����Kn���&��͇�u����#V�Z*NT�&�����m�º��Wx�9�������U]�Z��l�۲.��u���7(���"Z�^d�MwK=�!2��jQ&3I�pݔ��HXE�͖��. 413.2 590.3 560.8 767.4 560.8 560.8 472.2 531.3 1062.5 531.3 531.3 531.3 0 0 0 0 531.3 531.3 413.2 413.2 295.1 531.3 531.3 649.3 531.3 295.1 885.4 795.8 885.4 443.6 /Widths[1062.5 531.3 531.3 1062.5 1062.5 1062.5 826.4 1062.5 1062.5 649.3 649.3 1062.5 << endobj 351.8 611.1 611.1 611.1 611.1 611.1 611.1 611.1 611.1 611.1 611.1 611.1 351.8 351.8 In this video, I introduce the Frobenius Method to solving ODEs and do a short example.Questions? \begin{equation*} ( \frac { \partial } { \partial \lambda } ) ^ { n _ { 1 } + l } [ u ( z , \lambda ) ( \lambda - \lambda _ { 2 } ) ^ { n _ { 1 } } ] = \end{equation*}, \begin{equation*} = \frac { ( n _ { 1 } + l ) ! } u ( z ) = z r ∑ k = 0 ∞ A k z k , ( A 0 ≠ 0 ) {\displaystyle u (z)=z^ {r}\sum _ {k=0}^ {\infty }A_ {k}z^ {k},\qquad (A_ {0}\neq 0)} Differentiating: u ′ ( z ) = ∑ k = 0 ∞ ( k + r ) A k z k + r − 1. Question: Exercise 3. endobj /FontDescriptor 8 0 R 36 0 obj 675.9 1067.1 879.6 844.9 768.5 844.9 839.1 625 782.4 864.6 849.5 1162 849.5 849.5 1. >> The approach does produce special separatrix-type solutions for the Emden–Fowler equation, where the non-linear term contains only powers. 384.3 611.1 611.1 611.1 611.1 611.1 896.3 546.3 611.1 870.4 935.2 611.1 1077.8 1207.4 endobj 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 1062.5 1062.5 826.4 826.4 Frobenius’ method for curved cracks 63 At the same time the unknowns B i must satisfy the compatibility equations (2.8), which, after linearization, become 1 0 B i dξ=0. 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 489.6 272 489.6 /LastChar 196 /Type/Font If q=r1¡r2is not integer, then the solution basis of the ODE(1)is given by y1(x) =xr1. 947.3 784.1 748.3 631.1 775.5 745.3 602.2 573.9 665 570.8 924.4 812.6 568.1 670.2 /BaseFont/NPKUUX+CMMI8 as a recursion formula for $c_{j}$ for all $j \geq 1$. 38 0 obj An adaption of the Frobenius method to non-linear problems is restricted to exceptional cases. /BaseFont/FQHLHM+CMBX12 The Frobenius method is a generalization of the treatment of the simpler Euler–Cauchy equation, \begin{equation} \tag{a4} L _ { 0 } ( u ) = 0, \end{equation}, where the differential operator $L_0$ is made from (a1) by retaining only the leading terms. For any $i = 1 , \dots , \nu$, the zero $\lambda _ { i }$ of the indicial polynomial has multiplicity $n _ { i } \geq 1$, but none of the numbers $\lambda _ { 1 } - \lambda _ { i } , \ldots , \lambda _ { i - 1 } - \lambda _ { i }$ is a natural number. 812.5 875 562.5 1018.5 1143.5 875 312.5 562.5] /Widths[660.7 490.6 632.1 882.1 544.1 388.9 692.4 1062.5 1062.5 1062.5 1062.5 295.1 >> Formula (a1) gives the differential operator in its Frobenius normal form if $a ^ { [ N ] } ( z ) \equiv 1$. 666.7 666.7 666.7 666.7 611.1 611.1 444.4 444.4 444.4 444.4 500 500 388.9 388.9 277.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 663.6 885.4 826.4 736.8 {\displaystyle u' (z)=\sum _ {k=0}^ {\infty } (k+r)A_ {k}z^ {k+r-1}} 791.7 777.8] 15 0 obj 545.5 825.4 663.6 972.9 795.8 826.4 722.6 826.4 781.6 590.3 767.4 795.8 795.8 1091 /Subtype/Type1 also Analytic function). 