Applied Mathematics, 2011, 2, 608618 doi:10.4236/am.2011.25081 Published Online May 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Meshless Method of Lines for Numerical Solution of Kawahara Type Equations Nagina Bibi, Syed Ikram Abbas Tirmizi*, Sirajul Haq Faculty of Engineering Sciences, Ghulam Ishaq Khan Institute, Topi, Pakistan Email: tirmizi@giki.edu.pk Received Feburary 22, 2011; revised March 26, 2011; accepted March 30, 2011 Abstract In this work, an algorithm based on method of lines coupled with radial basis functions namely meshless method of lines (MMOL) is presented for the numerical solution of Kawahara, modified Kawahara and KdV Kawahara equations. The motion of a single solitary wave, interaction of two and three solitons and the phenomena of wave generation is discussed. The results are compared with the exact solution and with the results in the relevant literature to show the efficiency of the method. Keywords: Kawahara Type Equations, Meshless Method of Lines (MMOL), Radial Basis Functions (RBFs) 1. Introduction In this paper we study numerical solution of Kawahara equation, Modified Kawahara equation and KdVKawa hara equation respectively given as: 0,, 0 tx xxxxxxxx uuu uuaxbt (1) 20,, 0 tx xxxxxxxx uuu uuaxbt (2) 0,, 0 t xxxxxxxxxx uu uuuuaxbt (3) A variety of physical phenomena like, magneto acous tic waves in plasma [1], shallow water waves with sur face tension [2] and capillary gravity water waves [3], are described by Kawahara Equation (1) and modified Kawahara Equation (2). KdVKawahara Equation (3) is used to describe the one dimensional evolution of small but finite amplitude long waves in various problems in fluid dynamics. This equation is a specific form of Ben neyLin equation [4,5]. Different analytic and numerical methods including the tanhfunction method [6], Adomian decomposition method [7], sinecosine method [8], variational iteration method, homotopy perturbation method [9], CrankNi colson Differential quadrature algorithms [10], Predic tor corrector methods [11], DualPetrov Galerkin method [12] and RBF collocation method [13] have been pro posed for solving the Kawahara type equations. We shall use method of lines approach [14,15] using RBFs for numerical solution of the above problems using radial basis functions. The method of lines [16] is pow erful and comprehensive approach for solving time de pendent partial differential equations (PDEs). This me thod involves two steps, first the approximation of spatial derivatives converting the PDEs to a system of ordinary differential equations (ODEs) and in the second step the resulting system of ODEs is integrated in time. One of the salient features of the MOL is the use of existing and well established numerical methods for ODEs. Coupling method of lines with radial basis functions makes it very simple by getting rid of mesh generation as compared to conventional mesh based methods. In 1990 Kansa [17, 18] for the first time used radial basis functions for nu merical solution of PDEs. Other researchers like Forn berg et al. [19], Hon and Wu [20], Franke and Schaback [21], Wong et al. [22], Chen and Tanaka [23] also con tributed a lot in this area. In order to solve the resultant collocation method Fasshauer modified the Kansa’s me thod to a Hermite type collocation method [24]. For non linear time dependent PDEs see also [25,26]. In 1971 Hardy [27], developed MQ to approximate geogra