Adv Syst Sci Appl 2021; 03:75–90 Published online at https://ijassa.ipu.ru. Applications of Sine-Cosine Wavelets Method For Solving Drinfel’d–Sokolov–Wilson System Naser Azizi1, Reza Pourgholi1* 1School of Mathematics and Computer Science, Damghan University, Damghan, Iran Abstract: In this article, we use the Sine-Cosine wavelets (SCWs) method to numerically solve the Drinfel’d–Sokolov–Wilson (DSW) system. For this purpose, we use an approximation of functions with the help of SCWs, and we approximate spatial derivatives using this method. The operational matrix based on SCWs has a large number of zero components, which ensures good system performance and provides acceptable accuracy even with fewer collocation points. In the end, to show the effectiveness and accuracy of the method in solving this system one numerical example is provided. Keywords: Drinfel’d–Sokolov–Wilson system, numerical method, sine-cosine wavelets method, operational matrix 1. INTRODUCTION Nonlinear coupled partial differential equations (PDEs) are very significant in a type of scientific field, especially in fluid mechanics, solid-state physics, plasma waves, plasma physics, and chemical physics. Since many nonlinear physical phenomena can be explained by the exact and numerical solutions of nonlinear equations, the attempt for finding the exact and numerical solutions to these phenomena is important. In this article, our main goal is to solve numerically the DSW system. A generalized form of the DSW system is given by:{ Ψt + αΦΦx = 0, Φt + βΦxxx + γΨΦx + δΨxΦ = 0, (1.1) where α, β, γ, and δ are some nonzero parameters. System (1.1) plays an important role in fluid dynamics [8,12] and is originally introduced by Drinfel’d and Sokolov [7] and Wilson [24] as a model of dispersive water waves. Many researchers have devoted considerable efforts by successfully implementing various methods to extract solitary wave solutions and other solutions of DSW system [1, 10, 17, 23, 25]. One way to solve equations numerically is to use wavelets. The basic idea of wavelets goes back to the early 1960s [4,5]. There are developments concerning the multiresolution analysis algorithm based on wavelets [6] and the construction of compactly supported orthonormal wavelet bases [16]. So far, several problems have been solved numerically using different wavelets, for example, we can refer to references [2, 3, 9, 13, 18, 19, 26, 27]. In this paper, we consider system (1.1) by using the SCWs method to find numerical solutions. SCW has been used and showed efficiency to solve various problems. To indicate this, we can refer to some ∗Corresponding author: pourgholi@du.ac.ir 76 N. AZIZI, R. POURGHOLI works. Razzaghi and Yousefi in [20] have employed a SCW to solve variational problems. Tavassoli Kajani et al. [15] for solving integro-differential equations have presented a method based on SCWs. A numerical evaluation of Hankel transform for seismology has been given in [14] using