351.8 935.2 578.7 578.7 935.2 896.3 850.9 870.4 915.7 818.5 786.1 941.7 896.3 442.6 /Subtype/Type1 Let y=Ún=0 ¥a xn+r. /Name/F10 /LastChar 196 Suppose one is given a linear differential operator, \begin{equation} \tag{a1} L = \sum _ { n = 0 } ^ { N } a ^ { [ n ] } ( z ) z ^ { n } \left( \frac { d } { d z } \right) ^ { n }, \end{equation}, where for $n = 0 , \ldots , N$ and some $r > 0$, the functions, \begin{equation} \tag{a2} a ^ { [ n ] } ( z ) = \sum _ { i = 0 } ^ { \infty } a _ { i } ^ { n } z ^ { i } \end{equation}. It is assumed that all $\nu$ roots are different and one denotes their multiplicities by $n_i$. \end{equation*}, 1) $\lambda _ { 1 } = \lambda _ { 2 }$. Note that aFrobenius series is generally not power series. >> Method of Frobenius: Equal Roots to the Indicial Equation We solve the equation x2y''+3 xy'+H1-xL y=0 using a power series centered at the regular singular point x=0. 472.2 472.2 472.2 472.2 583.3 583.3 0 0 472.2 472.2 333.3 555.6 577.8 577.8 597.2 295.1 826.4 501.7 501.7 826.4 795.8 752.1 767.4 811.1 722.6 693.1 833.5 795.8 382.6 /Widths[791.7 583.3 583.3 638.9 638.9 638.9 638.9 805.6 805.6 805.6 805.6 1277.8 384.3 611.1 675.9 351.8 384.3 643.5 351.8 1000 675.9 611.1 675.9 643.5 481.5 488 708.3 795.8 767.4 826.4 767.4 826.4 0 0 767.4 619.8 590.3 590.3 885.4 885.4 295.1 Note that neither of the special cases below does exclude the simple generic case above. << 9 0 obj with $l = 0 , \dots , n _ { j } - 1$ and $\lambda = \lambda _ { j }$, are $n_j$ linearly independent solutions of the differential equation (a3). This fact is the basis for the method of Frobenius. The poles are compensated for by multiplying $u ( z , \lambda )$ at first with powers of $\lambda - \lambda _ { i }$ and differentiation by the parameter $\lambda$ before setting $\lambda = \lambda _ { i }$. /Widths[300 500 800 755.2 800 750 300 400 400 500 750 300 350 300 500 500 500 500 There is a theorem dealing endobj 24 0 obj /FirstChar 33 /FirstChar 33 The leading term $b _ { l0 } ( \operatorname { log } z ) ^ { l } z ^ { \lambda _ { i } }$ is useful as a marker for the different solutions. << >> In this case the leading behavior of y(x) as x ! /Subtype/Type1 n: 2. If r 1 −r 2 ∈ Z, then both r = r 1 and r = r 2 yield (linearly independent) solutions. /FontDescriptor 23 0 R Complications can arise if the generic assumption made above is not satisfied. << 687.5 312.5 581 312.5 562.5 312.5 312.5 546.9 625 500 625 513.3 343.8 562.5 625 312.5 380.8 380.8 380.8 979.2 979.2 410.9 514 416.3 421.4 508.8 453.8 482.6 468.9 563.7 0 0 0 0 0 0 0 0 0 0 777.8 277.8 777.8 500 777.8 500 777.8 777.8 777.8 777.8 0 0 777.8 /Type/Font a 0; n= 1;2;:::: (37) In the latter case, the solution y(x) has a closed form expression y(x) = x 15 X1 n=0 ( 1)n 5nn! Frobenius Method ( All three Cases ) - Free download as PDF File (.pdf), Text File (.txt) or read online for free. 2. 495.7 376.2 612.3 619.8 639.2 522.3 467 610.1 544.1 607.2 471.5 576.4 631.6 659.7 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 312.5 312.5 342.6 An adaption of the Frobenius method to non-linear problems is restricted to exceptional cases. 589.1 483.8 427.7 555.4 505 556.5 425.2 527.8 579.5 613.4 636.6 272] 652.8 598 0 0 757.6 622.8 552.8 507.9 433.7 395.4 427.7 483.1 456.3 346.1 563.7 571.2 Computation of the polynomials $p _ { j } ( \lambda )$. 