The Sinc-Collocation Method for Solving the Telegraph Equation

Transcription

1 Te Sinc-Collocation Metod for Solving te Telegrap Equation E. Hesameddini *1, E. Asadolaifard Department of Matematics, Faculty of Basic Sciences, Siraz University of Tecnology, Siraz, Iran *1 Abstract- Tis work illustrates te application of te sinccollocation metod to te second-order linear yperbolic telegrap equation in one-space dimension. Te exponential rate of convergence makes tis metod useful for approximating te solution of tis equation. umerical results sow te efficiency of tis metod. Keywords- Sinc-Collocation Metod; Telegrap Equation I. ITRODUCTIO Partial differential equations (PDEs) appear frequently in many areas suc as pysics, engineering, computer sciences, etc. Generally finding an analytic solution for tese equations is difficult. Many numerical metods and some approximate analytical solutions are proposed for tis kind of equations suc as some effective ways wic were suggested by Hesameddini and Latifizade [11, 1]. Due to [15], te yperbolic PDEs are te basis for fundamental equations of atomic pysics. In recent years, many different metods were developed for te numerical solution of te second-order yperbolic equations, for instance see [3-7]. One of te yperbolic PDEs is te telegrap equation. Telegrap equation is used in signal analysis for transmission and propagation of electrical signal [15]. Recently, it is found tat te telegrap equation is more suitable tan ordinary diffusion equation in modeling reaction diffusion for suc brances of sciences [4]. Tis equation is used in modeling several problems suc as wave propagation [8], random walk teory [9], etc. Also, te second-order linear yperbolic telegrap equation as been considered by [3, 15]. In [3], te autors used te collocation points and tin plate splines radial basis functions to approximate te solution of tis equation. In [15], te autors proposed a numerical sceme based on te sifted Cebysev tau metod, to solve it. In [14], tis equation was solved numerically by cubic B-spline quasiinterpolation. An effective approac to obtain te solution of equations is te sinc metod. Sinc metods were originally introduced by Frank Stonger in [4]. Up to now many autors used tis metod to approximate te solution of many equations [17-]. In tis paper, we approximate te solution of te second-order linear yperbolic telegrap equation by using te sinc-collocation metod. Te reminder of paper is organized as follows: we review some properties of sinc function and sinccollocation metod in Section. In Section 3, we will apply sinc-collocation metod to te second-order linear yperbolic telegrap equation. Finally, numerical results are given in Section 4. II. SIC FUCTIO AD SIC-COLLOCATIO METHOD In tis section, some properties of te sinc function and sinc-collocation metod will be presented. More properties of te sinc function can be found in [, 4]. Te sinc function is defined on te wole real line by sin(πx) sinc(x) = x 0, πx (1) 1 x = 0. If f(x) is a function tat is defined on te real axis, ten for any > 0, te Wittaker cardinal expantion of f(x) is defined as follows: c(f, )(x) = x k f(k)sinc( ), () k= wenever tis series converges. In te following definition, we caracterize a class of functions were teir Wittaker cardinal expantion converge to tem. Definition 1. Let be a positive constant. Te Paley- Wiener class of functions B() is te family of entire functions f suc tat on te real line f L (R) and in te complex plane f is of exponential type π, i.e., f(z) k exp ( π z ) for some k > 0. But tere are functions f B() were te approximation of f by its cardinal series as errors tat decrease exponentially. One is led to seek a less restrictive class of functions tan B() were te exponential decay of te error can be maintained. For any > 0, te translated sinc functions wit evenly spaced nodes are given by: x k s(k, )(x) = sinc k = 0, ±1, ±,,(3) Tey are based on te infinite strip D d in te complex plane, D d = w = u + iv: v < d п. (4) To construct approximation on te interval Γ= [a, b], te conformal map (z) = ln ( z a ) will be used. b z 13

