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Chapter 4 Power Series Solutions Alberto J. Benavides, Xiaohong Cui, Adewale Awoniyi Department of Chemical Engineering Texas A&M University, College Station, TX 1 Agenda 4.4 4.5 4.6 Legendre Functions Single Integrals; Gamma Function Bessel Functions 2 4.4. Legendre Functions Alberto J. Benavides, Xiaohong Cui, Adewale Awoniyi Department of Chemical Engineering Texas A&M University, College Station, TX 3 4.4.1 Legendre Polynomials A special form a second order differential equation which is encountered often when solving physical problems is: This equation is known as the Legendre Equation. The general solution to the Legendre equation is given as a function of two Legendre functions as follows: where Adapted from: http://www.mhtl.uwaterloo.ca/courses/m e755/web_chap5.pdf 4/9/2015 4 4.4.1 Legendre Polynomials If n = 0,1,2,3,…the Pn(x) functions are called Legendre Polynomials of order n and are given by Rodrigues’s formula: The first four Legendre Polynomials are: P0 ( x ) 1 P1 ( x ) x 1 (3 x 2 1) 2 1 P3 ( x ) (5 x 3 3 x ) 2 1 P4 ( x ) (35x 4 30x 2 3) 8 P2 ( x ) Plot of the first four Legendre Polynomials From: http://www.mae.ufl.edu/~uhk/MATHFUNC.htm Adapted from: http://www.mhtl.uwaterloo.ca/courses/m e755/web_chap5.pdf 4/9/2015 5 4.4.2 Orthogonality of Legendre Polynomials Proof of the orthogonality relation If Pm and Pn are solutions of Legendre’s equation, then (1) (2) Integrating the combination Pm (1) - Pn (2) Using integration by parts ` As Pm,n and their derivatives are finite at x±1 (i.e. they are regular there). Hence if n ≠ m From: http://www.ucl.ac.uk/~ucahdrb/MATHM242/OutlineCD2.pdf 4/9/2015 6 4.4.3 Generating Function and Properties The above equation is called the generating function for the Pn’s What is ? We can evaluate the integral using Rodrigues’s formula. We have Integrating by parts we get Note that differentiating (x2-1)n anything less than n times leaves an expression that has (x2-1) as a factor so that the rest of the terms vanishes. Similarly, integrating by parts n times gives From: http://www.ucl.ac.uk/~ucahdrb/MATHM242/OutlineCD2.pdf 4/9/2015 7 4.4.3 Generating Function and Properties The (2n)th derivative of the polynomial (x2-1)n, which has degree 2n is (2n)!. Thus Using the transformation s= (x+1)/2 to transform the integral and then use a reduction formula to show that The final result is: From: http://www.ucl.ac.uk/~ucahdrb/MATHM242/OutlineCD2.pdf 4/9/2015 8 Example Consider the Legendre Equation: (1 x 2 ) y' '2xy' ( 1) y 0 1. Find 2 linearly independent solutions in the form of power series about x= 0. 2. Compute the radius of convergence of the series. Explain why it is possible to predict the radius without actually deriving the series. 3. Show that α = 2n, with n an integer and n≥0, the series for one of the solutions reduces to an even polynomial of degree 2n 4. Show that α = 2n+1, with n an integer and n≥0, the series for one of the solutions reduces to an odd polynomial of degree 2n+1 5. Show that the first 4 polynomial solutions Pn(x) (known as Legendre polynomials) ordered by their degree and normalized so that Pn(1) =1 are P0 1 P1 x 1 P2 (3x 2 1) 2 1 P3 (5 x 3 3x) 2 From: Introduction to Applied Mathematics or Advanced Mathematical Methods for Scientists and Engineers - Mauch 4/9/2015 9 Solution 1. First we write the differential equation in the general form: (1) Since the coefficients of y’ and y are analytic in a neighborhood of x = 0, We can find two Taylor series solutions about that point. We find the Taylor series for y and its derivatives. From: Introduction to Applied Mathematics or Advanced Mathematical Methods for Scientists and Engineers.-Mauch 4/9/2015 10 Solution Index shifting was used to explicitly write two forms that we will need for y’’. Note that we can take the lower bound of summation to be n=0 for all the sums. The terms added by this operation are zero. We substitute the Taylor series in to Equation (1). We equate coefficients of xn to obtain the recurrence relation From: Introduction to Applied Mathematics or Advanced Mathematical Methods for Scientists and Engineers.