Ain Shams Engineering Journal (2014) xxx, xxx–xxx Ain Shams University Ain Shams Engineering Journal www.elsevier.com/locate/asej www.sciencedirect.com ENGINEERING PHYSICS AND MATHEMATICS A new approach for solving Duffing equations involving both integral and non-integral forcing terms * S Balaji Department of Mathematics, SASTRA University, Thanjavur 613 401, India Received 12 February 2014; accepted April 2014 KEYWORDS Abstract In this paper a Legendre wavelet operational matrix of derivative (LWOM) is used to solve the Duffing equation involving both integral and non-integral forcing terms with separated boundary conditions This operational matrix method together with Gaussian quadrature formula converts the given Duffing equation into system of algebraic equations, which indeed makes computation of solution easier The applicability and simplicity of the proposed method is demonstrated by some examples and comparison with other recent methods It is to be noted that, to the best of our knowledge, no wavelet based method applied for solving Duffing equations so far Duffing equation; Legendre wavelet; Operational matrix of derivative; Integral and non-integral forcing terms Ó 2014 Production and hosting by Elsevier B.V on behalf of Ain Shams University Introduction The general form of Duffing equation involving both integral and non-integral forcing terms with separated boundary conditions is given by Z x u00 xị ỵ ru0 xị ỵ fx; u; u0 ị ỵ kx; s; usịịds ẳ 0; 1ị 06x61 p0 u0ị q0 u0 0ị ẳ a; p1 u1ị ỵ q1 u0 1ị ẳ b * Tel.: +91 4362 264 101 E-mail address: balaji_maths@yahoo.com Peer review under responsibility of Ain Shams University Production and hosting by Elsevier ð2Þ where f: [0, 1] · R2 fi R, k: [0, 1] · [0, 1] · R fi R are continuous functions and r e R – {0}, p0, p1, q0, q1, a, b e R The Duffing equation is a well-known nonlinear equation of applied science, and its mathematical model is used as a powerful tool to describe many physical and practical phenomena such as classical oscillator in chaotic phenomena, periodic orbit extraction, nonlinear mechanical oscillators, and prediction of diseases [1–5] Solutions of Duffing equation have been studied through variety of numerical methods by many researchers [6–10] A generalized quasilinearization technique has been investigated in [11] for the solution of Duffing equation involving both integral and non-integral forcing terms with separated boundary conditions The analytic approximation of the forced Duffing equation with integral boundary conditions has been investigated in [12] through quasilinearization technique This method provides a sequence of approximate solutions, converges monotonically and quadratically to the unique solution Yao [13] presented an iterative reproducing kernel method for solving Duffing equation given by Eqs (1) and (2) In this method exact solution is represented 2090-4479 Ó 2014 Production and hosting by Elsevier B.V on behalf of Ain Shams University http://dx.doi.org/10.1016/j.asej.2014.04.001 Please cite this article in press as: Balaji S, A new approach for solving Duffing equations involving both integral and non-integral forcing terms, Ain Shams Eng J (2014), http://dx.doi.org/10.1016/j.asej.2014.04.001 in the form of series Geng [14] developed an improved variational iteration method for solving (1) and (2) This improved method avoids unnecessary repeated computation of unknown parameters in the initial solution In the recent years, wavelets theory is one of the growing and predominantly new methods in the area of mathematical and engineering research It has been applied in wide range of engineering sciences and mathematical sciences in thriving manner for solving variety of linear and non-linear differential and partial differential equations due to they build a connection with fast numerical algorithms [15–17], this is due to wavelets admit