phi cal surfaces. In Franke’s [28] review paper, the MQ was ranked the best in some thirty scattered data interpolation methods. For solving PDEs the convergence proofs in applying the RBF’s is given by Wu [29]. The accuracy of RBFs methods depends on the choice of a parameter called shape parameter involved in infinitely smooth RBFs like, Multiquadric (MQ), Gaussian (GA), Inverse multiquadric (IMQ) radial basis functions. Tarwater [30] found that the Root Mean Square of error decreases up to certain limit and then increases rapidly when .c
N. BIBI ET AL.609 Golberg, Chen, and Karur [31] and Hickernell and Hon [32] applied the technique of cross validation to obtain an optimal value of the shape parameter. However the methods based on globally supported ra dial basis functions (GSRBFs) approach face the prob lem of illconditioning of the dense interpolation matrix. In order to overcome these difficulties several alterna tives including domain decomposition [33], precondi tioning [34] and use of compactly supported RBF [35] have been introduced. Another useful approach in this regard is locally supported RBFs instead of globally supported RBFs. The readers are recommended to visit Wu and Liu [36], C. K. Lie et al. [37], R. Vertnik and Bozidar Sarler [38]. It is worth mentioning that global, infinitely differentiable RBFs typically interpolate sm ooth data with spectral accuracy [3942] and the shape parameter can be adjusted with the number of centers in order to produce a interpolation matrix which is well conditioned enough to be inverted in finite precision ari thmetic [43].The globally supported RBFs were used early on and for the problems whose size does not ex ceed 400500 data points. These methods should still be the method of choice [44]. This paper is organized in three sections. In Section 2 the numerical scheme is explained and Section 3 con tains the numerical examples for the justification of the method and we conclude in Section 4. 2. Numerical Scheme For implementation of numerical method, we consider the Kawahara Equation (1) with the following initial and boundary conditions ,0 ,,uxf xaxb (4) 1 ,,,uatg tubtgt 2 . (5) To apply meshless method of lines, we first use radial ba sis functions to approximate space derivatives. The problem domain [a,b] is divided into nodes ,1,2,, i iN Out of these points1 i are interior points d and ,2,,xi N ,1an i iN are the boundary points. ,uxtThe approximate solution of is given by T 1 ,, N nn jj j uxtu xttxx λ, (6) where T T 12 ,,, N xx x We have used the following RBFs 22, jj xxc MQ 2 exp , jj cx xGA 22 1, j j IMQ xx c where 1, 2,,jN and c being shape parameter. The Equation (6) in the matrix form is uAλ (7) where TT 12 12 ,,,, ,,, NN ututu t uλ and 11 211 T 112 222 T 2 T 12 ... .... .... ... .... N N N NNNN xx x xxx x x x xx x A Using Equation (7) in Equation (6) it follows that T1 ,() ()uxtxx Au Su, (8) where T1 xx A Applying Equation (8) to Equation (1), and collocating at each nodei , we get system of first order ODEs d0, d 1, 2,, i ix ixxx ixxxxxi uux xx t iN SuS uSu (9) where ii ut u 12 ,,, ,1,2,, ,,1,2, , xixi xiNxi jx iji SxSxS xiN xSxijN x S S In the similar fashion 12 ,,, 1, 2,, xxx ixxx ixxx iNxxx i xSxSx Sx iN , S 3 3,,1,2, , jxxx ij i Sx SxijN x 12 ,,, 1, 2,, xxxxxixxxxxixxxxxiNxxxxx i xS xSxSx iN , S 5 5,,1,2,, jxxxxx ij i Sx Sxij x N In order to write the above system of equations in terms of column vectors, let T 123 1 ... , NN uu uuu U , xjxi N Sx S Copyright © 2011 SciRes. AM