2 Te map carries te eye-saped region to D E = z = x + iy: arg z a < d b z π. (5) Te basic functions on te Γ = [a, b] for z D d are derived from te composite translated sinc functions s j (z) = s(j, )o (z) (z) j = sinc, (6) were te inverse map of w= (z) is; z = 1 (w) a + bew =. (7) 1 + ew Hence te range of 1 on te real line is as follows; Γ = {ψ(u) = 1 (u) D E : < u < }. (8) Te sinc grid points z k Γ in D d will be denoted by x k and will be sown as; x k = 1 a + bek (k) = k 1 + ek = 0, ±1, ±,. (9) Te class of functions suc tat te known exponential error estimates exist for sinc interpolation is denoted by B(D E ) and is defined by te following definition. Definition. Let B(D E ) denote a family of functions F wic are analytic in D E and satisfy; F(z)dz 0, u ±, ψ(u+l) were L={iv: v < d π }, and on te boundary of D E (denoted by D E ), satisfy F(z)dz < (F) =. D E In te following teorem te interpolation rule based on te sinc function will be presented. Teorem 1. If F B(D E ), ten for all x Γ we ave; k= (F ) ) πd F(x) (F ) πdsin( πd F(x k )s(k, )o (x) e πd. Moreover, if F(x) c 1 e α (x), x Γ, for some positive constants c 1 and α, and coosing = πd α πd ln, ten, F(x) = k= F(x k )s(k, )o (x) + O(exp (πdα) 1 ). As we see in Teorem 1, tis metod as exponential rate of convergence so te error of computations decreases exponentially. Teorem. Let be a conformal one-to-one map of te simply connected domain D d onto D E. Ten (0) δ kj = [s(k, )o (x)] x=xj 1 k = j, = 0 k j. (1) d δ kj = d [s(k, )o (x)] x=x j = 1 0 k = j, ( 1) j k k j. j k () δ kj = d d [s(k, )o (x)] x=x j = 1 π 3 ( 1) j k (j k) k = j, k j. (10) (11) (1) One can see te proofs of Teorems 1 and in [4]. Based on te metod wic is described in [1], te approximate solution for y(x), in equation y n = g x, y, y,, y (n 1) x [0, 1], (13) were te boundary conditions are given by te values of Function y or its pt derivatives in Points 0 and 1, may be written as y (x) = u (x) + p(x), were u (x) = k= c k w(x)s k (x), w(x) = x p (x 1) p and p(x)=a 0 + a 1 x + + a n x n. Te unknown coefficients {c k } k= and a 0, a 1,, a n are determined by substituting y (x) in te Equation (13) and evaluating results at te sinc points; x j = ej 1+e j j = 1,,. III. APPLICATIO OF THE SIC-COLLOCATIO METHOD TO THE TELEGRAPH EQUATIO In tis paper we consider te second-order yperbolic problem wit constant coefficients as follows: u u + α t t + βu = u + f(x, t), x 0 < x < 1, 0 < t T. (14) Wit te initial conditions given by; u(x, 0) = g 1 (x) 0 < x < 1, (15) u t (x, 0) = g (x) 0 < x < 1, (16) and Diriclet boundary conditions; u(0, t) = 1 (t) 0 < t T, (17) u(1, t) = (t) 0 < t T, (18) 14