the exact representation of a variety of function and operators Recently, one can see the application of Legendre wavelets for solving variety of problems involving both engineering and applied sciences in [18–22] In this paper, we shall present a method based on the Legendre wavelet operational matrix of derivative for solving Duffing equation involving both integral and non-integral forcing terms with separated boundary conditions We have adopted this method to solve Duffing equation not only due to its emerging application of but also due to its greater convergence region It is to be noted that, to the best of our knowledge, no wavelet based method applied for solving Duffing equations so far The rest of the paper is organized as follows: Properties of Legendre wavelets and its operational matrix of derivative are introduced in Section In Section applicability of the proposed method for solving dumping equations is described The numerical experiments are presented in Sections and ends with brief conclusion Properties of Legendre wavelets and its operational matrix of derivative A family of functions constituted by wavelets, constructed from dilation and translation of a single function called mother wavelet When the parameters a of dilation and b of translation vary continuously, following are the family of continuous wavelets [23] xÀb ; a; b R; a0: wa;b xị ẳ jaj1=2 w a If the parameters a and b are restricted to discrete values as Àk a ¼ aÀk ; b ¼ nb0 a0 ; a0 > 1; b0 > and n, and k are positive integers, following are the family of discrete wavelets: wk;n xị ẳ ja0 jk=2 wak0 x À nb0 Þ; where wk,n(x) form a wavelet basis for L2(R) In particular, when a0 = 2, and b0 = 1, wk,n(x) forms an orthonormal basis [23] Legendre wavelets wn;m xị ẳ wk; n^; m; xị have four arguments; n^ ¼ 2n À 1; n ¼ 1; 2; 3; ; 2kÀ1 , k can assume any positive integer, m is the order for Legendre polynomials and t is the normalized time They are defined on the interval [0, 1) as ( q n^ỵ1 m ỵ 122k=2 Pm ð2k x À n^Þ; for n^2À1 k t < 2k ; wnm xị ẳ 0; otherwise; where m = 0, 1, , MÀ1 and n = 1, 2, 3, , 2kÀ1 [24,25] q The coefcient m ỵ 12 is for orthonormality, the dilation S Balaji parameter is a = 2Àk and translation parameter is b ¼ n^2Àk Pm(x) are the well-known Legendre polynomials of order m defined on the interval [À1, 1], and can be determined with the aid of the following recurrence formulae: P0 xị ẳ 1; P1 xị ẳ x; m xPm xị mỵ1 Pm1 xị; Pmỵ1 xị ẳ 2mỵ1 mỵ1 m ẳ 1; 2; 3; : The Legendre wavelet series representation of the function f(x) defined over [0, 1) is given by fxị ẳ X X cnm wnm xị; 3ị nẳ1 mẳ0 where cnm = ặf(x), wnm(x)ổ, in which Ỉ.,.ỉ denotes the inner product If the infinite series in Eq (3) is truncated, then Eq (3) can be written as fðxÞ ffi 2kÀ1 M À1 X X cnm wnm xị ẳ CT Wxị: 4ị nẳ1 mẳ0 where C and W(x) are 2kÀ1M · matrices given by C ¼ ½c10 ; c11 ; ; c1MÀ1 ; c20 ; c21 ; ; c2MÀ1 ; ; c2kÀ1 ; c2kÀ1 ; ; c2kÀ1 MÀ1 ; WðxÞ ẳ ẵw10 xị; w11 xị; ; w1M1 xị;w20 xị; w21 ðxÞ; ; w2MÀ1 ðxÞ; .; w2kÀ1 ðxÞ; w2kÀ1 ðxÞ; ; w2kÀ1 MÀ1 ðxÞ; T ð5Þ The derivative of the vector W(x) defined in [21] can be expressed by dWxị ẳ DWxị: dx 6ị where D is the 2k(M + 1) · 2k(M + 1) operational matrix of derivative given by F B0 B D¼B B @ 0 F ÁÁÁ ÁÁÁ 0C C C C A ð7Þ F in which F is (M + 1) · (M + 1) matrix and its (i, j)th element is defined as follow kỵ1 p > 2i 1ị2j 1ị i ẳ 2; .; M ỵ 1ị;j ẳ 1;2; : otherwise 8ị By using Eq (6) the operational matrix for nth derivative can be derived as dn Wxị ẳ Dn Wxị: dxn ð9Þ where Dn is the nth power of matrix D and this operational matrix of derivative has been successfully applied in [22] for optimal control in a convective–diffusive fluid problem For M = 2, k = 0, the operational matrix of derivative is given by Please cite this article in press as: Balaji S, A new approach for solving Duffing equations involving both integral and non-integral forcing terms, Ain Shams Eng J (2014), http://dx.doi.org/10.1016/j.asej.2014.04.001 Solving Duffing equations involving both integral and non-integral forcing terms 0 where wk,n(x) form a wavelet basis for L2(R) In particular, when a0 = 2, and b0 = 1, wk,n(x) forms P an orthonormal basis [23] By Eq (10), let uxị ẳ M1 iẳ1 c1i w1i xị be the solution of Eqs (1) and (2) where c1i = Ỉu(x), w1i(x)ỉ, for k = in which Ỉ.,.