5. Análisis energético y exergético de los diferentes ciclos.
5.2.2. El ciclo de Linde con dos compresores adiabáticos en serie.
The following Tables 3.1 and 3.2 show the comparisons of the absolute errors of Boussinesq-Burgers’ equations obtained by using two terms and three terms approximations of HPM and OHAM at different values of c, k, b, x and t. To show the effectiveness and accuracy of proposed schemes, L2 and L error norms have been presented in Tables 3.3 and 3.4 respectively.
Table 3.1 The absolute errors in the solutions of Boussinesq-Burgers’ equations using two terms approximation for HPM and OHAM at various points with ,
2 1 c k1,b2 and , 01653 . 1 1 C D10.934599 obtained by eq. (1.33).
86
x ,t
HPM Exact u
u vExactvHPM uExactuOHAM vExactvOHAM (0.1, 0.1) 4.03765E-5 5.49815E-5 5.43565E-5 5.30293 E-5 (0.1, 0.2) 1.57743E-4 2.24577E-4 3.17229E-5 8.55529E-6 (0.1, 0.3) 3.46200E-4 5.15476E-4 6.20012E-5 1.91443E-4 (0.1, 0.4) 5.99524E-4 9.33935E-4 2.20592E-4 5.01892E-4 (0.1, 0.5) 9.11191E-4 1.48572E-3 4.37526E-4 9.45668E-4 (0.2, 0.1) 3.45343E-5 6.17326E-5 6.27431E-5 3.12906E-5 (0.2, 0.2) 1.33936E-4 2.51026E-4 6.06192E-5 6.49797E-5 (0.2, 0.3) 2.91679E-4 5.73647E-4 1.52649E-7 2.94577E-4 (0.2, 0.4) 5.00967E-4 1.03482E-3 1.11857E-4 6.62723E-4 (0.2, 0.5) 7.54752E-4 1.63915E-3 2.68365E-5 1.17404E-3 (0.3, 0.1) 2.80601E-5 6.75811E-5 7.13609E-5 8.76219E-6 (0.3, 0.2) 1.07664E-4 2.73710E-4 9.11785E-5 1.21023E-4 (0.3, 0.3) 2.31765E-4 6.23007E-4 6.64983E-5 3.93977E-4 (0.3, 0.4) 3.93103E-4 1.11945E-3 4.58083E-6 8.14078E-4 (0.3, 0.5) 5.84236E-4 1.76633E-3 8.71313E-5 1.38462E-3 (0.4, 0.1) 2.10557E-5 7.23025E-5 8.00687E-5 1.40750E-5 (0.4, 0.2) 7.93485E-5 2.91751E-4 1.22900E-5 1.75296E-4 (0.4, 0.3) 1.67436E-4 6.61637E-4 1.35937E-4 4.86954E-4 (0.4, 0.4) 2.77729E-4 1.18453E-3 1.26768E-4 9.51621E-4 (0.4, 0.5) 4.02528E-4 1.86226E-3 1.03094E-4 1.57112E-3 (0.5, 0.1) 1.36435E-5 7.57082E-5 8.87114E-5 3.67202E-5 (0.5, 0.2) 4.94924E-5 3.04426E-4 1.55217E-4 2.26450E-4 (0.5, 0.3) 9.98476E-5 6.87978E-4 2.07217E-4 5.71013E-4 (0.5, 0.4) 1.56938E-4 1.22742E-3 2.52482E-4 1.07147E-3 (0.5, 0.5) 2.12957E-4 1.92305E-3 2.98817E-4 1.72811E-3
Table 3.2 The absolute errors in the solutions of Boussinesq-Burgers’ equations using three terms approximation for HPM and OHAM at various points with , 1, 2
2 1 k b c and , 9162929 . 3 , 9786175 . 0 2 1 C C D11.0514603 ,D24.209964 obtained by eq. (1.33). x ,t HPM Exact u
u vExactvHPM uExactuOHAM vExactvOHAM (0.1, 0.1) 9.11428E-7 1.19150E-6 3.15534E-6 5.85344E-7 (0.1, 0.2) 7.40859E-6 9.41690E-6 7.33961E-7 2.12165E-6 (0.1, 0.3) 2.53911E-5 3.13655E-5 1.36454E-6 1.12982E-5 (0.1, 0.4) 6.10825E-5 7.32950E-5 3.08338E-6 3.43727E-5 (0.1, 0.5) 1.21007E-4 1.40972E-4 2.06021E-5 7.71116E-5 (0.2, 0.1) 1.02449E-6 1.06292E-6 3.23055E-6 8.39207E-7 (0.2, 0.2) 8.29954E-6 8.34741E-6 1.05314E-6 3.45590E-6 (0.2, 0.3) 2.83495E-5 2.76201E-5 7.84907E-9 2.08340E-6 (0.2, 0.4) 6.79737E-5 6.41010E-5 6.84685E-6 8.49823E-6 (0.2, 0.5) 1.34218E-4 1.22412E-4 2.86627E-5 3.29106E-5 (0.3, 0.1) 1.12268E-6 8.94544E-7 3.34664E-6 2.35740E-6 (0.3, 0.2) 9.06757E-6 6.96343E-6 1.53865E-6 9.48698E-6
87