3 Were α, β are known constant coefficients, f, g 1, g, 1 and are known functions and function u is unknown. Discretizing time derivative of tis equation by using a classic finite difference formula and space derivatives by te θ-weigted (0 θ 1) sceme between successive two time levels n and n + 1 one obtains; u u n + u n 1 δt + α u u n δt + θ βu u xx f(x, t ) + (1 θ) βu n u n xx f(x, t n ) = 0, (19) Were u n = u(x, t n ), t = t n + δt and δt is a time step size. ow te sinc-collocation metod wit respect to te space variable wic is represented in [1] and described briefly latter, will be used. We modify te sinc basis functions as w(x)s k (x), were w(x)=1. ow an approximate solution for u (x) as u (x) = y (x) + p (x) were p (x) = a 0 + a 1 x + a x and y (x) = k= c k w(x)s k (x) = k= c k s k (x) will be considered. Substituting u (x) in (14) and evaluating te result at te sinc points x j = ej j = 1,,, te e j +1 equation is reduced to a system of algebraic equations as follows: (1 + αδt + θβδt ) c k s k x j + a 0 θδt c k k= k= + a 1 x j + a x j = ( + αδt β(1 θ)δt ) (1) j δ k,j + j δk,j + a c n k s k x j + a n 0 + a n 1 x j + a n x j + k= (1 θ)βδt c k n k= + a n (1) j δ k,j + j δk,j c k n 1 s k x j + a 0 n 1 + a 1 n 1 x j k= + a n 1 x j θf x j, t (1 θ)f x j, t n. (0) Applying te initial conditions result in: u (0) = u(0, t ) = 1 (t ) = y (0) + p (0) = a 0 (1) u (1) = u(1, t ) = (t ) = y (1) + p (1) = a 0 + a 1 + a. () By te Equations (0)-(), one can find unknown coefficient {c k } k= and a 0, a 1, a. At te first step, u n and u n 1 will be computed by using te initial conditions as follows: u(x, 0) = g 1 (x) u 0 (x) = c 0 k s k (x) + a a 0 1 x + a 0 x (3) k= u t (x, 0) = g (x) u 0 t (x) = u1 (x) u 0 (x) = g δt (x) u 1 (x) = δtg (x) + u 0 (x) = c 1 k s k (x) + k= a a 1 1 x + a 1 x. IV. UMERICAL RESULTS To sow te efficiency of te sinc-collocation metod in comparison wit te exact solution of te telegrap equations in te form of (14)-(18), some examples are presented. In te following examples we coose α = 1/4 and d = π/ wic leads to = π/ and δt = 0.1. Example 1. Consider te yperbolic telegrap Equation (14)-(18) wit α = 1, β = 1, f(x, t) = 0, g 1 (x) = e x, g (x) = e x, 1 (t) = e t, (t) = e 1 t. Te exact solution of tis equation is u(x, t) = e x t. In Table 1, te results of te present metod are compared wit te exact solution for t = 0.4. TABLE I COMPARISO OF THE VALUES OF U(X, T) AD ABSOLUTE ERROR AT T = 0.4 x j Exact solution Sinc solution Absolute error

4 Example. Considering te following telegrap equation u t + u t + u = u x + x + t 1 0 < x < 1, 0 < t T Wit te given initial and boundary conditions as follows; u(x, 0) = x u t (x, 0) = 1 u(0, t) = t u(1, t) = 1 + t, Te exact solution of tis equation is u(x, t) = x + t. Wen t = 0.3 te following numerical solution will be resulted: Fig. 1 Plot of te absolute error for Example 1 wen t = 0.4 Also wen x = 0.01 one obtains: TABLE II COMPARISO OF THE VALUES OF U(X, T) AT X = 0.01 AD ABSOLUTE ERROR Exact Sinc Absolute Solution Solution Error TABLE III COMPARISO OF THE VALUES OF U(X, T) AD ABSOLUTE ERROR AT T = 0.3 Exact solution Sinc solution Absolute error Fig. Plot of analytic and Sinc-Collocation solutions of Example 1 at x = 0.01 Fig. 3 Plot of te absolute error for Example 1, wen x = 0.01 Fig. 4 Plot of te absolute error for Example, wen x = 0.01 Te most advantage of tis metod is tat it provides te solution of te problem at any point. However te finite difference metod provides te solution of equation on mes point only. Considering absolute error in te above examples, we see tat te sinc-collocation metod is an accurate metod for approximating te solution of te telegrap equation. ote tat te accuracy of tis metod can be improved by increasing. V. COCLUSIO In tis work we used sinc-collocation metod for solving te second-order linear yperbolic telegrap equations. Tis approac reduced te PDE equation to a linear system of algebraic equations. Te obtained results sowed tat tis metod can solve tese kinds of problems effectively. 16

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