ỉ denotes the inner product 0 0 0 C C B pffiffiffi B 0 A and D2 ¼ @ D ¼ @2 0A pffiffiffipffiffiffi pffiffiffi 12 0 Solution of Duffing equations involving both integral and nonintegral forcing terms uxị ẳ n X huxị; w1i xịiw1i xị iẳ1 Consider the Dufng equation given in (1) and (2) Using Legendre wavelet approximations we have let uxị ẳ CT Wxị 10ị jẳ1 By using Eqs (9) and (10), we have u0 xị ẳ CT DWxị and u00 xị ẳ CT D2 Wxị 11ị n X bj huxị; wxj ịi ẳ jẳ1 n X bj bj ẳ Therefore Eq (1) becomes CT D2 Wxị ỵ rCT DWxị ỵ fx; CT Wxị; CT DWxịị Z x kx; s; CT Wsịịds ẳ ỵ Let bj = P ặu(x), w(x)ổ where w(x) = w1i(x) Let sn ẳ njẳ1 bj wxj ị be a sequence of partial sums Then, * + n X huxị; sn i ẳ uxị; bj wxj ị jẳ1 12ị n X ẳ jbj j2 j¼1 By means of the transformation x s ¼ z ỵ 1ị Further 13ị and by using Gaussian quadrature rule, we have Z x n x x xX kðx; s; CT WðsÞÞds % xi k x; z ỵ 1ị; CT W z ỵ 1ị iẳ1 2 ð14Þ Here the weights xi can be calculated with the help of the formula Z x Y z zj dz xi ẳ 15ị 16j6n zi À zj j–i Now apply Eqs (14), (15) into (12), we have CT D2 Wxị ỵ rCT DWxị þ fðx; CT WðxÞ; CT DWðxÞÞ n x x xX xi k x; z ỵ 1ị; CT W z ỵ 1ị ẳ ỵ iẳ1 2 ð16Þ Also, initial and boundary conditions from Eq (2) yields p0 CT W0ị q0 CT DW0ị ẳ a; p1 CT W1ị ỵ q1 CT DW1ị ẳ b 17ị k1 To find the solution u(x), we first collocate Eq (16) into M points at x, by taking suitable collocation points as following 2i ỵ 1ịp i ẳ 1; 2; ; 2kÀ1 M: xi ¼ cos ð18Þ 2k M These equations together with Eq (17) generate 2kÀ1 M + nonlinear equations which can be solved using Newton’s iterative method Consequently u(x) given in Eq (10) can be calculated 3.1 Convergence analyses Let wk;n ðxÞ ¼ ja0 jk=2 wðak0 x À nb0 Þ;
2 X n ksn sm k ẳ bj wxj ị jẳmỵ1 * + n n X X ẳ bi wxi ị; bj wxj ị iẳmỵ1 ẳ n n X X jẳmỵ1 bi bj hwxi ị; wxj ịi iẳmỵ1jẳmỵ1 ẳ n X jbj j2 jẳmỵ1 P As n 1, from Bessel’s inequality, we have j¼1 jbj j is convergent It implies that {sn} is a Cauchy sequence and it converges to s (say) Also hs À uðxÞ; wxj ịi ẳ hs; wxj ịi huxị; wxj ịi ẳ Lt sn ; wxj ị bj n!1 ẳ Lt hsn ; wxj ịi bj n!1 * + n X ẳ Lt bj wxj ị; wxj Þ À bj n!1 j¼1 ¼ bj À bj ¼ 0: Which is possible only if u(x) = s i.e both u(x) and sn converges to the same value, which indeed give the guarantee of convergence of LWOM Numerical experiments In this section in order to demonstrate the applicability of the proposed method, we have solved two Dumping equations, studied in [13,14], involving both integral and non-integral forcing terms The obtained results are compared with the corresponding experimental results obtained by the methods presented in [13,14] Please cite this article in press as: Balaji S, A new approach for solving Duffing equations involving both integral and non-integral forcing terms, Ain Shams Eng J (2014), http://dx.doi.org/10.1016/j.asej.2014.04.001 S Balaji Example 4.1 Consider the Duffing equation Rx 00 u ðxÞ þ u0 ðxÞ þ uðxÞu0 ðxÞ þ su2 ðsÞds ¼ fðxÞ; uð0Þ À u0 ð0Þ ¼ 0; where fðxÞ ẳ x6 1.25 u1ị ỵ u0 1ị ẳ 0: x5 À x4 3 LWOM 1.2 ỵ 3x 2x 3x: 1.15 u (x) It can be easily seen that the exact solution is u(x) = + x À x2 We have solved this problem using the approach described in the previous section Here the solution u(x) is approximated by u(x) = CTW(x) = c10w10 + c11w11 + c12w12 then the above equation is transformed into following form T C D Wxi ị ỵ CT DWxi ị ỵ CT DWxi ịCT D2 Wðxi Þ > > > > n x < xi X xi i ỵ xj z ỵ 1ịCT W z ỵ 1ị ẳ fxị jẳ1 2 > > > > : T T C Wð0Þ À C DW0ị ẳ 0; CT W1ị ỵ CT DW1ị ẳ 0: Exact 0