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Applications of Taylor Series - University of Tennessee

Applications of Taylor Series Jacob Fosso-Tande Department of Physics and Astronomy, University of Tennessee 401 Nielsen Physics Building 1408 Circle Drive (Completed 1st October, 2008; submitted 23rd October, 2008). Polynomial functions are easy to understand but complicated functions, infinite polynomials, are not obvious. Infinite polynomials are made easier when represented using Series : complicated functions are easily represented using Taylor 's Series . This representation make some functions properties easy to study such as the asymptotic behavior. Differential equations are made easy with Taylor Series . Taylor 's Series is an essential theoretical tool in computational science and approximation. This paper points out and attempts to illustrate some of the many Applications of Taylor 's Series expansion. Concrete examples in the physical science division and various engineering fields are used to paint the Applications pointed out.

Applications of Taylor Series Jacob Fosso-Tande Department of Physics and Astronomy, University of Tennessee 401 A.H. Nielsen Physics Building 1408 Circle Drive

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Transcription of Applications of Taylor Series - University of Tennessee

1 Applications of Taylor Series Jacob Fosso-Tande Department of Physics and Astronomy, University of Tennessee 401 Nielsen Physics Building 1408 Circle Drive (Completed 1st October, 2008; submitted 23rd October, 2008). Polynomial functions are easy to understand but complicated functions, infinite polynomials, are not obvious. Infinite polynomials are made easier when represented using Series : complicated functions are easily represented using Taylor 's Series . This representation make some functions properties easy to study such as the asymptotic behavior. Differential equations are made easy with Taylor Series . Taylor 's Series is an essential theoretical tool in computational science and approximation. This paper points out and attempts to illustrate some of the many Applications of Taylor 's Series expansion. Concrete examples in the physical science division and various engineering fields are used to paint the Applications pointed out.

2 INTRODUCTION evaluating definite integrals of these functions difficult because the fundamental theorem of calculus cannot be Taylors Series is an expansion of a function into an used. However, a Series representation of this function infinite Series of a variable x or into a finite Series plus a eases things up. Suppose we want to evaluate the defi- remainder term[1]. The coefficients of the expansion or of nite integral the subsequent terms of the Series involve the successive Z 1 . derivatives of the function. The function to be expanded sin x2 dx (5). should have a nth derivative in the interval of expansion. 0. The Series resulting from Taylors expansion is referred this integrand has no anti-derivative expressible in terms to as the Taylor Series . The Series is finite and the only of familial functions. However, we know how to find its concern is the magnitude of the remainder.

3 Given the Taylor Series : interval of expansion a 5 5 b the Lagrangian form of the remainder is given as follows: t3 t5 t7. sin t = t + + + .. (6). n (x a) (n) 3! 5! 7! Rn = f ( ) (1). n! if, we substitute t = x2 , then a, is the reference point. The f (n) ( ) is the nth derivative x6 x10 x14. at a. When the expanding function is such that: sin x2 = x2 + + + .. (7). 3! 5! 7! lim Rn = 0, (2) The Taylor Series can then be integrated: n . the Taylors Series of the expanding function becomes Z 1 x3 x7 x11 x15. sin x2 dx = + + + + .. X n 0 3 7 3! 11 5! 15 7! (x a) (n). f (x) = f (a) . (3) (8). n=0. n! This is an alternating Series and by adding all the terms, the Series converges to [1]. Taylor Series specifies the value of a function at one point, x. Setting the derivative operator, D = d/dx, the Taylor expansion becomes: UNDERSTANDING ASYMPTOTIC BEHAVIOR.

4 X hn Dn f (x + h) = f (x) = ehD f (x) [2] (4) Sometimes the Taylor Series is used to describe how a n! n=0 function behaves in a sub domain [2]. The electric field obeys the inverse square law. Taylor Series could also be written in the context of a complex variable kq E= (9). r2. EVALUATING DEFINITE INTEGRALS Where E is the electric field, q is the charge, r is the distance away from the charge and k is some constant Some functions have no anti-derivative which can be of proportionality. Two opposite charges placed side by expressed in terms of familiar functions. This makes side, setup an electric dipole moment such that we can 2. consider the electric field far away from the dipole mo- geometry can be computed. ment. Taylor 's Series is used to study this behavior. U (x, y) U1 + Ux,1 (x x1 ) Uy,1 (y y1 ). kq kq E= 2 + 2 (10) 1 (1) 2 (1). (x r) (x + r) + Ux,x (x x1 ) + Ux,y (x x1 ) (y y1 ) (16).