1. This could happen if r 1 = r 2, or if r 1 = r 2 + N. In the latter case there might, or might not, be two Frobenius solutions. Since a change x-x 0 ↦ x of variable brings to the case that the singular point is the origin, we may suppose such a starting situation. /Length 1951 For instance, with r= /LastChar 196 The Set-Up The Calculations and Examples The Main Theorems Inserting the Series into the DE Getting the Coe cients Observations Coe cients We have, rst of all, F (r )=r (r 1 )+p 0 r +q 0 =0 ; the indicial equation. ( \operatorname { log } z ) ^ { l } z ^ { \lambda _ { 2 } } + \ldots, \end{equation*}. The cut along some ray is introduced because the solutions $u$ are expected to have an essential singularity at $z = 0$. (3.6) 4. 756.4 705.8 763.6 708.3 708.3 708.3 708.3 708.3 649.3 649.3 472.2 472.2 472.2 472.2 /Type/Font 324.7 531.3 531.3 531.3 531.3 531.3 795.8 472.2 531.3 767.4 826.4 531.3 958.7 1076.8 Let $1 \leq j \leq \nu$ and let $\lambda _ { i }$ be a zero of the indicial polynomial of multiplicity $n_i$ for $i = 1 , \dots , j - 1$. The method of Frobenius starts with the guess, \begin{equation} \tag{a6} u ( z , \lambda ) = z ^ { \lambda } \sum _ { k = 0 } ^ { \infty } c _ { k } ( \lambda ) z ^ { k }, \end{equation}, with an undetermined parameter $\lambda \in \mathbf{C}$. /Type/Font 726.9 726.9 976.9 726.9 726.9 600 300 500 300 500 300 300 500 450 450 500 450 300 The functions, \begin{equation*} u ( z , \lambda _ { 1 } ) = z ^ { \lambda _ { 1 } } + \ldots, \end{equation*}, \begin{equation*} \frac { \partial u } { \partial \lambda } ( z , \lambda _ { 1 } ) = ( \operatorname { log } z ) z ^ { \lambda _ { 1 } } \end{equation*}, 2) $\lambda _ { 1 } - \lambda _ { 2 } \in \mathbf{N}$. When the roots of initial When the roots of initial equation are real, there is a Frobeni us solution for the larger of the tw o roots. endobj Method of Frobenius Example First Solution Second Solution (Fails) What is the Method of Frobenius? 708.3 708.3 826.4 826.4 472.2 472.2 472.2 649.3 826.4 826.4 826.4 826.4 0 0 0 0 0 /Subtype/Type1 /FontDescriptor 20 0 R >> The functions, \begin{equation*} ( \frac { \partial } { \partial \lambda } ) [ u ( z , \lambda ) ( \lambda - \lambda _ { 2 } ) ] = z ^ { \lambda_2 } + \ldots , \end{equation*}. Press (1989). The second solution can contain logarithmic terms in the higher powers starting with $( \operatorname { log } z ) z ^ { \lambda _ { 1 } }$. 761.6 489.6 516.9 734 743.9 700.5 813 724.8 633.9 772.4 811.3 431.9 541.2 833 666.2 \end{equation}, Here, one has to assume that $a ^ { 2_0 } \neq 0$ to obtain a regular singular point. /Subtype/Type1 /BaseFont/TBNXTN+CMTI12 In the former case there’s obviously only one Frobenius solution. are $n_i$ linearly independent solutions of the differential equation (a3). The approach does produce special separatrix-type solutions for the Emden–Fowler equation, where the non-linear term contains only powers. All solutions have expansions of the form, \begin{equation*} u _ { i l } = z ^ { \lambda _ { i } } \sum _ { j = 0 } ^ { l } \sum _ { k = 0 } ^ { \infty } b _ { j k } ( \operatorname { log } z ) ^ { j } z ^ { k }. 0 0 0 613.4 800 750 676.9 650 726.9 700 750 700 750 0 0 700 600 550 575 862.5 875 The Method of Frobenius (4.4) Handout 2 on An Overview of the Fobenius Method : 16-17: Evaluation of Real Definite Integrals, Case III Evaluation of Real Definite Integrals, Case IV: The Method of Frobenius - Exceptional Cases (4.4, 4.5, 4.6) 18-19: Theorems for Contour Integration Series and … Consider roots r1;r2of the indicial equation(3). The Frobenius method is useful for calculating a fundamental system for the homogeneous linear differential equation, \begin{equation} \tag{a3} L ( u ) = 0 \end{equation}. 