the SCWs approach. Amir and Umer Saeed in [22] have used SCWs to solve the fractional nonlinear oscillator equations. In the present article, we intend to use the SCWs method to numerically solve the DSW system (1.1) with the initial conditions Ψ(x, 0) = f1(x), Φ(x, 0) = f2(x), x ∈ [0, 1], (1.2) and the boundary conditions Ψ(0, t) = g1(t), Φ(0, t) = g2(t), t ∈ [0, tfin], Φ(1, t) = k2(t), Φx(0, t) = w2(t), t ∈ [0, tfin], (1.3) where tfin represents the final time. The differentiable functions fi(x), gi(t), for i = 1, 2, k2(t), and w2(t) are known. The structure of this article is as follows: In Section 2, we describe the properties of the SCWs. In the following, expanding functions into the SCWs series and operational matrix of them for the numerical solutions are discussed. In Section 3, the procedure of implementation of the SCWs method, for system (1.1) with specified initial and boundary conditions (1.2) and (1.3) is presented. The numerical performance of the method is made in Section 4, and finally, concluding remarks are given in Section 5. 2. PROPERTIES OF SCWS Wavelets are useful mathematical functions constructed from the dilation and translation of a single function called the mother wavelet, which can be denoted by ω. Assuming that the expansion parameter η and the translation parameter ν are considered, we have the continuous wavelets family as follows [11]: Wη,ν(x) = |η|− 1 2ω (x− ν η ) , η, ν ∈ R, µ 6= 0. If the parameters η and ν are restricted to take values η = η−κ0 and ν = rν0a −κ 0 , a family of discrete wavelets is obtained as: Wκ,r(x) = |η0| κ 2ω(ηκ0x− rν0), (2.4) where η0 > 1, ν0 > 0, and r and κ are positive integers. The set {Wκ,r(x)} in (2.4), forms a wavelet basis for L2(R). Especially, if η0 = 2 and ν0 = 1, the set {Wκ,r(x)} forms an orthonormal basis. SCWs are defined on interval x ∈ [0, 1) as [14]: Wr,s(x) = 2 κ+1 2 Gs(2 κx− r)χ[ r 2κ , r+1 2κ ), (2.5) where κ = {0} ∪ N, r = 0, 1, 2, . . . , 2κ − 1, and χ[ r 2κ , r+1 2κ ) denotes the characteristic function given as χ[ r 2κ , r+1 2κ ) = { 1, x ∈ [ r 2κ , r+1 2κ ), 0, elsewhere. (2.6) Also, Gs(x) =  1√ 2 , s = 0, cos(2sπx), s = 1, 2, . . . , `, sin(2(s− `)πx), s = `+ 1, `+ 2, . . . , 2`, (2.7) Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) APPLICATIONS OF SINE-COSINE WAVELETS METHOD FOR SOLVING DRINFELD-SOKOLOV-WILSON 77 where ` is any positive integer. SCWs have compact support and are an orthonormal basis forL2([0, 1)). The orthonormal basis functions for SCWs by assuming κ = 1 and ` = 1 are obtained as follows: for 0 ≤ x < 1 2 =⇒ W0,0(x) = √ 2, W0,1(x) = 2 cos(4πx), W0,2(x) = 2 sin(4πx), for 1 2 ≤ x < 1 =⇒ W1,0(x) = √ 2, W1,1(x) = 2 cos(2π(2x− 1)), W1,2(x) = 2 sin(2π(2x− 1)). (2.8) So, with the collocation points xm = 2m− 1 2N , m = 1, 2, . . . ,N = 2κ(2`+ 1), (2.9) the graphs of Wr,s(x) for κ = ` = 1, are shown in Fig. 2.1. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 W0,0(x) W0,1(x) W0,2(x) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 W1,0(x) W1,1(x) W1,2(x) Fig. 2.1. The graphs of Wr,s(x) for κ = ` = 1. 