expansion method - İstanbul Ticaret Üniversitesi
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expansion method - İstanbul Ticaret Üniversitesi
İstanbul Ticaret Üniversitesi Fen Bilimleri Dergisi Yıl: 14 Sayı: 28 Güz 2015 s. 85-92 Research Article TRAVELING WAVE SOLUTIONS OF SOME NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS BY USING EXTENDED (G ′ G ) - EXPANSION METHOD Doğan KAYA 23 Asıf YOKUŞ 1 ABSTRACT In this study, we use extended (G ′ G ) -expansion method which is suggested in (Wang et al., 2008). By this method, we find traveling wave solutions of nonlinear Burgers and KdV equations. It is obtained graphics of these solutions by using Mathematica. Keywords: Burgers equation, KdV equation, extended (G ′ G ) -expansion method, traveling wave solutions. Araştırma Makalesi GENİŞLETİLMİŞ (G ′ G ) -AÇILIM METODU KULANILARAK BAZI LİNEER OLMAYAN KISMİ DİFERANSİYEL DENKLEMLERİN YÜRÜYEN DALGA ÇÖZÜMLERİ ÖZ Bu çalışmada bir genişletilmiş (G ′ G ) –açılım metodu inşa edildi. Biz bu metodu lineer olmayan denklemi, Burgers ve KdV denkleminin yürüyen dalga çözümünü bulmak için bu metodu uyguladık. Biz ayrıca bu metodun diğer lineer olmayan denklemlere uygulanabilir olduğunu gösterdik. Anahtar kelimeler: Burgers denklemi; KdV denklemi; Makale Gönderim Tarihi: 03.03.2016 (G ′ G ) -açılım metodu; yürüyen dalga çözümleri. Kabul Tarihi : 06.04.2016 Corresponding Author/Sorumlu Yazar: Fırat Üniversitesi, Fen Fakültesi, Aktüerya Bilimleri Bölümü, e-mail/e-ileti: asfyokus@firat.edu.tr İstanbul Ticaret Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü, dogank@ticaret.edu.tr 1 2 Asıf YOKUŞ, Doğan KAYA 1. INTRODUCTION In this paper, we used the (G ′ G ) -expansion method. Firstly, this method is introduced in (Wang et al., 2008). Then, a new extended version of this method was presented in (Guo and Zhou, 2010; Guo and Zhou, 2010) have suggested a new expansion to obtain traveling wave solution of nonlinear partial differential equation (in short, PDE) in his paper. We find the traveling wave solutions of the Burgers equation and KdV equation by using Guo and Zhou’s recommendations. To find exact solutions of nonlinear partial differential equations always impressed scientists. It is too important to find exact solutions of nonlinear partial differential equations. These equations are mathematical models of complex physical occurrences that arise in engineering, chemistry, biology, mechanics and physics. Solution of a nonlinear partial differential equation gives much knowledge to the scientists about the nature of these events. Therefore, various effective methods have been developed to understand the mechanisms of these physical models, to help physicists and engineers and to ensure knowledge for physical problems and its applications. There are many methods to obtain traveling wave solutions of nonlinear partial differential equation in literature (Fan, 2000; Clarkson, 1989; Wazwaz, 2005; Parkes et al., 1996; Fan, 2000; Elwakil et al., 2002; Zheng et al., 2003; He et al., 2006; Inan, 2010). 