N. BIBI ET AL. Copyright © 2011 SciRes. AM 610 , , xxxjxxxi NN xxxxxjxxxxx i N Sx Sx S S Equation (9) then can be written as follows d dx xxx xxxxx t 0 UUSU SU SU (10) Rewriting Equation (10) as d dG t UU (11) where x xxx xxxxx G UUSUSUSU and the sy mbol (*) denotes component by component multiplica tion of two vectors. The initial condition is 00 0 012 ,,, N tuxuxux U (12) From the boundary conditions described in (5) we get 11 2 and N ut gtut gt (13) Now we use classical fourth order RungeKutta sch eme to solve Equations (11)(13), namely 1234 1 12 1 324 2( ) 6 ,2 , 2 nn nn nn tKK KK t KG KGK t 3 GKKGt UU UU UUK N (14) The RK4 scheme does not face problem of stability as long as the time step Δt is chosen sufficiently small (see Collatz [45]) which is so in our case. The rule of thumb in selection of the time step Δt for stability is as follows [46]: “The method of lines is stable if the eigenvalues of the (linearized) spatial discretization operator, scaled by time step Δt, lie in the stability region of the time discretization operator”. So the method is stable if all the eigenvalues of the Jacobian matrix in (11) lie inside the stability region R (i.e. , 1,2,, jj 2.78 0 j t ) of RK4 scheme. For further details regarding stability of RungeKutta fourth order method, see Lambert [47] and Jain [48]. As far as the selection of the shape parameter c is concerned in this work, first we have to find an inter val for c in which matrix A of radial basis functions is invertible and then select a value from that interval which gives us the most accurate results. 3. Numerical Application In this section the numerical results for Kawahara, modi fied Kawahara and KdV Kawahara equations are pre sented. The root mean square norm and maximum error norms are calculated by the formulas 2 2 2 1 N ex numexnum j Luuxuju j (15) max ex numexnum j Luuuju j (16) For Kawahara equation the lowest three conserved quantities, defined in [49] are also calculated using 2 12 2232 3 1 d, d, 2 131051 d 8962 xx IuxIux x uuu x Example 3.1 Consider the Kawahara equation 0 txxxxxxxxx uuu uu with the following initial and boundary conditions 4 105 , 0sech 169 o uxkx x (17) ,0;,uatubt 0 (18) The exact solution given in [10] is, 4 105 36 , sech 169169o uxtk xtx (19) where 11 213 k For numerical computation we take [a,b] = [–20,30], 2,1, 51 o xxN . The simulation is carried out up to time t = 25. L2 and L∞ norms are calculated at t = 0, 5, 15 and 25, using MQ and GA radial basis functions with the value of shape parameter found to be in neighbor hood of 4 as shown in Figure 1 and similarly for GA the value is found to be in neighborhood of 0.3. We have searched the optimal value of the shape parameter by plotting maximum error verses shape parameter with step 0.01.The three conserved quantities are also shown in the Table 1.The amplitudes and peak position of the solitary waves are also calculated. The results with present me thod using MQ are better than the polynomial based dif ferential quadrature (PDQ) method [10] and are very close to cosine expansion based differential quadrature (CDQ) method [10]. While the results obtained by GA are better than both methods mentioned in [10]. In Fig ure 2 the forward motion of the solitary wave in com parison with exact solution (19) at different time levels is shown. The Point wise rate of convergence in space is calcu lated using the following formula:
N. BIBI ET AL. 611 Table 1. Results for Kawahara equation in comparison with [10]. Method Time 3 210L 3 10L 1 I 2 I 3 I Height Peak Position CPU time(s) MQ(c = 4.3) GA(c = 0.27) PDQ[10] (Δt = 0.1) CDQ[10] 0 5 15 25 0 5 15 25 0 5 15 25 0 5 15 25 0 0.09468 0.15362 0.16818 0 0.10075 0.10113 0.13160 0 1.986 2.543 2.851 0 0.151 0.156 0.159 0 0.04669 0.05939 0.04660 0 0.034297 0.03830 0.03990 0 0.921 1.045 0.863 0 0.043 0.049 0.076 5.97359 5.97348 5.97343 5.97355 5.973599 5.973662 5.973675 5.973532 5.97357 5.97060 5.97014 5.97353 5.97357 5.97372 5.97364 5.97350 1.27250 1.27250 1.27250 1.27250 1.272502 1.272502 1.272502 1.272502 1.27250 1.27250 1.27250 1.27250 1.27250 1.27250 1.27250 1.27250 –0.16458 –0.16458 –0.16458 –0.16458 –0.16458 –0.16458 –0.16458 –0.16458 –0.16458 –0.16458 –0.16458 –0.16457 –0.16458 –0.16458 –0.16458 –0.16458 0.62130 0.62119 0.62038 0.61880 0.621301 0.621201 0.620382 0.618802 0.62130 0.62102 0.62047 0.61872 0.62130 0.62122 0.62037 0.61877 2 3 5 7 2 3 5 7 2 3 5 7 2 3 5 7 0.094 0.188 0.328 0.171 0.172 0.281 1 2 345 67 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Sha pe parame t er for MQ error Figure 1. Error verses shape parameter c for Example 3.1. Figure 2. Travelling wave solution of Kawahara equation (solid lines showing numerical solution and dot (.) showing exact solution. 1 10 10 1 log log ii hh ii uU uU hh where and i h U represents the exact solution and the numerical solution respectively and hi is spatial step size. We calculate spatial rate of convergence by keeping time step u t0.001 fixed and varying the number of collo cation points (F = 20, 40, 80). From the Table 2 we can see that the order of convergence decreases with the smaller spatial step size. In all numerical examples we have used MQ and GA in order to calculate order of convergence. Example 3.2 Consider the modified Kawahara equation 20 txxxxxxxxx uuuuu with the following initial and boundary conditions 2 , 0sechux Dkx (20) 2 2 , sech; , sech uatDka Bt ubtDkb Bt (21) The above conditions are extracted from the exact so lution given in [50], 2 ,sechuxtDkx Bt (22) where 311 ,, 25 25 10 DkB 4 neighborhood of 0.5 and 0.0001. The L2 and L∞ norms cal culated at t = 0, 5, 15, 25 are shown in Table 3. The calculation is carried out by taking [a,b] = [–30,30] with x = 1.We use MQ, GA and IMQ radial basis with shape parameter found to be in the neighborhood of 3 for MQ as shown in Figure 3 and for GA and IMQ it is in the Copyright © 2011 SciRes. AM
N. BIBI ET AL. 612 en N Order Table 2. Spatial rate of convergce at for Example 3.1, t = 25. L∞ order L2 MQ 2.8100 10–2 9.1143 8.2164 1.0180 10–1 9.2509 8.6358 20 40 80 GA 20 40 80 1× 5.07036×10–5 –5 3.56316×10 1.4909 ×10–2 4 5.01268×10–5 4.77379×10–5 0.5089 0.0704 8× 1.67092×10–4 –4 1.28499×10 5.3877 –2 5×10 1.35443×10–4 1.25773×10–4 0.3788 0.1068 ults for ed Kawation. method time(s) Table 3. ResModifihara equa time L∞ L2 I1 I 2 CPU M GA(c 0.43) IMQ(c = 0.0001) 6.1995×10–5 5.3996×10–5 1.806 10–1 1.7896×10–4 1.0804×10–4 5.2570×10–1 –8.4 2.68 2. 0.18 0. 0. Q(c = 2.9) = 0 5 15 25 0 5 15 25 0 5 15 25 0 1.0717×10–4 1.2130×10–4 0 8.6124×10–5 7.8371×10–5 0 59× 5.18281×10–1 7.63711×10–1 0 2.7337×10–4 3.4855×10–4 0 1.9575×10–4 2.5257×10–4 0 5.2819×10–1 2.2681×100 8525 –8.48524 –8.48487 –8.48464 –8.48525 –8.48510 –8.48472 –8.48442 –8.48525 –8.48525 –8.48134 –8.47434 328 2.683176 2.68296 2.68275 2.68328 2.68317 2.68296 2.68275 683281 2.683281 2.683307 2.683351 7 0.313 0.453 171 0.313 0.543 203 0.343 0.469 11.5 22.5 33.5 44.5 55.5 6 0 0. 