2.1. Expanding functions into the SCWs Using the set of SCWs, any function Υ(x) ∈ L2([0, 1)) can be approximated as an infinite series of these functions as follows: Υ(x) = ∞∑ r=0 2∑̀ s=0 ar,sWr,s(x), (2.10) where ar,s =< Υ,Wr,s >= ∫ 1 0 Υ(x)Wr,s(x) dx. By truncating the infinite series (2.10) at levels r = 2κ − 1 and s = 2`, we obtain an approximate representation for Υ(x) as Υ(x) ' 2κ−1∑ r=0 2∑̀ s=0 ar,sWr,s(x) = ATΓ(x), (2.11) where A and Γ are (N × 1)-vectors and are introduced as follows: A = [ a0,0, a0,1, . . . , a0,2`, a1,0, a1,1, . . . , a1,2`, . . . . . . , a2κ−1,0, a2κ−1,1 . . . , a2κ−1,2` ]T , Γ = [ W0,0,W0,1, . . . ,W0,2`,W1,0,W1,1, . . . ,W1,2`, . . . . . . ,W2κ−1,0(x),W2κ−1,1 . . . ,W2κ−1,2` ]T . (2.12) Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) 78 N. AZIZI, R. POURGHOLI The SCWs matrix ΓN×N at the collocation points (2.9), is given as follows: ΓN×N = [ Γ ( 1 2N ) ,Γ ( 3 2N ) , . . . ,Γ (2N − 1 2N )] , in other words ΓN×N =  W0,0( 1 2N ) W0,0( 3 2N ) . . . W0,0(2N−1 2N ) W0,1( 1 2N ) W0,1( 3 2N ) . . . W0,1(2N−1 2N ) ... ... . . . ... W0,2`( 1 2N ) W0,2`( 3 2N ) . . . W0,2`( 2N−1 2N ) W1,0( 1 2N ) W1,0( 3 2N ) . . . W1,0(2N−1 2N ) W1,1( 1 2N ) W1,1( 3 2N ) . . . W1,1(2N−1 2N ) ... ... . . . ... W1,2`( 1 2N ) W1,2`( 3 2N ) . . . W1,2`( 2N−1 2N ) ... ... . . . ... ... ... . . . ... W2κ−1,0( 1 2N ) W2κ−1,0( 3 2N ) . . . W2κ−1,0(2N−1 2N ) W2κ−1,1( 1 2N ) W2κ−1,1( 3 2N ) . . . W2κ−1,1(2N−1 2N ) ... ... . . . ... W2κ−1,2`( 1 2N ) W2κ−1,2`( 3 2N ) . . . W2κ−1,2`( 2N−1 2N )  . In particular, for κ = ` = 1, the SCWs matrix Γ6×6 is given as follows: Γ6×6 =  √ 2 √ 2 √ 2 0 0 0 1 −2 1 0 0 0√ 3 0 − √ 3 0 0 0 0 0 0 √ 2 √ 2 √ 2 0 0 0 1 −2 1 0 0 0 √ 3 0 − √ 3  . 2.2. The operational matrix of SCWs Due to the vector form (2.12), the integration of Γ(x) can be calculated as follows:∫ x 0 Γ(t) dt = QΓ(x), where Q is N ×N operational matrix given by Q = 1 2κ+ 1 2  F S · · · S 0 F · · · S ... ... . . . ... 0 0 · · · F  , where S and F are (2`+ 1)× (2`+ 1) matrices (see [15]). Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) APPLICATIONS OF SINE-COSINE WAVELETS METHOD FOR SOLVING DRINFELD-SOKOLOV-WILSON 79 3. APPLICATION OF THE METHOD In this section, we use the SCWs method for finding the approximate solutions of system (1.1) with specified initial and boundary conditions (1.2) and (1.3). For this, dividing the interval [0, tfin] into M equal parts of length ~t = tfin M and denoting tn = (n− 1)~t, n = 1, 2, . . . , (M+ 1), we expand Ψ̇′ and Φ̇′′′ in terms of SCWs as, Ψ̇′(x, t) ∼= 2κ−1∑ r=0 2∑̀ s=0 ar,sWr,s(x) = ATΓ(x), (3.13) Φ̇′′′(x, t) ∼= 2κ−1∑ r=0 2∑̀ s=0 br,sWr,s(x) = BTΓ(x), (3.14) where prime and dot mean differentiation concerning x and t, respectively. Now, we consider the following two cases: Case 1: Considering the equation (3.13): By integrating this equation once concerning t from tn to t and once concerning x from 