2. AN ANALYSIS OF THE METHOD AND APPLICATIONS Before starting to give the (G ′ G ) - expansion method, we will give a simple description of the (G ′ G ) -expansion method. For doing this, one can consider in a two-variables general form of nonlinear ordinary differential equations (1) Q(u , u t , u x , u xx , ) = 0 , and transform Eq. (1) with u (x, t ) = u (ξ ) , ξ = x − Vt , where V is constant. After transformation, we get nonlinear ordinary differential equations for u (ξ ) Q' (u ′, u ′′, u ′′′,) = 0. (2) The solution of the Eq. (2) we are looking for is expressed as − i +1 G′ i i i − − 1 2 m 1 G′ G G′ G′ G′ u (ξ ) = a0 + ∑ ai + bi σ 1 + + ci + d i , 2 G G i =1 G 1 G′ µG σ 1+ µ G (3) where G = G (ξ ) satisfies the second order linear ordinary differential equation in the form (4) G ′′ + µG = 0, 86 İstanbul Ticaret Üniversitesi Fen Bilimleri Dergisi Güz 2015 where ai , , bi , , ci , , di , and µ are constants to be determined later, the positive integer m can be determined by balancing the highest order derivative and with the highest nonlinear terms into Eq. (2). Substituting solution (3) into Eq. (2) and using Eq. (4) yields a set of algebraic equations for same order of (G ′ G ) ; then all coefficients same order of (G ′ G ) have to vanish. After this separated algebraic equation, we can find ai , , bi , , ci , , di , and µ constants. General solutions of the Eq. (4) have been well known us, then substituting ai , , bi , , ci , , di , and the general solutions of Eq. (4) into (3) we have more traveling wave solutions of Eq. (1) (Guo et al., 2010; Yokus, 2011; Inan, 2010). Example 1. Let’s consider classical Burgers equation (Gorguis, 2006) (5) u t + uu x + u xx = 0. For doing this example, we can use transformation u ( x, t )= u (ξ ) , ξ= x − Vt then Eq. (5) become 1 (6) c − Vu + u 2 + δ u ′ = 0, when balancing u 2 with u′ 2 then gives m = 1 . Therefore, we may choose −1 1 G′ 2 G′ G′ u (ξ ) = a0 + a1 + b1 σ 1 + + c1 + d1 µG G G 1 1 G′ 2 σ 1 + µ G , (7) substituting Eq. (7) into (6) yields a set of algebraic equations for a0 , a1 , a2 , b1 , c1 , d1 , µ , σ and V . These systems are a 02 b 2σ + c + a1c1 + b1 d1 − a 0V + c1δ − a1δµ + 1 = 0, d1δσ = 0, 2 2 2 c 2 + 2c1δµ d12 a1 (a 0 − V ) = 0, = 0, c1 (a 0 − V ) = 0, = 0, 2 2σ d1δσ a12 µ + 2a1δµ − b12σ b1c1 = 0, = 0, a 0 d1 − d1V = 0, = 0, µ 2µ b δσ a1 d1 − b1δσ = 0, 1 = 0, a 0 b1 − b1V = 0, a1b1 = 0. c1 d1 = 0, µ (8) From the solutions system, we obtain the following with the aid of Mathematica. Case 1: a 0 = − 2 c − 8δ 2 µ , a1 = 2δ , b1 = 0, c1 = −2δµ , d 1 = 0, V = − 2 c − 8δ 2 µ , (9) 87 Asıf YOKUŞ, Doğan KAYA substituting Eq. (9) into (7) we have three types of traveling wave solutions of Eq. (5): ( ( ( )) ( ( µ A Cos µ x + 2 c − 8δ 2 µ t − A Sin µ x + 2 c − 8δ 2 µ t 2 1 u1 (ξ ) = − 2 c − 8δ 2 µ + 2δ 2 2 A1Cos µ x + 2 c − 8δ µ t + A2 Sin µ x + 2 c − 8δ µ t ( )) ( ( ( ( ( ( )) )) ( ( ( ( ( ( µ A Cos µ x + 2 c − 8δ 2 µ t − A Sin µ x + 2 c − 8δ 2 µ t 1 2 − 2δµ A1Cos µ x + 2 c − 8δ 2 µ t + A2 Sin µ x + 2 c − 8δ 2 µ t )) )) ))) ))) − −1 . Fig 1: Traveling wave solution of Eq. (5) for case 1 when A1 = A2 = 1, c = 0.6, µ = 0.1 and δ = 0.2. Case 2: a 0 = − 2 c − 2δ 2 µ , a1 = 2δ , b1 = 0, c1 = 0, d 1 = 0, V = − 2 c − 2δ 2 µ , (10) substituting Eq. (10) into (7) we have three types of traveling wave solutions of Eq. (5) as following ( ( ( ( ( )) ( ( ( ( µ A Cos µ x + 2 c − 2δ 2 µ t − A Sin µ x + 2 c − 2δ 2 µ t 2 1 u 2 (ξ ) = − 2 c − 2δ 2 µ + 2δ A1Cos µ x + 2 c − 2δ 2 µ t + A2 Sin µ x + 2 c − 2δ 2 µ t )) Fig 2: Traveling wave solution of Eq. (5) for case 2 when 88 )) ))). İstanbul Ticaret Üniversitesi Fen Bilimleri Dergisi Güz 2015 A1 = A2 = 1, c = 0.6, µ = 0.1 and δ = 0.2. Case 3: a0 = 2 c − 2δ 2 µ , a1 = 0, c1 = −2δµ , b1 = 0, d1 = 0, (11) 2 c − 2δ 2 µ , V = substituting Eq. (11) into (7) we obtain three types of traveling wave solutions of Eq. (5) as following ( ( ( ( ( )) ( ( ( ( µ A Cos µ x − 2 c − 2δ 2 µ t − A Sin µ x − 2 c − 2δ 2 µ t 2 1 u 3 (ξ ) = 2 c − 2δ µ − 2δµ A1Cos µ x − 2 c − 2δ 2 µ t + A2 Sin µ x − 2 c − 2δ 2 µ t 2 )) )) ))) −1 . Example 2. Let’s consider KdV equation (Kaya D. 2002), (12) ut + uu x + δ u xxx = 0. For doing this example, we can use transformation u ( x, t )= u (ξ ) , ξ= x − Vt then Eq. (12) become − Vu ′ + uu ′ + δu ′′′ = 0. (13) When balancing u ′u , u ′′′ then gives m = 2 . Therefore, we may choose −1 1 G′ 2 G′ G′ u (ξ ) = a0 + a1 + b1 σ 1 + + c1 + d1 µG G G 1 1 G′ 2 σ 1 + µ G 2 2 G′ G′ 1 G′ G′ + a2 + b2 σ 1 + + c2 + d 2 G G µ G G G′ G 2 + −1 1 G′ 2 σ 1 + µ G , (14) substituting Eq. (14) into (13) yields a set of algebraic equations for a0 , a1 , a2 , b1 , b2 , c1 , c2 , d1 , d 2 , µ , σ and V. These systems are a 02 b 2σ + c + a1c1 + a 2 c 2 + b1 d1 + b2 d 2 − a 0V + 2c 2δ + 2a 2δµ 2 + 1 = 0, 2 2 c 22 + 6c 2δµ 2 = 0, 2 c1c 2 + 2c1δµ 2 = 0, a0 c1 + a1c 2 + b1 d 2 − c1V + 2c1δµ = 0, a1 a 2 + 2a1δ + b1b2σ µ = 0, 2σ b σ b σ a + = 0, + a 0 a 2 + a 2V + 8a 2δµ + 2 2 2µ (15) d1 d 2 = 0, σ 2 2 3d 2δσ 2 = 0, 2 1 3d1δσ 2 = 0, µ2 = 0, 2 c12 d2 + a 0 c 2 − c 2V + 8c 2δµ = 0, 