A Matrix Approach to Solving Hyperbolic Partial Differential Equations Using Bernoulli Polynomials

Transcription

1 Filomat 3: DOI 12298/FIL164993E Pulished y Faculty of Sciences and Mathematics University of iš Seria Availale at: A Matrix Approach to Solving Hyperolic Partial Differential Equations Using Bernoulli Polynomials Kura Erdem Bicer a Salih Yalcinas a Celal Bayar University Science and Art Faculty Mathematics Department Manisa Turkey Celal Bayar University Science and Art Faculty Mathematics Department Manisa Turkey Astract The present study considers the solutions of hyperolic partial differential equations For this an approximate method ased on Bernoulli polynomials is developed This method transforms the equation into the matrix equation and the unknown of this equation is a Bernoulli coefficients matrix To demostrate the validity and applicaility of the method an error analysis developed ased on residual function Also examples are presented to illustrate the accuracy of the method 1 Introduction Hyperolic partial differential equations are important for a variety of reasons The defining properties of hyperolic prolems include well posed Cauchy prolems finite speed of propagation and the existence of wave like structures with infinitely varied form The infinite variety of wave forms make hyperolic equations the preferred mode for sending information for example hearing sight television and radio Well posed Cauchy prolems with finite speed lead to hyperolic equations Since the fundamental laws of physics must respect the principles of relavity finite speed is required This together with causality require hyperolicity Thus there are many equations from Physics Those which are most fundamental tend to have close relationship with Lorentzian geometry A source of countless mathematical and technological prolems of hyperolic type are equations of fluid Dynamics [1 Also there are many models prolems of wave equation such as Virating string fixed at oth the ends electromagnetic waves longitudinal viration in a ar Hyperolic partial differential equations have attracted attention and investigation of new methods to solve these equations Characteristics method [2 is used commonly to solve these equations Also several numerical methods is used such as Homotopy perturation method Adomian decomposition Taylor matrix method Spline methods Parameters spline methods Bernstein Ritz-Galerkin Method etc [3-[9 In this study we try to solve this prolem using Bernoulli matrix method Also this method has een used to solve high-order linear differential-difference equations linear delay difference equations with variale coefficients and mixed linear Fredholm integro-differential-difference equations [1-[12 Also there are so many studies aout Bernoulli polynomials and its properties [13-[15 In this paper we will consider the second-order linear hyperolic partial differential-equation [16-[17 21 Mathematics Suject Classification Primary 35L1 58J45; Secondary 11B68 4C5 Keywords Second-order hyperolic equations Bernoulli and Euler numers and polynomials Matrix methods Received: 3 July 215; Accepted: 25 August 215 Communicated y Gradimir Milovanović and Yilmaz Simsek addresses: kura84@gmailcom Kura Erdem Bicer salihyalcinas@cuedutr Salih Yalcinas

2 K Erdem Bicer S Yalcinas / Filomat 3: Ax y 2 u x + Bx y 2 u 2 x y + Cx u y 2 + Dx y u y2 x 1 +Ex y u + Fx yu = Gx y x y 1 y with the Dirichlet oundary conditions ux = f 1 x x 1 u y = 1 y y 1 with the eumann oundary conditions u y x = f 2 x x and with the nonlinear integral conditions ux ydx = 2 y y 1 5 If B 2 4AC > then 1 is called a hyperolic equation Our purpose is to otain an approximate solution of 1 with the conditions and 5 in the following Bernoulli polynomial form [18 ux y = a rs B rs x y r= s= 6 a rs unknown Bernoulli coefficients and B rs x y = B r xb s y Here B r x and B s y are r-th and s-th degree Bernoulli polynomials respectively For two variales functions Bernoulli collocation points can e defined y x i = i y j = j i = 1 j = 1 7 on the x y 1 2 Fundamental Matrix Relations Second Order Linear Partial Differential Equations The approximate solution of 1 in terms of Bernoulli polynomials can e expressed as ux y = a rs B rs x y r= s= The matrix equation for this expression can e written as ux y = BxQyA 8 so that B = [ B x B 1 x B x 1 +1

3 K Erdem Bicer S Yalcinas / Filomat 3: B y B y B y B y Q = B y B y and unknown Bernoulli coeffients A = [ T a a 1 a a 1 a 11 a 1 a a 1 a And also we can write the Bernoulli polynomials B n x in the matrix form as follows B T x = ζx T x Bx = Xxζ T Bx = [ B x B 1 x B x Xx = 1 x x and ζ = k are the Bernoulli numers There is a matrix relation etween Bx and B 1 x as B 1 x = BxT T T =