5 2. 1 (1) 2. An electric field further away from the dipole is obtained + Uy,y (y y1 ). from (10) after expanding the terms in the denominator. 2. kq kq E= . (11). r 2 r 2. x2 1 x x2 1 + x FIG. 1: Dipole of optimized water molecule[8]. Taylor 's Series can be used to expand the denominators if (x r). kq 2r 3r2 4r3. =1+. r 2. + 2 + 3 (12). 1 x x x x kq 2r 3r2 4r3. =1 . r 2. + 2 3. (13). 1+ x x x x One now obtain: 4rq E . (14). x3. In the field of physics and chemistry, there is a great geometry autosym units angstrom need for geometric optimization of physical systems. In O chemistry, as an example, the quasi-newton method make H H use of a two variable Taylor 's Series to approximate the equilibrium geometry of a cluster of atoms [3]. Consider TABLE I: Nwchem[9]cartesian coordinates of water molecule U,the geometry of a molecule, and assume it is a function of only two variables, x and y, let x1 and y1 be the initial coordinates, if terms higher than the quadratic terms are The superscript denotes the first approximation to the neglected then the Taylor Series is as follows: Hessian matrix elements at or near the equilibrium ge- ometry.

6 The molecular geometry has a 3N-6 dimensional U vector when internal coordinates are considered and by U (x, y) U (x1 , y1 ) + |(x ,y ) (x x1 ). x 1 1 3N when only cartesian coordinates are used. N is the U 1 2U 2 number of atoms in the molecule. In cartesian coordi- + |(x1 ,y1 ) (y y1 ) + |(x ,y ) (x x1 ) (15). y 2 x2 1 1 nates, rotation and translation accounts for the six in 1 2U 2U 3N-6. Once the Hessian matrix elements are determined, 2. + | (x ,y ) (y y1 ) + |(x ,y ) (x x1 ) (y y1 ) the molecular properties can be extracted via the Tay- 2 y 2 1 1 x y 1 1. lor's Series expansion. The U(x, y) function fits well around the equilibrium po- The position of the atom in the molecule constantly sition, the quadratic approximation works well around shift from the equilibrium position. an overall atomic the minimum. If U were accurately a quadratic function behavior , in the course of vibration, is modeled on of the coordinates in the region near (x1 , y1 ), then the the Lennard-Jones 6-12 potential.

7 The dynamics of elements of the Hessian matrix (second partial deriva- the vibrations can be study by expanding the poten- tives) will be constant in this region. Accurate ab initio tial in a Taylor 's Series . The second derivative of the self consistent field calculation of the second derivatives Taylor 's Series expansion correspond to the gradient of is very time consuming, thus the optimization usually the,(harmonic) potential curve of a short range vibration starts with an approximation of the Hessian and then around the equilibrium position,re . proceeds to improve on this approximation. If U(x,y) is 12 6. written in the form below then, the first approximation to V (r) = 4 [ (17). the Hessian matrix element at (or near) the equilibrium r r 3. EXAMPLES OF Applications OF Taylor bust and high accuracy methode that is use to study abi- Series trary shapes) (CFD) algorithm is optimally made accu- rate for the unsteady Incompressible Navier-Stokes (INS).]

8 The Gassmann relations of poroelasticity provide a equation via Taylor Series (TS) operation followed by connection between the dry and the saturated elastic pseudo-limit process. A spatially finite element democra- moduli of porous rock and are useful in a variety of tization in the implementation of the INS termed Taylor petroleum geoscience Applications [4]. Because some un- Weak Statement (TWS) generates a CFD algorithm for certainty is usually associated with the input parameters, analysis. The TWS algorithm phase velocity and am- the propagation of error in the inputs into the final mod- plification factor error function are then derived for lin- uli estimates is immediately of interest. Two common ap- ear and bi-linear basis implementations assembled at the proaches to error propagation include: a first-order Tay- generic node. The lower order error terms are affected as lor's Series expansion and Monte-Carlo methods.

9 The a result of a subsequent TS expansion in wave number Taylor 's Series approach requires derivatives, which are space. obtained either analytically or numerically and is usually limited to a first-order analysis. The formulae for ana- lytical derivatives were often prohibitively complicated CONCLUSION. before modern symbolic computation packages became prevalent but they are now more accessible [4]. We have probe through the complexity of Taylor Series A numerical method for simulations of nonlinear sur- and shown evidence of its extensive and very effective face water waves over variable bathymetry (study of Applications . The effectiveness in error determination, underwater depth of third dimension of lake or ocean function optimization, definite integral resolution, and floor) and which is applicable to either two- or three- limit determination is evidence of the Taylor Series being dimensional flows, as well as to either static or mov- an enormous tool in physical sciences and in Computa- ing bottom topography, is based on the reduction of the tional science as well as an effective way of representing problem to a lower-dimensional Hamiltonian system in- complicated functions.

10 Volving boundary quantities alone. A key component of this formulation is the Dirichlet-Neumann operator (used in analysing boundary conditions fluid dynamics and ACKNOWLEDGMENTS. crystal growth) which, in light of its joint analyticity properties with respect to surface and bottom deforma- I want to acknowledge the original topic proposal by tions, is computed using its Taylor 's Series representation. Dr. Adriana Moreo, her advice and her willingness to The new stabilized forms for the Taylor terms, are effi- have me be not only a participant but a contributor to the ciently computed by a pseudo spectral method using the success of this course. I again felt the pain of what it is fast Fourier transform [5]. to read, understand and communicate complex scientific The current-mode pseudo-exponential circuit based is information which is another step in my training as an optimized using the n-order Taylor 's Series expansion.


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