-Mauch 4/9/2015 11 Solution We can solve this difference equation to determine the an’s (a0 and a1 are arbitrary) We will find the fundamental solutions at x=0, that is the set {y1,y2} that satisfies For y1 we take a0=1 and a1=0; for y2 we take a0=0 and a1=1. The rest of the coefficients are determined from the recurrence relation. From: Introduction to Applied Mathematics or Advanced Mathematical Methods for Scientists and Engineers.-Mauch 4/9/2015 12 Solution 2. We determine the radius of convergence of the series solutions with the ratio test. = = Therefore. Thus we see that the radius of convergence of the series is 1. We knew that the radius of convergence would be at least one, because the nearest singularities of the coefficients of (1) occur at x=±1, a distance of 1 from the origin. This implies that the solutions of the equation are analytic in the unit circle about x=0. The radius of convergence of the Taylor series expansion of an analytic function is the distance to the nearest singularity. From: Introduction to Applied Mathematics or Advanced Mathematical Methods for Scientists and Engineers-Mauch. 4/9/2015 13 Solution 3. If α=2n then a2n+2=0 in our first solution. From the recurrence relation, we see that all subsequent coefficients are also zero. The solution becomes an even polynomial. 4. If α=2n+1 then a2n+3=0 in our second solution. From the recurrence solution, we see that all subsequent coefficients are also zero. The solution becomes an odd polynomial. From: Introduction to Applied Mathematics or Advanced Mathematical Methods for Scientists and Engineers-Mauch. 4/9/2015 14 Solution 5. From our solution above, the first four polynomials are To obtain the Legendre polynomials we normalize these to have unity at x=1 From: Introduction to Applied Mathematics or Advanced Mathematical Methods for Scientists and Engineers-Mauch. 4/9/2015 15 4.5. Single Integrals; Gamma Function Alberto J. Benavides, Xiaohong Cui, Adewale Awoniyi Department of Chemical Engineering Texas A&M University, College Station, TX 16 4.5.1 Singular Integrals; Gamma Function An integral is said to be singular (or improper) if one or both integration limits are infinite and/or if the integrand is unbounded on the interval; otherwise, it is regular (or proper). We can group improper integrals into two: and Adapted from: http://www.mcae.umn.edu/acadsupport/documents/ptest_000.pdf 4/9/2015 17 4.5.1 Singular Integrals The “p-test” is used to determine whether or not the integral converges an diverges and it can be summarized as follows. The following series is called the p-series: The p-series acts similar to the Type I improper integral. Therefore: Adapted from: http://www.mcae.umn.edu/acadsupport/docu ments/p-test_000.pdf 4/9/2015 18 The Integral Test can be summarized as follows: For proof of this, consider the following graphs, If integral diverges, so does series 4/9/2015 If integral converges, so does series Adapted from: http://faculty.otterbein.edu/TJames/math 1800/presentations/9.3%20The%20Integ ral%20Test.pdf 19 4.5.1 Singular Integrals The Comparison Tests: Suppose that ∑ an and ∑ bn are series with positive terms and suppose that an ≤ bn for all n. Then Example 1: Determine if the following series converges or diverges. Solution: We have and we know that converges. Hence by the comparison test, it also converges. From: http://www.math.northwestern.edu/~mlerma/cour ses/math214-2-04f/notes/c2-convertest.pdf 4/9/2015 20 4.5.1 Singular Integrals Example 2: Determine if the following converges or diverges. Solution: We consider . Here p=3. Since p >1. This integral converges. Looking at our integral, we see that for positive x, From the comparison test, since converges, so does . From: http://www.mcae.umn.edu/acadsupport/documents/ptest_000.pdf 4/9/2015 21 4.5.1 Singular Integrals Absolute Convergence A series is values said to be absolutely convergent if the series of absolute converges. Example: Consider the series The series absolutely converges since converges. From: http://de2de.synechism.org/c 5/sec56.pdf 4/9/2015 22 4.5.2 Gamma Function The gamma function is given by the improper integral The gamma function satisfied the recursive property: The recursive relationship can be used to compute the gamma function of all real numbers (except the non-positive integers) by