935.2 351.8 611.1] Regular and Irregular Singularities As seen in the preceding example, there are situations in which it is not possible to use Frobenius’ method to obtain a series solution. 2 Frobenius Series Solution of Ordinary Differential Equations At the start of the differential equation section of the 1B21 course last year, you met the linear first-order separable equation dy dx = αy , (2.1) where α is a constant. endobj /Widths[272 489.6 816 489.6 816 761.6 272 380.8 380.8 489.6 761.6 272 326.4 272 489.6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 458.3 458.3 416.7 416.7 Let $\mathbf{N}$ denote the set of natural numbers starting at $1$ (i.e., excluding $0$). How to Calculate Coe cients in the Hard Cases L. Nielsen, Ph.D. Solve the hypergeometric equation around all singularities: 1. x ( 1 − x ) y ″ + { γ − ( 1 + α + β ) x } y ′ − α β y = 0 {\displaystyle x(1-x)y''+\left\{\gamma -(1+\alpha +\beta )x\right\}y'-\alpha \beta y=0} /FirstChar 33 All the three cases (Values of 'r' ) are covered in it. Here, $\epsilon > 0$, and for an equation in normal form, actually $\epsilon \geq r$. x��ZYo�6~�_�G5�fx�������d���yh{d[�ni"�q�_�U$����c�N���E�Y������(�4�����ٗ����i�Yvq�qbTV.���ɿ[�w��`:�`�ȿo��{�XJ��7��}׷��jj?�o���UW��k�Mp��/���� This page was last edited on 12 December 2020, at 22:42. /BaseFont/IMGAIM+CMR8 https://encyclopediaofmath.org/index.php?title=Frobenius_method&oldid=50967, R. Redheffer, "Differential equations, theory and applications" , Jones and Bartlett (1991), F. Rothe, "A variant of Frobenius' method for the Emden–Fowler equation", D. Zwillinger, "Handbook of differential equations" , Acad. 489.6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 611.8 816 The indicial polynomial is simply, \begin{equation*} \pi ( \lambda ) = ( \lambda + 2 ) ( \lambda + 1 ) a ^ { 2_0 } + ( \lambda + 1 ) a ^ { 1_0 } + a ^ { 0_0 } = \end{equation*}, \begin{equation*} = a ^ { 2 } o ( \lambda - \lambda _ { 1 } ) ( \lambda - \lambda _ { 2 } ). /Name/F6 795.8 795.8 649.3 295.1 531.3 295.1 531.3 295.1 295.1 531.3 590.3 472.2 590.3 472.2 272 272 489.6 544 435.2 544 435.2 299.2 489.6 544 272 299.2 516.8 272 816 544 489.6 /Type/Font In this case, define $m_j$ to be the sum of those multiplicities for which $\lambda _ { i } - \lambda _ { j } \in \mathbf{N}$. 295.1 826.4 531.3 826.4 531.3 559.7 795.8 801.4 757.3 871.7 778.7 672.4 827.9 872.8 Hence, \begin{equation*} m _ { j } = \sum \{ n _ { i } : 1 \leq i < j \ \text{ and } \ \lambda _ { i } - \lambda _ { j } \in \mathbf{N} \}. /Subtype/Type1 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 272 761.6 462.4 /Name/F1 481.5 675.9 643.5 870.4 643.5 643.5 546.3 611.1 1222.2 611.1 611.1 611.1 0 0 0 0 For the case r= 1, we have a n = a n 1 5n+ 6 = ( 1)na 0 Yn k=1 (5j+ 1) 1; n= 1;2;:::; (36) and for r= 1 5, we have a n = a n 1 5n = ( 1)n 5nn! 694.5 295.1] /Name/F3 This is a method that uses the series solution for a differential equation, where we assume the solution takes the form of a series. The other solution takes the form y2(t) = y1(t)lnt + tγ1 + 1 ∞ ∑ n = 0dntn. /LastChar 196 in the domain $\{ z \in \mathbf{C} : | z | < \epsilon \} \backslash ( - \infty , 0 ]$ near the regular singular point at $z = 0$. But P and Q cannot be arbitrary: (x−x 0)P(x) and (x−x 0)2Q(x) must be analytic at x 0. 593.8 500 562.5 1125 562.5 562.5 562.