4 K Erdem Bicer S Yalcinas / Filomat 3: Similarly the matrix relation etween Bx and its second derivative can e expressed as B 2 x = B 1 xt T = BxT T 2 If we continue to differentiate consecutively the i-th derivative of Bx is B i x = B i 1 xt T = BxT T i 9 Accordingly the j-th derivative of Qy is otained Q 1 y = QyT Q 2 y = Q 1 yt = QyT 2 Q j y = Q j 1 yt = QyT j 1 T = T T T T T T Here T is lock diagonale matrix which dimention is Consequently we otain a matrix relation as u ij x y = B i xq j ya = BxT T i QyT j A 11 for the truncated Bernoulli series with two variales u ij x y = r= s= a rs B ij rs x y On the other handwith the help of 8 and 11 the unknown function and its partial derivatives in 1 can e written as follows u x y = ux y = BxQyA u 1 x y = u x = BxTT QyA u 2 x y = 2 u x 2 = BxTT 2 QyA u 1 x y = u = BxQyTA 12 y u 2 x y = 2 u y 2 = BxQyT2 A u 11 x y = 2 u x y = BxTT QyTA The corresponding matrix form of the conditions and 5 respectively is otained as follows ux = BxQA = f 1 x x 1 13 u y = BQyA = 1 y y 1 14 u y x = BxQTA = f 2 x x 1 15

5 K Erdem Bicer S Yalcinas / Filomat 3: and BxQyAdx = Xxdx ζt QyA 16 = Lζ T QyA = 2 y y 1 L = [ Collocation Method We sustitute the matrix relations 12 into 1 to construct fundamental matrix equation So we otain the matrix equation Ax ybxt T 2 QyA + Bx ybxt T QyTA+ Cx ybxqyt 2 A + Dx ybxt T QyA+ 17 Ex ybxqyta + Fx ybxqya = Gx y By using in 17 collocation points defined y 7 the system of the matrix equation is otained as Ax i y j Bx i T T 2 Qy j A + Bx i y j Bx i T T Qy j TA+ Cx i y j Bx i Qy j T 2 A + Dx i y j Bx i T T Qy j A+ Ex i y j Bx i Qy j TA + Fx i y j Bx i Qy j A = Gx i y j i = 1 j = 1 18 or riefly the fundamental matrix equation ecomes ABT T 2 QA + BBT T QTA + CBQT 2 A+ DBT T QA + EBQTA + FBQA = G corresponds to a system of linear algeraic equations with unknown Bernoulli coefficients a a 1 a a 1 a 11 a 1 a a 1 a Therefore we can write the fundamental matrix equation 19 corresponding to 1 as WA = G 2 W = ABT T 2 Q + BBT T QT + CBQT 2 + DBT T Q + EBQT + FBQ Also we express the matrix forms of conditions and 16 y using collocation points as UA = [F 1 VA = [G 1 UA = [F 2 VA = [G

6 K Erdem Bicer S Yalcinas / Filomat 3: Here U = [ U U 1 U T [ V = V V 1 V T U = [ T [ U U 1 U V = V V 1 V T F 1 = [ f 1 x f 1 x 1 f 1 x T G1 = [ 1 y 1 y 1 1 y T F 2 = [ f 2 x f 2 x 1 f 2 x T G2 = [ 2 y 2 y 1 2 y T i = 1 j = 1 U i = Bx i Q = [ u i1 u i2 u i+1 2 V j = BQy j = [ v j1 v j2 v j+1 2 U i = Bx i QT = [ u i1 u i2 u i+1 2 V j = Lζ T Qy j = [ v j1 v j2 v j+1 2 To otain the approximate solution of 1 under conditions and 5 we form the augmented matrix U ; F 1 V ; G 1 [ W G = U ; F 2 25 V ; G 2 W ; G The unknown Bernoulli coefficients are otained as A = W 1 G [ W; G is generated y using the Gauss elimination method and then removing zero rows of gauss eliminated matrix By sustituting the determined coefficients into 6 we otain the Bernoulli polynomial solution ux y = a rs B rs x y r= s= 26 4 Residual Correction and Error Estimation In this section the residual correction method [19-[2 an efficient error estimation will e given for the Bernoulli collocation method For our purpose we can define the residual function with two variales of the Bernoulli collocation method as Rx y = Lu f 27 u which is the Bernoulli polynomial solution defined y 6 is the approximate solution of the prolem 1-5 hence u satisfies the prolem L [ u x y = Ax y 2 u + Bx y 2 u x 2 x y + Cx y 2 u + Dx y u y 2 x +Ex y u y + Fx yu = Gx y + R x y 28 Also the error function e x y can e defined as e x y = ux y u x y 29