0 5 0.1 0. 1 5 0.2 0. 2 5 0.3 0. 3 5 0.4 0. 4 5 0.5 Shape paramet er for MQ error Figure 3. Error verses shape parameter c for Example 3.2 A and MQ are showing better accuracy than IMQ. The dVKawahara equation with the following initial and boundary conditions . G order of convergence in space decreases with increasing N as shown in Table 4. The solitary wave profile at dif ferent time levels in comparison with the exact solution is shown in Figure 4. Example 3.3 Consider the K 0 txxxxx xxxxx uuuu uu 4 105 1 ,0 sechuxx x 169213 o (23) 4 1051205 , sech 169 169 213 o uxaat x 4 1051205 , sech 169 169 213 o uxbbt x (25) initial condition and boundary conditions are extracted from the exact solution given in [51]. 4 1051205 , sech 169 169 213 o uxtxt x (26) The calculation is carried out by tak ing [a,b] = [0,200] with Δx = 1. The discrete root mean square error and maximum error norm L∞ are calculated us GA a norm L2 ing MQ, nd IMQ for time t = 1 up to 5. From the results shown in Table 5 we can see that the both MQ and GA are showing very good agreement with the exact solution. Shape parameter verses error plot for MQ is shown in Figure 5. The spatial rate of convergence is shown in Table 6. The order of convergence decreases by in creasing collocation points for a fixed time step Δt = 0.001. The forward movement of the solitary wave at different time levels in comparison with the exact solu tion (26) is shown in Figure 6, same as in [52]. Example 3.4 Considering Equation (1) for interaction of two posi tive solitary waves with the following initial condition 224 1 , 0sech4 ii i ux Ax x . (24) i A We solve the problem by using MQ and GA RBFs taking [a,b] = [–50,100] with N = 201. The calculation is carried Copyright © 2011 SciRes. AM
N. BIBI ET AL. 613 Table 4. Spatial rate of converge N L∞ order nce at for Example 3.2, t = 5. L2 order MQ 24 48 –1 2.422 10–5 –2 –1 5.670 –5 –2 7.6180 96 GA 24 48 96 2.56860 × 10 1.49043 × 10–3 7.4291 5.20872×10 2.65141×10–3 85 × 3.47477 × 10 –4 5.00876 × 10 2.17409 × 10–5 5.9428 6.1163 4.5259 16×10 9.95424×10 –4 9.19460×10 5.46571×10–5 5.5472 6.7583 4.0723 Tab Resle 5.ults for Kdwahara eq thod L2 I1 plitudeCPU (s) V Kauation at t = 5. me L∞ I2 Amtime MQ(c = 2.6) GA(c = 0.2) IMQ(c = 0.0001) 3.7697×1 9 141 3.390 3.641 1.0977 × 10–4 4.7924 × 10–6 5.0111 × 10–1 1.8465×10–5 1.5920×100 5.97368 5.95790 1.27250 1.27250 0.621200 0.619041 0–4 5.9735.272500.62123.266 Tab ratevergt forple le 6.atial Sp of cone ance Exam 3.3, t = 5. N L∞ order L2order MQ 40 80 1.568 –4 3.731 –4 4.8553 160 GA 40 80 160 –2 3.12550×10 1.62584×10–3 4.2648 1.33333×10 4.60613×10–3 26×10 2.6983×10 –4 –3 1.4710×10 1.2344×10–5 2.1471 4.1971 3.5749 –1 04×10 1.1620×10 –3 –2 3.2768×10 2.4624×10–4 2.7242 4.7639 3.7341 Figure 4. Solitary wave solution of modified Kawahara equation in comparison with exact solution (solid lines show numerical solution and dot (.) showing exact solution). out up to time t = 50 by taking time step Δt = 0.001. For our numerical calculations the values of the parameters involved in above equation are chosen as: 121 2 4108 0, 20,,, 105 xxA A The two solitary waves propagate towards right as the time progresses. The process of interaction is shn in Figure 7. During this process the larger wave catches up th ow e smaller one and