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 606.7 816 748.3 679.6 728.7 811.3 765.8 571.2 Under these assumptions, the $N$ functions, \begin{equation*} u ( z , \lambda _ { 1 } ) = z ^ { \lambda _ { 1 } } + \ldots , \ldots , u ( z , \lambda _ { N } ) = z ^ { \lambda _ { N } } +\dots \end{equation*}. The point $z = 0$ is called a regular singular point of $L$. SINGULAR POINTS AND THE METHOD OF FROBENIUS 291 AseachlinearcombinationofJp(x)andJ−p(x)isasolutiontoBessel’sequationoforderp,thenas wetakethelimitaspgoeston,Yn(x)isasolutiontoBessel’sequationofordern.Italsoturnsout thatYn(x)andJn(x)arelinearlyindependent.Thereforewhennisaninteger,wehavethegeneral /Type/Font also Fuchsian equation). a 0x 1000 1000 1055.6 1055.6 1055.6 777.8 666.7 666.7 450 450 450 450 777.8 777.8 0 0 656.3 625 625 937.5 937.5 312.5 343.8 562.5 562.5 562.5 562.5 562.5 849.5 500 574.1 Suppose $\lambda _ { 1 } - \lambda _ { 2 } \in \mathbf{N}$. This case is an example of a CASE III equation where the method of Frobenius will yield both solutions to the differential equation. /Widths[1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 1000 1000 1000 777.8 /Name/F2 /FirstChar 33 n≥2. << If r1¡r2= 0, the solution basis of the ODE(1)is given by y1(x) =xr1. << (You should check that zero is really a regular singular point.) 638.4 756.7 726.9 376.9 513.4 751.9 613.4 876.9 726.9 750 663.4 750 713.4 550 700 500 500 500 500 500 500 500 300 300 300 750 500 500 750 726.9 688.4 700 738.4 663.4 P1 n=0Anx. 833.3 1444.4 1277.8 555.6 1111.1 1111.1 1111.1 1111.1 1111.1 944.4 1277.8 555.6 1000 Example 3: x = 0 is an irregular point of the flrst order equation Ly = x2y0 +y = 0 The solution of this flrst order linear equation can be obtained by means of … 1062.5 826.4] /FontDescriptor 32 0 R 1277.8 811.1 811.1 875 875 666.7 666.7 666.7 666.7 666.7 666.7 888.9 888.9 888.9 The method of Frobenius gives a series solution of the form y(x) = X∞ n=0 an (x −c)n+s where p or q are singular at x = c. Method does not always give the general solution, the ν = 0 case of Bessel’s equation is an example where it doesn’t. There is at least one Frobenius solution, in each case. %PDF-1.2 /BaseFont/BPIREE+CMR6 The easy generic case occurs if the indicial polynomial has only simple zeros and their differences $\lambda _ { i } - \lambda _ { j }$ are never integer valued. { l ! } Indeed (a1) and (a2) imply, \begin{equation*} L ( u ( z , \lambda ) ) = \end{equation*}, \begin{equation*} = [ \sum _ { i = 0 } ^ { \infty } \sum _ { n = 0 } ^ { N } a _ { i } ^ { n } z ^ { n + i } ( \frac { \partial } { \partial z } ) ^ { n } ] [ \sum _ { k = 0 } ^ { \infty } c _ { k } ( \lambda ) z ^ { \lambda + k } ] = \end{equation*}, \begin{equation*} = \sum _ { i = 0 } ^ { \infty } \sum _ { k = 0 } ^ { \infty } c _ { k } ( \lambda ) z ^ { i } \sum _ { n = 0 } ^ { N } a _ { i } ^ { n } z ^ { n } \left( \frac { \partial } { \partial z } \right) ^ { n } z ^ { \lambda + k } = \end{equation*}, \begin{equation*} = \sum _ { i = 0 } ^ { \infty } \sum _ { k = 0 } ^ { \infty } c _ { k } ( \lambda ) z ^ { i } p _ { i } ( \lambda + k ) z ^ { \lambda + k } = \end{equation*}, \begin{equation*} = z ^ { \lambda } \sum _ { j = 0 } ^ { \infty } z ^ { j } \left[ \sum _ { i + k = j } c _ { k } ( \lambda ) p _ { i } ( \lambda + k ) \right] = \end{equation*}, \begin{equation*} = c _ { 0 } z ^ { \lambda } \pi ( \lambda ) + \end{equation*}, \begin{equation*} + z ^ { \lambda } \sum _ { j = 1 } ^ { \infty } z ^ { j } \left[ c _ { j } ( \lambda ) \pi ( \lambda + j ) + \sum _ { k = 0 } ^ { j - 1 } c _ { k } ( \lambda ) p _ { j - k } ( \lambda + k ) \right]. endobj 597.2 736.1 736.1 527.8 527.8 583.3 583.3 583.3 583.3 750 750 750 750 1044.4 1044.4 A similar method of solution can be used for matrix equations of the first order, too. ( \operatorname { log } z ) ^ { l } z ^ { \lambda _ { j } } + \ldots, \end{equation*}. 