3d 2δσ = 0, 1 = 0, 2 2 2 σ µ − 3d1δσ a b σ + 6a 2δ + = 0, 2µ 2 6d 2δσ 2 µ 2 2 = 0, 2 2 6d1δσ 2 µ − 2b1δσ 2 (σ ) 2 µ (σ ) 2 µ 3 2 a0 a1 + a 2 c1 + b2 d1 − a1V + 2a1δµ + b1b2σ = 0, d 2 = 0, c 2 d 2 + 2d 2δµ 2 = 0, 2 1 3d1δσ 2 3 = 0, a 2 b2 + 2b2δ = 0, c 2 d 1 + c1 d 2 = 0, = 0, − d1δµσ = 0, d 2δµσ = 0, 3 3 (σ ) 2 (σ ) 2 89 Asıf YOKUŞ, Doğan KAYA − 2b2δσ 2 (σ ) µ 3 = 0, − 2 d 2δσ (σ ) µ 3 2 − b2δσ 2 (σ ) 3 = 0, − 2 a0 d1 + a1 d 2 − d1V + b1δµσ = 0, b1c2 = 0, a 0 b1 + b2 c1 − b1V = 0, b2δσ 2 (σ ) µ 3 2 8b2δσ = 0, 2 = 0, − 2b1δσ 2 (σ ) 3 2 µ 2 3b1δσ = 0, µ a1d1 + a 2 d 2 + 3b2δµσ = 0, = 0, 5b2δσ µ = 0, a 2 d 1 + 4b1δσ = 0, c1b1 + b2 c2 = 0, a1b1 + a0 b2 − b2V + 2b2δµ = 0, a 2 b1 + a1b2 = 0, From the solutions of the system, we obtain the following with the aid of Mathematica. Family 1: a0 = −8δµ − 2 c + 128δ 2 µ 2 , a1 = 0, a2 = −12δ b1 = 0, b2 = 0 c1 = 0, c2 = −12δµ 2 , (16) d1 = 0, d 2 = 0, V = − 2 c + 128δ 2 µ 2 . Substituting Eq. (16) into (14) we have three types of traveling wave solutions of Eq. (12) as following u1 (ξ ) = −8δµ − 2 c + 128δ 2 µ 2 )) ))) ( ( ( ( )) )) ( ( ( ( µ ( A Cos ( µ ( x + 2 c + 128δ µ t ) ) − A Sin ( µ ( x + 2 c + 128δ µ t ) ) ) A Cos ( µ ( x + 2 c + 128δ µ t ) ) + A Sin ( µ ( x + 2 c + 128δ µ t ) ) ( µ A Cos µ x + 2 c + 128δ 2 µ 2 t − A Sin µ x + 2 c + 128δ 2 µ 2 t 2 1 − 12δ 2 2 2 2 A1Cos µ x + 2 c + 128δ µ t + A2 Sin µ x + 2 c + 128δ µ t − 12δµ 2 2 −2 2 2 2 2 2 1 2 2 1 2 . 2 2 Family 2: a 0 = −8δµ − 2 c + 8δ 2 µ 2 , a1 = 0, a 2 = −12δ b1 = 0, b2 = 0 d1 = 0, d 2 = 0, V = − 2 c + 8δ µ . 2 2 c1 = 0, c 2 = 0, (17) Substituting Eq. (17) into (14) we find three types of traveling wave solutions of Eq. (12) as following 90 İstanbul Ticaret Üniversitesi Fen Bilimleri Dergisi Güz 2015 u2 ( ξ ) = −8δµ − 2 c + 8δ 2 µ 2 ( ( ( )) ( ( µ A Cos µ x + 2 c + 8δ 2 µ 2 t − A Sin µ x + 2 c + 8δ 2 µ 2 t 2 1 − 12δ 2 2 2 2 A1Cos µ x + 2 c + 8δ µ t + A2 Sin µ x + 2 c + 8δ µ t )) ( ( )) ( ( ))) . 2 Family 3: a 0 = −8δµ − 2 c + 8δ 2 µ 2 , a1 = 0, a 2 = 0, b1 = 0, b2 = 0 c1 = 0, c 2 = −12δµ 2 , (18) d1 = 0, d 2 = 0, V = − 2 c + 8δ 2 µ 2 . Substituting Eq. (18) into (14) we write three types of traveling wave solutions of Eq. (12) as following −8δµ − 2 c + 8δ 2 µ 2 u3 (ξ ) = ( ( ( )) ( ( µ A Cos µ x + 2 c + 8δ 2 µ 2 t − A Sin µ x + 2 c + 8δ 2 µ 2 t 2 1 − 12δµ 2 2 2 2 2 A1Cos µ x + 2 c + 8δ µ t + A2 Sin µ x + 2 c + 8δ µ t ( ( )) ( ( )) ))) −2 . 3. CONCLUSIONS In this study, we consider to solve the traveling wave solutions of the classical Burgers equation and KdV equation by using the extended (G ′ G ) -expansion method. The method can be applied to many other nonlinear equations or coupled ones. 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