then the both waves separate from each other maintaining their original shape. From Table 7, we can see that the invariants of motion remain almost conserved as time increases. The variation in the three 1234567 0 0.2 0.4 1.4 0.6 0.8 1 1.2 error S hape p aram eter for MQ Figure 5. Error verses shape parameteric for Example 3.3. conserved quantities is found to be in the range: onsider Equation (1) with the initial condition 1 23 1 MQ 40.5092540.48389, 45.8361445.85093, 32.3721932.15991, GA 40.5092540.41284, I II I 23 45.8361445.84364, 32.3721832.14082.II Example 3.5 For interaction of three solitary waves we c 324 1 , 0sech4 i ii i A uxAx x Copyright © 2011 SciRes. AM
N. BIBI ET AL. 614 We use multiquadric and Gaussian radial basis func tions in our numerical simulations to solve this problem. The spatial domain is selected as [a,b] = [–30,120] with Δx = 0.75. The calculation is carried out up to time t = 50 taking time step Δt = 0.001. The values of the pa rameters used in above equation are selected as: 123 123 41086 20, 0, 20,,,, 105 xxx AAA The three solitary waves propagate towards right. The process of interaction is shown in Figure 8. During this rocess the taller wave moves faster and catches up the m p saller waves and then the three waves separate from each other. The shape of the three solitary waves after collision is maintained. From Table 8 we can see that the invariants of motion remain almost conserved as time increases. The variation in the three conserved quantities Figure 6. Solitary wave motion of KdV Kawahara equation (solid lines show numerical solution and dot (.) showing exact solution). 50 050 100 0 0. 5 1 1. 5 2 2. 5 3 3. 5 4 x u(x,t) t=0 GA MQ 50 050 100 0 0. 5 1 1. 5 2 2. 5 3 3. 5 4 4. 5 x u(x,t) t=25 GA MQ = 0 = 25 u (x,t) u (x,t) Figure 7. Interaction of two solitons for Kawahara equation Example 3.4. Table 7. Invariants for interaction of two solitons for Example 3.4. MQ GA –50 –50 time I1 I2 I3 I1 I 2 I 3 0 10 20 30 40 50 40.509259 40.507987 40.499950 40.5207361 40.550 45.836141 45.839240 45.8427481 45.8359623 45.8443065 –32.37219 –32.12856 –32.73918 –32.58956 –32.1 40.509259 40.492695 40.466653 571742 40.579624 45.836141 45.837775 45.839353 45.840373 45.841964 –32.37218 –32.12820 –32.73746 –32.59560 –32.12278 2.14082 5240 40.4838916 45.8509384 –32.1599140.412842 45.843648 –3 0125 40. is found to be in the range: Example 3.6 he phenomena of wave generation for Equatio). We consider the following initial conion 1 II 2 3 12 3 MQ 51.4726351.56859,51.10049.15838, 33.63806 33.06973, GA 51.4726351.47389,51.1004951.10452, 33.63806 33.39361. I II I 51 In this example we will show t n (1 dit 41 , 0sech213 o uxx x Computational domain [–40,130] with h = 0.625 is considered. The scheme is run up to time t = 18. We take Copyright © 2011 SciRes. AM
N. BIBI ET AL.615 10 . The sing solilitee s tary waves. The serves, hosi of three waves are calcariovel shown in Table W oflead itsoci from the other oo re n in three con ies i be wing range: Table 8. Invariants for interaction of three solitons for Example 3.5. MQ GA tary waves. The serves, hosi of three waves are calcariovel shown in Table W oflead itsoci from the other oo re n in three con ies i be wing range: Table 8. Invariants for interaction of three solitons for Example 3.5. MQ GA letary wave sps in to throliwavoli wav conconed quantitied quantitieight and peight and ption twtion tw ulated at vulated at vus time leus time les as servs as serv 9. 