450 500 300 300 450 250 800 550 500 500 450 412.5 400 325 525 450 650 450 475 400 /FontDescriptor 11 0 R /FirstChar 33 295.1 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 295.1 are holomorphic for $| z | < r$ and $a ^ { N_ 0} \neq 0$ (cf. /BaseFont/XZJHLW+CMR12 /LastChar 196 >> >> 3. Suppose the roots of the indicial equation are r 1 and r 2. /FontDescriptor 29 0 R Commonly, the expansion point can be taken as, resulting in the Maclaurin series (1) /FontDescriptor 26 0 R The Euler–Cauchy equation can be solved by taking the guess $z = u ^ { \lambda }$ with unknown parameter $\lambda \in \mathbf{C}$. Section 8.4 The Frobenius Method 467 where the coefficients a n are determined as in Case (a), and the coefficients α n are found by substituting y(x) = y 2(x) into the differential equation. In the Frobenius method one examines whether the equation (2) allows a series solution of the form. The coefficients have to be calculated by requiring that, \begin{equation} \tag{a7} L ( u ( z , \lambda ) ) = \pi ( \lambda ) z ^ { \lambda }. Keywords: Frobenius method; Power series method; Regular singular 1 Introduction In mathematics, the Method of Frobenius [2], named for Ferdinand Georg Frobenius, is a method to nd an in nite series solution for a second-order ordinary di erential equation of the form x2y00+p(x)y0+q(x)y= 0 … \end{equation*}. /Name/F5 767.4 767.4 826.4 826.4 649.3 849.5 694.7 562.6 821.7 560.8 758.3 631 904.2 585.5 1111.1 1511.1 1111.1 1511.1 1111.1 1511.1 1055.6 944.4 472.2 833.3 833.3 833.3 833.3 Putting $\lambda = \lambda _ { i }$ in (a6), obtaining solutions of (a3) can be impossible because of poles of the coefficients $c_j ( \lambda )$. 33 0 obj Application of Frobenius’ method In order to solve (3.5), (3.6) we start from a plausible representation of B x,B y that is >> Introduction The “na¨ıve” Frobenius method The general Frobenius method Remarks Under the hypotheses of the theorem, we say that a = 0 is a regular singular point of the ODE. \end{equation}, This requirement leads to $c _ { 0 } \equiv 1$ and, \begin{equation} \tag{a8} c _ { j } ( \lambda ) = - \sum _ { k = 0 } ^ { j - 1 } \frac { c _ { k } ( \lambda ) p _ { j - k } ( \lambda + k ) } { \pi ( \lambda + j ) } \end{equation}. Computation of the polynomials $p _ { j } (\lambda)$. /Name/F4 View Notes - Lecture 5 - Frobenius Step by Step from ESE 319 at Washington University in St. Louis. FROBENIUS SERIES SOLUTIONS 5 or a n = a n 1 5n+ 5r+ 1; n= 1;2;:::: (35) Finally, we can use the concrete values r= 1 and r= 1 5. 777.8 777.8 1000 1000 777.8 777.8 1000 777.8] /Type/Font all with $\lambda = \lambda _ { 2 }$ and $l = 0 , \dots , n _ { 2 } - 1$, are $n_{2}$ linearly independent solutions of the differential equation (a3). Frobenius’ method for solving u00+ b(x) x u0+ c(x) x2 u = 0 (with b;canalytic near 0) is slightly more complicated when the indicial equation ( 1) + b(0) + c(0) = 0 has repeated roots or roots di ering by an integer. 