knowing only the value of the gamma function between 1 and 2. Example 1: For α = 5, and using the recursive property: Values of can be found using this sort of Table Adapted from: jekyll.math.byuh.edu/courses/m321/han douts/gammaproperties.pdf 4/9/2015 23 4.5.2 Gamma Function Example 2: Now with α = 23/3, and using the recursive property Adapted from: jekyll.math.byuh.edu/courses/m321/han douts/gammaproperties.pdf 4/9/2015 24 4.5.2 Gamma Function When α = n and n is a positive integer, then the gamma function is related to the factorial function by The gamma function evaluated at α = ½ is Next, we can extend the usefulness of the gamma function so that it can be computed for negative non-integer numbers. Substituting α+1 with α in the recursive relationship, we get: and solving for Adapted from: jekyll.math.byuh.edu/courses/m321/han douts/gammaproperties.pdf 4/9/2015 25 4.5.2 Gamma Function Example 3: Suppose that α = -5/6 Using linear interpolation between 1.16 and 1.17 yields . Hence, which is close to the actual value of -6.679579. Many integrals can be evaluated by making suitable changes of variables so as to reduce them to gamma function integrals. Example 4: Solution: Adapted from: jekyll.math.byuh.edu/courses/m321/han douts/gammaproperties.pdf 4/9/2015 26 4.5.2 Gamma Function Evaluating the integral, we get 4/9/2015 27 4.5.2 Gamma Function Table of the Gamma Function From: jekyll.math.byuh.edu/courses/m321/han douts/gammaproperties.pdf 4/9/2015 28 4.6. Bessel Functions Alberto J. Benavides, Xiaohong Cui, Adewale Awoniyi Department of Chemical Engineering Texas A&M University, College Station, TX 29 4.6 Bessel Functions One of the varieties of special functions which are encountered in the solution of physical problems is the class of functions called Bessel functions. They are solutions to a very important differential equation, the Bessel equation: Bessel's equation arises when finding separable solutions to Laplace's equation and the Helmholtz equation in cylindrical or spherical coordinates. In solving problems in cylindrical coordinate systems, one obtains Bessel functions of integer order (α = n); in spherical problems, one obtains half-integer orders (α = n + ½). For example: •Electromagnetic waves in a cylindrical waveguide •Heat conduction in a cylindrical object •Modes of vibration of a thin circular (or annular) artificial membrane •Diffusion problems on a lattice •Solutions to the radial Schrödinger equation (in spherical and cylindrical coordinates) for a free particle •Solving for patterns of acoustical radiation Adapted from: http://en.wikipedia.org/wiki/Bessel_function 4/9/2015 30 4.6 Bessel Functions of the first kind Since this is a second-order differential equation, there must be two linearly independent solutions. Depending upon the circumstances, however, various formulations of these solutions are convenient, and the different variations are described below. Bessel functions of the first kind, denoted as Jv(x), are solutions of Bessel's differential equation that are finite at the origin (x = 0) for integer v, and diverge as x approaches zero for negative non-integer v. The solution type (e.g., integer or non-integer) and normalization of Jv(x) are defined by its properties below. It is possible to define the function by its Taylor series expansion around x = 0: Γ(z) is the gamma function, a generalization of the factorial function to non-integer values. Adapted from: http://en.wikipedia.org/wiki/Bessel_function 4/9/2015 31 4.6.1 v ≠ integer Shaped like oscillating sine or cosine functions that decay proportionally to 1/√x. Taken from: http://www.dplot.com/fct_besselj.htm For non-integer v, the functions Jv(x) and J−v(x) are linearly independent, and are therefore the two solutions of the differential equation. That is y(x) = AJv(x)+BJ-v(x). On the other hand, for integer order v=n, the following relationship is valid (note that the Gamma function becomes infinite for negative integer arguments). This means that the two solutions are no longer linearly independent. In this case, the second linearly independent solution is then found to be the Bessel function of the second kind, as discussed below. Adapted from: Calculus: A Physical & Intuitive