0 to x, we have Ψ′(x, t) = (t− tn)ATΓ(x) + Ψ′(x, tn), (3.15) Ψ̇(x, t) = ATQΓ(x) + g′1(t), (3.16) Now, integrating equation (3.16) once concerning t from tn to t, we obtain Ψ(x, t) = (t− tn)ATQΓ(x) + [g1(t)− g1(tn)] + Ψ(x, tn). (3.17) Case 2: Considering the equation (3.14): By integrating this equation once concerning t from tn to t and three times concerning x from 0 to x, we obtain Φ′′′(x, t) = (t− tn)BTΓ(x) + Φ′′′(x, tn), (3.18) Φ′(x, t) = (t− tn)BTQ2Γ(x) + Φ′(x, tn) + [w2(t)− w2(tn)] + x[Φ̇′′(0, t)− Φ̇′′(0, tn)], (3.19) Φ(x, t) = (t− tn)BTQ3Γ(x) + Φ(x, tn) + [g2(t)− g2(tn)] + x[w2(t)− w2(tn)] + x2 2 [Φ̇′′(0, t)− Φ̇′′(0, tn)], (3.20) Φ̇(x, t) = BTQ3Γ(x) + g′2(t) + xw′2(t) + x2 2 Φ̇′′(0, t). (3.21) By using the boundary condition Φ(1, t) = k2(t) equations (3.19)-(3.21) are changed as follows: Φ′(x, t) = (t− tn)BTQ2Γ(x) + Φ′(x, tn) + (1− 2x)[w2(t)− w2(tn)] + 2x[k2(t)− k2(tn)] − 2x[g2(t)− g2(tn)], (3.22) Φ(x, t) = (t− tn)BTQ3Γ(x) + Φ(x, tn) + (1− x2)[g2(t)− g2(tn)] + x(1− x)[w2(t)− w2(tn)] + x2[k2(t)− k2(tn)], (3.23) Φ̇(x, t) = BTQ3Γ(x) + (1− x2)g′2(t) + x(1− x)w′2(t) + x2k′2(t). (3.24) Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) 80 N. AZIZI, R. POURGHOLI Discretizing the results (3.15)-(3.17) and (3.18) and (3.22)-(3.24), by assuming x→ xm and t→ tn+1, we have Ψ′(xm, tn+1) = ~tATΓ(xm) + Ψ′(xm, tn), (3.25) Ψ̇(xm, tn+1) = ATQΓ(xm) + g′1(tn+1), (3.26) Ψ(xm, tn+1) = ~tATQΓ(x) + [g1(tn+1)− g1(tn)] + Ψ(xm, tn), (3.27) Φ′′′(xm, tn+1) = ~tBTΓ(xm) + Φ′′′(xm, tn), (3.28) Φ′(xm, tn+1) = ~tBTQ2Γ(xm) + Φ′(xm, tn) + (1− 2xm)[w2(tn+1)− w2(tn)] + 2xm[k2(tn+1)− k2(tn)]− 2xm[g2(tn+1)− g2(tn)], (3.29) Φ(xm, tn+1) = ~tBTQ3Γ(xm) + Φ(xm, tn) + (1− x2 m)[g2(tn+1)− g2(tn)] + xm(1− xm)[w2(tn+1)− w2(tn)] + x2 m[k2(tn+1)− k2(tn)], (3.30) Φ̇(xm, tn+1) = BTQ3Γ(xm) + (1− x2 m)g′2(tn+1) + xm(1− xm)w′2(tn+1) + x2 mk ′ 2(tn+1), (3.31) where, xm’s are the collocation points that are introduced in (2.9). To linearized the nonlinear terms ΦΦx, ΨxΦ, and ΨΦx in system (1.1), we use the linearization form given by Rubin and Graves [21] as follows: ΦΦx = Φx(x, tn)Φ(x, tn+1)− Φx(x, tn)Φ(x, tn) + Φ(x, tn)Φx(x, tn+1), (3.32) ΨxΦ = Φ(x, tn)Ψx(x, tn+1)− Φ(x, tn)Ψx(x, tn) + Ψx(x, tn)Φ(x, tn+1), (3.33) ΨΦx = Φx(x, tn)Ψ(x, tn+1)− Φx(x, tn)Ψ(x, tn) + Ψ(x, tn)Φx(x, tn+1). (3.34) Using linear expressions (3.32)-(3.34), the discrete form of system (1.1) considering xm and tn+1 is as follows: Ψ̇(xm, tn+1) + αΦ′(xm, tn)Φ(xm, tn+1) + αΦ(xm, tn)Φ′(xm, tn+1) = αΦ′(xm, tn)Φ(xm, tn), Φ̇(xm, tn+1) + βΦ′′′(xm, tn+1) + δΦ(xm, tn)Ψ′(xm, tn+1) + δΨ′(xm, tn)Φ(xm, tn+1) +γΦ′(xm, tn)Ψ(xm, tn+1) + γΨ(xm, tn)Φ′(xm, tn+1) = δΦ(xm, tn)Ψ′(xm, tn) +γΦ′(xm, tn)Ψ(xm, tn). (3.35) Now, by using equations (3.25)-(3.31), system (3.35) leads to{ AT θ1 + BT θ2 = H1(xm, tn), AT θ3 + BT θ4 = H2(xm, tn), (3.36) where the matrices θi, i = 1, 2, 3, 4 are matrices with dimensions N ×N as follows: θ1 = QΓ(xm), θ2 = [ αΦ′(xm, tn)~tQ3Γ(xm) + αΦ(xm, tn)~tQ2Γ(xm) ] , θ3 = [ δΦ(xm, tn)~tΓ(xm) + γΦ′(xm, tn)~tQΓ(xm) ] , θ4 = [ Q3Γ(xm) + β~tΓ(xm) + δΨ′(xm, tn)~tQ3Γ(xm) + γΨ(xm, tn)~tQ2Γ(xm) ] , Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) APPLICATIONS OF SINE-COSINE WAVELETS METHOD FOR SOLVING DRINFELD-SOKOLOV-WILSON 81 and the matricesHi, i = 1, 2 are matrices with dimensions N × 1 as follows: H1(xm, tn) =− αΦ(xm, tn)Φ′(xm, tn)− g′1(tn+1)− αΦ′(xm, tn) [ (1− x2 m)[g2(tn+1)− g2(tn)] + xm(1− xm)[w2(tn+1)− w2(tn)] + x2 m[k2(tn+1)− k2(tn)] ] − αΦ(xm, tn) [ (1− 2xm)[w2(tn+1)− w2(tn)] + 2xm[k2(tn+1)− k2(tn)] − 2xm[g2(tn+1)− g2(tn)] ] , H2(xm, tn) =− βΦ′′′(xm, tn)− δΦ(xm, tn)Ψ′(xm, tn) − γΦ′(xm, tn)[g1(tn+1)− g1(tn) + Ψ(xm, tn)] − [ (1− x2 m)g′2(tn+1) + xm(1− xm)w′2(tn+1) + x2 mk ′ 2(tn+1) ] − δΨ′(xm, tn) [ (1− x2 m)[g2(tn+1)− g2(tn)] + xm(1− xm)[w2(tn+1)− w2(tn)] + x2 m[k2(tn+1)− k2(tn)] ] − γΨ(xm, tn) [ (1− 2xm)[w2(tn+1)− w2(tn)] + 2xm[k2(tn+1)− k2(tn)]− 2xm[g2(tn+1)− g2(tn)] ] . The matrix-vector form of system (3.36) is as follows:[ θ1 θ2 θ3 θ4 ] 2N×2N [ A B ] 2N×1 = [ H1 H2 ] 2N×1 (3.37) From (3.37), the coefficients vector A and B can be calculated. With these coefficients and using the equations (3.27) and (3.30), the approximate solutions are successively obtained. 4. NUMERICAL EXPERIMENTS In this section, we apply the SCWs method to obtain the numerical solutions of the DSW system (1.1). To compare the obtained numerical results, we use the following solutions that obtained by Arnous et al. ( [1]): Ψ(x, t) = 6c γ+2δ sech2 (√ c βk2 (k(x− ct)− ξ0) ) , Φ(x, t) = ± √ 12c2 α(γ+2δ) sech (√ c βk2 (k(x− ct)− ξ0) ) , where c and k are arbitrary real constants. To show the effectiveness and accuracy of the proposed method, we considered an example with α = β = γ = δ = 1, tfin = 1, ~t = 0.01. Remark 4.1: For describing the error, we introduce the infinity-norm of absolute error and the root mean square (RMS) error norm as follows: LΨ ∞ = ||Ψ(xm, t)−Ψ∗(xm, t)||∞ = max 1≤m≤N |Ψ(xm, t)−Ψ∗(xm, t)|, RMSΨ = [ 1 2N 2N∑ m=1 ( Ψ(xm, t)−Ψ∗(xm, t) )2 ] 1 2 , (4.38) Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) 82 N. AZIZI, R. POURGHOLI where Ψ∗ is the approximate solution of Ψ. Similarly, the LΦ ∞ and RMSΦ are obtained according to formulas (4.38). The numerical results for Ψ(x, t) and Φ(x, t) at time t = 1 when ` = κ = 1 are reported in Table 4.1. For different values of κ, Table 4.2 stated the calculated errors (4.38) at time t = 0.5, also, the execution times for these values are given in Table 4.3. Difference between exact and numerical solutions Ψ and Φ at time t = 1 are shown in Figs. 4.2-4.4 and 4.5-4.7, respectively. Table 4.1. The numerical results for Ψ(x, t) and Φ(x, t) at t = 1 when ` = κ = 1. xm Ψ(xm, 1) Ψ∗(xm, 1) |Ψ(xm, 1)−Ψ∗(xm, 1)| Φ(xm, 1) Φ∗(xm, 1) |Φ(xm, 1)− Φ∗(xm, 1)| 0.0833333 0.153514 0.153517 2.358647e− 06 0.159955 0.159953 2.434473e− 06 0.25 0.157382 0.157384 2.526247e− 06 0.161958 0.161949 8.477983e− 06 0.416667 0.160643 0.160646 2.873385e− 06 0.163627 0.163619 7.684178e− 06 0.583333 0.163242 0.163244 