888.9 888.9 888.9 888.9 666.7 875 875 875 875 611.1 611.1 833.3 1111.1 472.2 555.6 An infinite series of the form in (9) is called a Frobenius series. You were also shown how to integrate the equation to … /FontDescriptor 14 0 R /FontDescriptor 17 0 R named for the German mathematician Georg Frobenius (1848—19 17), who discovered the method in the 1870s. 18 0 obj 275 1000 666.7 666.7 888.9 888.9 0 0 555.6 555.6 666.7 500 722.2 722.2 777.8 777.8 /Subtype/Type1 << In fact Frobenius method is just an extension from the power series method that you add an additional power that may not be an integer to each term in a power series or even add the log term for the assumptions of the solution form of the linear ODEs so that you can find all groups of the linearly independent solutions that in cases of cannot find all groups of the linearly independent solutions … /Name/F9 endobj 699.9 556.4 477.4 454.9 312.5 377.9 623.4 489.6 272 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 531.3 826.4 826.4 826.4 826.4 0 0 826.4 826.4 826.4 1062.5 531.3 531.3 826.4 826.4 1444.4 555.6 1000 1444.4 472.2 472.2 527.8 527.8 527.8 527.8 666.7 666.7 1000 1000 /Type/Font In the following we solve the second-order differential equation called the hypergeometric differential equation using Frobenius method, named after Ferdinand Georg Frobenius. Method of Frobenius. /LastChar 196 2n 2, so Frobenius’ method fails. \end{equation*}. are a fundamental system of solutions of (a3). 720.1 807.4 730.7 1264.5 869.1 841.6 743.3 867.7 906.9 643.4 586.3 662.8 656.2 1054.6 P1 n=0anx. This is the extensive document regarding the Frobenius Method. The Frobenius method has been used very successfully to develop a theory of analytic differential equations, especially for the equations of Fuchsian type, where all singular points assumed to be regular (cf. Theorem 1 (Frobenius). EnMath B, ESE 319-01, Spring 2015 Lecture 4: Frobenius Step-by-Step Jan. 23, 2015 I expect you to 30 0 obj 761.6 272 489.6] • Back to Frobenius method for second solutions in three cases –n = = 0, the double root – Integer = n 0, roots differ by an integer, J-n(x) = (-1)nJ n(x) – Non-integer , easiest case, J and J- are two linearly independent solutions • General case for second solution [0,1] 2( ln() m m n Because of (a7), one finds $c _ { 0 } \equiv 1$ and the recursion formula (a8). The next two theorems will enable us to develop systematic methods for finding Frobenius solutions of ( eq:7.5.2 ). /BaseFont/KNRCDC+CMMI12 5 See Joseph L. Neuringera, The Frobenius method for complex roots of the indicial equation, International Journal of Mathematical Education in Science and Technology, Volume 9, Issue 1, 1978, 71–77. endobj /Widths[342.6 581 937.5 562.5 937.5 875 312.5 437.5 437.5 562.5 875 312.5 375 312.5 Solution of the polynomials $ p _ { 2 } \in \mathbf N... } \in \mathbf { N } $ holomorphic differential equation Step by Step from ESE 319 at University. R 1 and r 2 least one Frobenius solution, in each case j 1. It is assumed that all $ j \geq 1 $ series is generally not power series ’ s only! The roots of the special cases below does exclude the simple generic case.. Method in the Frobenius method to non-linear problems is restricted to exceptional cases zero is really a regular singular (! { 1 } = frobenius method cases _ { 1 } - \lambda _ { 1 } = _. Step from ESE 319 at Washington University in St. Louis 1 and r 2 j \geq 1 $ 0! 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