Approach - Morris Kline 4/9/2015 32 4.6.2 v = integer Since for the case n = 1,2,… the two solutions are no longer linearly independent, therefore we must use the Frobenius regular singular point solution theorem in order to find a solution. For v = n = integer the indicial roots r ±n differ by an integer; y2(x) can be found from y1(x) which results in what is know as the Weber’s Bessel function of the second kind, or order n: That is, for v = n = integer(x), the solution of the Bessel equation will be given by: y(x) = AJn(x)+BYn(x). Taken from: http://www.dplot.com/fct_besseljyhtm Adapted from: Calculus: A Physical & Intuitive Approach - Morris Kline 4/9/2015 33 4.6.2 Bessel Functions of the second kind If we define: For non-integer v, then the limit of Yv(x) as v n gives the same result as Yn(x). That is Nn(x) = Yn(x). Nv(x) is known as the Neumann function of order n Adapted from: Calculus: A Physical & Intuitive Approach - Morris Kline 4/9/2015 34 4.6.3 Hankel functions Another important formulation of the two linearly independent solutions to Bessel's equation are the Hankel functions of the first and second kind Hv(1)(x) and Hv(2)(x), defined by: where j is the imaginary unit. These linear combinations are also known as Bessel functions of the third kind; they are two linearly independent solutions of Bessel's differential equation. Adapted from: Calculus: A Physical & Intuitive Approach - Morris Kline 4/9/2015 35 4.6.4 General solution of the Bessel Equation As expected any two of the following functions are linearly independent solutions of the Bessel equation: Thus, the general solution can be written as a linear combination of any two of the above functions. Usually the general solution is written as one of these forms: Adapted from: Calculus: A Physical & Intuitive Approach - Morris Kline 4/9/2015 36 4.6.5 Modified Bessel Equation The Bessel functions are valid even for complex arguments x, and an important special case is that of a purely imaginary argument. In this case, the solutions to the Bessel equation are called the modified Bessel functions (or occasionally the hyperbolic Bessel functions) of the first and second kind, and are defined by any of these equivalent alternatives: These are chosen to be real-valued for real and positive arguments x. The series expansion for In(x) is thus similar to that for Jn(x), but without the alternating (−1)k factor. In(x) and Kn(x) are the two linearly independent solutions to the modified Bessel's equation: Adapted from: Calculus: A Physical & Intuitive Approach - Morris Kline 4/9/2015 37 4.6.5 Modified Bessel Equation Unlike the ordinary Bessel functions, which are oscillating as functions of a real argument, In and Kn are exponentially growing and decaying functions, respectively. Like the ordinary Bessel function Jn, the function In goes to zero at x = 0 for α > 0 and is finite at x = 0 for α = 0. Analogously, Kn diverges at x = 0. Modified Bessel Functions of the First Kind Taken from: http://mathworld.wolfram.com/ModifiedBesselFunctionoft heFirstKind.html 4/9/2015 Modified Bessel Functions of the Second Kind Taken from: http://mathworld.wolfram.com/ModifiedBesselFunctionoftheS econdKind.html 38 4.6.6 Equations reducible to Bessel Equations There are many equation which can be reduced to Bessel or modified Bessel equations by changes of variables and then solved in closed form in terms of Bessel or modified Bessel functions. There it is the true power of Bessel equations! Example 1. The Airy differential equation known in astronomy and physics has the form: y’’ – xy = 0 Example 2. The differential equation of type It can be also reduced to the Bessel equation. Its solution is given by the Bessel functions of the fractional order +/- 3 differs from the Bessel equation only by a factor a2 before x2 and has the general solution in the form: y(x) = C1Jv(x)+C1Yv(x). Adapted from: http://www.math24.net/bessel-equation.html 4/9/2015 39 4.6.6 Equations reducible to Bessel Equations (1) Example 3. (2) Taken from: Textbook Of Engineering Mathematics By Debashis Dutta 4/9/2015 40 4.6.6 Equations reducible to Bessel Equations Examples 3.i, 3.ii Taken from: Textbook Of Engineering Mathematics By Debashis Dutta 4/9/2015 41