2.279712e− 06 0.164945 0.164933 1.248603e− 05 0.75 0.165133 0.165128 5.211826e− 06 0.165898 0.165906 7.514665e− 06 0.916667 0.166281 0.166280 1.685403e− 06 0.166474 0.166457 1.740170e− 05 Table 4.2. The calculated errors (4.38) at time t = 0.5 with ` = 1. Ψ(x, 0.5) Φ(x, 0.5) L∞ κ = 1 1.333830e− 06 8.630715e− 06 κ = 2 6.055654e− 08 1.078673e− 06 κ = 3 2.810132e− 09 7.847945e− 08 κ = 4 8.968863e− 11 5.304455e− 09 κ = 5 3.105035e− 12 3.172111e− 10 κ = 6 9.736049e− 14 2.018670e− 11 RMS κ = 1 7.661924e− 07 5.199718e− 06 κ = 2 3.315228e− 08 5.115186e− 07 κ = 3 1.000311e− 09 2.933696e− 08 κ = 4 3.271928e− 11 2.026272e− 09 κ = 5 9.981293e− 13 1.217716e− 10 κ = 6 3.154537e− 14 7.797156e− 12 Table 4.3. The execution times for different values of κ with ` = 1. κ = 1 κ = 2 κ = 3 κ = 4 κ = 5 κ = 6 CPU time (s) 102.835546 189.641444 373.585049 753.232887 1519.095320 3163.873643 Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) APPLICATIONS OF SINE-COSINE WAVELETS METHOD FOR SOLVING DRINFELD-SOKOLOV-WILSON 83 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x -6 -5 -4 -3 -2 -1 0 1 2 3 Ψ (x ,1 ) − Ψ ∗ (x ,1 ) ×10 -6 (ℓ = 1, κ = 1) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 Ψ (x ,1 ) − Ψ ∗ (x ,1 ) ×10 -7 (ℓ = 1, κ = 2) Fig. 4.2. Difference between exact and numerical solutions Ψ at time t = 1, when ` = 1 and κ = 1, 2. Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) 84 N. AZIZI, R. POURGHOLI 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x -12 -10 -8 -6 -4 -2 0 2 4 6 8 Ψ (x ,1 ) − Ψ ∗ (x ,1 ) ×10 -9 (ℓ = 1, κ = 3) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x -4 -3 -2 -1 0 1 2 3 Ψ (x ,1 ) − Ψ ∗ (x ,1 ) ×10 -10 (ℓ = 1, κ = 4) Fig. 4.3. Difference between exact and numerical solutions Ψ at time t = 1, when ` = 1 and κ = 3, 4. Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) APPLICATIONS OF SINE-COSINE WAVELETS METHOD FOR SOLVING DRINFELD-SOKOLOV-WILSON 85 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x -1.5 -1 -0.5 0 0.5 1 Ψ (x ,1 ) − Ψ ∗ (x ,1 ) ×10 -11 (ℓ = 1, κ = 5) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x -4 -3 -2 -1 0 1 2 3 Ψ (x ,1 ) − Ψ ∗ (x ,1 ) ×10 -13 (ℓ = 1, κ = 6) Fig. 4.4. Difference between exact and numerical solutions Ψ at time t = 1, when ` = 1 and κ = 5, 6. Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) 86 N. AZIZI, R. POURGHOLI 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x -2 -1.5 -1 -0.5 0 0.5 1 Φ (x ,1 ) − Φ ∗ (x ,1 ) ×10 -5 (ℓ = 1, κ = 1) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Φ (x ,1 ) − Φ ∗ (x ,1 ) ×10 -6 (ℓ = 1, κ = 2) Fig. 4.5. Difference between exact and numerical solutions Φ at time t = 1, when ` = 1 and κ = 1, 2. Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) APPLICATIONS OF SINE-COSINE WAVELETS METHOD FOR SOLVING DRINFELD-SOKOLOV-WILSON 87 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x -2 -1.5 -1 -0.5 0 0.5 1 Φ (x ,1 ) − Φ ∗ (x ,1 ) ×10 -7 (ℓ = 1, κ = 3) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x -12 -10 -8 -6 -4 -2 0 2 4 6 8 Φ (x ,1 ) − Φ ∗ (x ,1 ) ×10 -9 (ℓ = 1, κ = 4) Fig. 4.6. Difference between exact and numerical solutions Φ at time t = 1, when ` = 1 and κ = 3, 4. Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) 88 N. AZIZI, R. POURGHOLI 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x -8 -6 -4 -2 0 2 4 6 Φ (x ,1 ) − Φ ∗ (x ,1 ) ×10 -10 (ℓ = 1, κ = 5) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x -5 -4 -3 -2 -1 0 1 2 3 4 Φ (x ,1 ) − Φ ∗ (x ,1 ) ×10 -11 (ℓ = 1, κ = 6) Fig. 4.7. Difference between exact and numerical solutions Φ at time t = 1, when ` = 1 and κ = 5, 6. Given the approximation function Υ(x) expressed in section 2, the solutions of the system (1.1), can be expanded as: Υ(x) = ∞∑ r=0 2∑̀ s=0 ar,sWr,s(x). (4.39) Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) APPLICATIONS OF SINE-COSINE WAVELETS METHOD FOR SOLVING DRINFELD-SOKOLOV-WILSON 89 In the investigated example, we approximate the solution of this equation as follows: Υ(x) ' 2κ−1∑ r=0 2∑̀ s=0 ar,sWr,s(x), (4.40) which is the truncating the infinite series (4.39). By substituting the solutions ar,s in (4.40), we get the error function E (x) as follows: E (x) = ∣∣∣∣∣Υ(x)− 2κ−1∑ r=0 2∑̀ s=0 ar,sWr,s(x) ∣∣∣∣∣. Therefore, as κ increases, the series (4.40) becomes larger and closer to the series (4.39), in other words, E (x) approaches zero. The obtained numerical results for Ψ and Φ confirm this. 5. CONCLUSION In this article, using the SCWs method and using the initial and boundary conditions (1.2) and (1.3), we solved the DSW system (1.1) numerically. Considering the obtained numerical results in Tables 4.1 and 4.2, Figs. 4.2-4.7, and also comparing these results with the exact solutions, it can be concluded that the presented method for solving the DSW system (1.1) is an efficient and high accuracy method. The strength of this method is the simplicity of calculations with low storage space. REFERENCES 1. Arnous, A., Mirzazadeh, M., & Eslami, M. (2016). Exact solutions of the Drinfel’d– Sokolov–Wilson equation using Bäcklund transformation of Riccati equation and trial function approach. Pramana, 86(6), 1153–1160. 2. Aziz, I., Khan, F., et al. (2014). A new method based on Haar wavelet for the numerical solution of two-dimensional nonlinear integral equations. Journal of Computational and Applied Mathematics, 272, 70–80. 3. Chen, C., & Hsiao, C. (1997). Haar wavelet method for solving lumped and distributed- parameter systems. IEE Proceedings-Control Theory and Applications, 144(1), 87–94. 4. Chui, C. K. (2016). 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Legendre wavelets approach for numerical solutions of distributed order fractional differential equations. Applied Mathematical Modelling, 70, 350–364. Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) Introduction Properties of SCWs Expanding functions into the SCWs The operational matrix of SCWs Application of the method Numerical experiments Conclusion