9. ith passageith passage time the time the ing ing e owing toe owing to faster vel faster velty gets farty gets far waves as sh waves as shwn in Figu wn in Figu 9. Variatio9. Variatio ed quantited quantits found tos found toin the folloin the follo time I1 I2 I3 I1 I 2 I 3 0 10 20 30 40 50 51.472631 51.631982 51.688762 51.664151 51.601572 51.568591 51.100493 51.094986 51.103862 51.127187 51.14098 51.15838 –33.63806 –33.40011 –34.23846 –35.7 –33.3 –33.06973 51.472631 51.835962 51.4 51.100493 51.098240 –33.63806 –33.39015 1904 8561 51.848746 51.690220 51.493752 73898 51.099504 51.101129 51.102653 51.104529 –34.21111 –35.15871 –33.83892 –33.39361 Table 9. Invariants for interaction o S f three solitons for Example 3.6. econd wave Third wave Leading wave time Height Position Height Position Height Position MQ 0 2 4 8 12 18 10 13.990228 13.645838 14.024270 1 13. 0 12.5 22.5 43.125  7.98375 8.471020 8.347924.375 3.427390 7.5 12.5 20 3.93545 90723 63.75 94.375 8.48316 8.34579 36.25 55 3.49294 3.50220 3  5.625 11.875  2.964111 3.247767  –0.625 1.875 5 40 200 4 20 4060 80100120 0 0.5 1 1.5 2 2.5 3 3.5 x u(x,t) t=0t=0 GA MQ 40 4.5 20 20 4006080100 120 0 0.5 1 1.5 2 2.5 3 3.5 4 GA x u t=2 (x,t) 5t=25 MQ 40 20 02040 60 80 100 120 0 0. 5 1 1. 5 2 2. 5 3 3. 5 4 4. 5 x u(x,t) t=40t=40 GA MQ 40 20 020406080 100 120 0 0. 5 1 1. 5 2 2. 5 3 3. 5 4 4. 5 x u(x,t) t=60t=60 GA MQ ree s = 0 Figure 8. Interaction of tholitons for Example 3.5. u (x,t) u (x,t) u (x,t) u (x,t) –40 –20 –40 –20 = 25 = 40 = 60 –40 –20 –40 –20 Copyright © 2011 SciRes. AM
N. BIBI ET AL. Copyright © 2011 SciRes. AM 616 20 020 40 60 80 100 0 1 2 3 4 5 6 7 8 9 10 MQ GA t=0 20 02040 60 80 100 2 0 2 4 6 8 10 12 14 MQ GA t=1 = 1 = 0 –20 –2 –20 20 020 40 6080100 2 0 2 4 6 8 10 12 14 MQ GA t=2 14 20 020 406080100 2 0 2 4 6 8 10 MQ GA 12 t=18 Figure 9. Generation of waves for Example 3.6. = 18 = 2 12 3 12 3 MQ 96.143654365,329.65039 329.65039, 875.41604701.42681, GA96.14365 96.14365,329.65039 329.65039, 875.41612706.85789. 96.1 I I I I 4. Conclusions In this paper, we have used method of lines coupled with radial basis functions for numerical solution of Kawahara type equations. The numerical results describing motion of single solitary wave, interaction of two and three soli tary wavend phenomena of wave generation have been discussed. The accuracy of the solution depends upon the choice of the shape parame lected experimentally. The numerical results using MQ and GA for Kawahara equation are better than the Crank Nicolson differential quadrature algorithm [10]. The in variants of motion remained conserved during the proc ess of computation for all cawo main advantages of this method are mesh less property and use of ODE solvers of high quality and their codes to approach the solution of PDEs. Also the presented method is simple, easy to implement because no mesh is required in the problem domain. Only radial distance between the nodes is used to approximate the solution. 5. Acknowledgements The first author is thankful to HEC Pakistan for financial help through Grant No. 063112367Ps3096. We are also thankful to the reviewers for their constructive ts. 6. References [1] T. Kawahara, “Oscillatory Solitary Waves in Dispersive Media,” Journal of the Physical Society of Japan, Vol. 33, s a ter, which has been se ses. T commen –20 –2 –2 –2 0
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