7 K Erdem Bicer S Yalcinas / Filomat 3: ux y is the exact solution of the prolem 1-5 Sustituting 29 into 1-5 and using we have the error differential equation L [ e x y = L [ ux y L [ u x y = R x y with the homogenous conditions ux = x 1 u y = y 1 u y x = x 1 ux ydx = y 1 or clearly the prolem Ax y 2 e x 2 + Bx y 2 e x y + Cx y 2 e + Dx y e y 2 x +Ex y e y + Fx ye = R x y 3 Solving the prolem 3 in the same way as Section 3 we get the approximation e M x y to e x y M which is the error function ased on the Residual function R x y Consequently y means of the polynomials u x y and e M x y M we otain the corrected Bernoulli polynomial solution u M x y = u x y + e M x y Also we construct the Bernoulli error function e x y = ux y u x y the corrected Bernoulli error function E M x y = e x y e M x y = ux y u M x y and the estimated error function e M x y 5 umerical Example Consider the second order linear hyperolic equation 2 u t 2 2 u x 2 = 2x te x t x 1 t 1 with Dirichlet oundary conditions given y ux = x 1 u t = t 1 eumann oundary condition u t x = xe x x 1 and the nonlinear integral condition ux tdx = 2te t 1 + te t t T The exact solution of the this prolem is ux t = xte x t [9 By using Bernoulli Collocation Method asolute error functions and corrected Bernoulli error functions are compared in Tale for the various values of M and As shown in the tale the etter results may e otained y increasing tha values of and M

8 K Erdem Bicer S Yalcinas / Filomat 3: Tale The Comparison of asolute error function e and corrected Bernoulli error function EM x i t i Asolute error function e Corrected Bernoulli error function EM Asolute error function e Corrected Bernoulli error function EM = 3 M = 4 = 7 M = e e e e e e e e e e e e e e e 8 687e e e e e e e e e e e e e e e e e e e e e 6 6 Conclusion In the present article Bernoulli matrix method is presented to solve numerically second order linear hyperolic equation Also an error analysis ased on Residual functions has een descriped To illustrate the accuracy and efficiency of the new method an example has een analyzedthe otained numerical results show that this method can solve the prolem effectively References [1 J Rauch Hyperolic Partial Differential Equations and Geometric Optics American Mathematical Society 212 [2 Smith umerical Methods for Partial Differential Equations Oxford Press 1978 [3 J Biazara H Ghazvinia Homotopy perturation method for solving hyperolic partial differential equations Computers and Mathematics with Applications [4 J Biazar H Erahimi An approximation to the solution of hyperolic equations y Adomian decomposition method and comparison with characteristics method Applied Mathematics and Computation [5 B Bülül M Sezer A Taylor matrix method for the solution of a two-dimensional linear hyperolic equation Applied Mathematics Letters [6 J Rashidinia R Jalilian V Kazemi Spline methods for the solutions of hyperolic equations Applied Mathematics and Computation [7 H Ding Y Zhang Parameters spline methods for the solution of hyperolic equations Applied Mathematics and Computation [8 SA Yousefi Z Barikin M Dehghan Bernstein Ritz-Galerkin method for solving an initial-oundary value prolem that comines eumann and integral condition for the wave equation umerical Methods for Partial Differential Equations [9 F Shakeri M Dehghan The method of lines for solution of the one-dimensional wave equation suject to an integral conservation condition Computers and Mathematics with Applications [1 K Erdem SYalçınaş Bernoulli Polynomial Approach to High-Order Linear Differential-Difference Equations AIP Conf Proc [11 K Erdem SYalçınaş umerical approach of linear delay difference equations with variale coefficients in terms of Bernoulli polynomials AIP Conf Proc [12 K Erdem S YalçınaşM Sezer A Bernoulli Polynomial Approach with Residual Correction for Solving Mixed Linear Fredholm Integro-Differential-Difference Equations Journal of Difference Equations and Applications [13 H M Srivastava H Ozden I Cangul Y Simsek A Unified Presentation of Certain Meromorphic Functions Related to the Families of the Partial Zeta Type Functions and the L-Functions Appl Math Comput [14 F Costaile F Dell accio MI Gualtier A new approach to Bernoulli polynomials Rendiconti di Matematica Series VII [15 MX He PE Ricci Differential equation of Appell polynomials via the factorization method Journal of Comp and App Math [16 R Dennemeyer Introduction to partial differential equations and oundary value prolems McGraw-Hill 1968 [17 PR Garaedian Partial Differential Equations Wiley 1964 [18 TM Apostol Introduction to Analytic umer Theory Springer-Verlag 1976 [19 FA Oliveira Collacation and residual correction umer Math [2 I Celik Collacation method and residual correction using Cheyshev series Appl Math Comput