(0.3, 0.3) 3.08802E-5 2.28283E-5 1.62153E-6 1.67671E-5 (0.3, 0.4) 7.38211E-5 5.24659E-5 1.08559E-5 2.02210E-5 (0.3, 0.5) 1.45333E-4 9.91676E-5 3.66844E-5 1.65572E-5 (0.4, 0.1) 1.20223E-6 6.91094E-7 3.50880E-6 3.92344E-6 (0.4, 0.2) 9.68307E-6 5.30519E-6 2.20317E-6 1.57786E-5 (0.4, 0.3) 3.28850E-5 1.71337E-5 3.52562E-6 3.22743E-5 (0.4, 0.4) 7.83973E-5 3.87483E-5 1.50654E-5 5.08384E-5 (0.4, 0.5) 1.53919E-4 7.19747E-5 4.45217E-5 6.96457E-5 (0.5, 0.1) 1.25997E-6 4.59601E-7 3.72005E-6 5.48493E-6 (0.5, 0.2) 1.01214E-5 3.43102E-6 3.05227E-6 2.21052E-5 (0.5, 0.3) 3.42835E-5 1.07397E-5 5.69591E-6 4.80354E-5 (0.5, 0.4) 8.15178E-5 2.34456E-5 1.94224E-5 8.22156E-5 (0.5, 0.5) 1.59629E-4 4.18318E-5 5.20368E-5 1.24363E-4
Table 3.3 L2andL error norm for Boussinesq-Burgers’ equations using two terms approximation for HPM and OHAM at various points of x.
x Homotopy Perturbation Method (HPM) Optimal Homotopy Asymptotic Method (OHAM) Error in case of two terms
approximation for u( tx, )
Error in case of two terms approximation for v( tx, )
Error in case of two terms approximation for u( tx, )
Error in case of two terms approximation for v( tx, )
2
L L L2 L L2 L L 2 L
0.1 5.16927E-4 9.11191E-4 1.48572E-3 1.48572E-3 2.22663E-4 4.37526E-4 4.86974E-4 9.45668E-4 0.2 4.30076E-4 7.54752E-4 9.11434E-4 1.63915E-3 1.35752E-4 1.11857E-4 6.17988E-4 1.17404E-3 0.3 3.35248E-4 5.84236E-4 9.83942E-4 1.76633E-3 7.13313E-5 9.11785E-5 7.41598E-4 1.38462E-3 0.4 2.34067E-4 4.02528E-4 1.03916E-3 1.86226E-3 1.15493E-4 1.35937E-4 8.53470E-4 1.57112E-3 0.5 1.28519E-4 2.12957E-4 1.07484E-3 1.92305E-3 2.13513E-4 2.98817E-4 9.50063E-4 1.72811E-3
Table 3.4 L2andL error norm for Boussinesq-Burgers’ equations using three terms approximation for HPM and OHAM at various points of x.
x Homotopy Perturbation Method (HPM) Optimal Homotopy Asymptotic Method (OHAM) Error in case of three
terms approximation for )
, ( tx u
Error in case of three terms approximation for
) , ( tx v
Error in case of three terms approximation for
) , ( tx u
Error in case of three terms approximation for
) , ( tx v 2 L L L2 L L2 L L 2 L
0.1 6.17644E-5 1.21007E-4 7.25523E-5 1.40972E-4 9.44786E-6 2.06021E-5 3.81056E-5 7.71116E-5 0.2 6.85690E-5 1.34218E-4 6.31305E-5 1.22412E-4 1.32663E-5 2.86627E-5 1.53122E-5 3.29106E-5 0.3 7.43079E-5 1.45333E-4 5.12978E-5 9.91676E-5 1.72034E-5 3.66844E-5 1.45583E-5 2.02210E-5 0.4 7.87576E-5 1.53919E-4 3.74272E-5 4.18318E-5 2.11601E-5 4.45217E-5 4.18116E-5 6.96457E-5 0.5 8.17386E-5 1.59629E-4 2.20314E-5 7.19747E-5 2.50625E-5 5.20368E-5 7.07837E-5 1.24363E-4
Graphical representation of results is very useful to demonstrate the efficiency and accuracy of the proposed methods for the discussed problem. The following Figures 3.1 and 3.2 cite the comparison graphically between the approximate solutions obtained by five terms HPM, three terms OHAM and exact solutions for different values of x and
88 . 5 . 0
t Figures 3.3 and 3.4 respectively show one soliton approximate solutions of u( tx, )
and v( tx, ), obtained by OHAM for Boussinesq-Burgers’ equations.
Figure 3.1 Comparison of five terms HPM solution and three terms OHAM solution with the exact solution of u( tx, ) for Boussinesq-Burgers’ equations when c0.5,k1,b2 and
. 5 . 0 t
Figure 3.2 Comparison of five terms HPM solution and three terms OHAM solution with the exact solution of v( tx, ) for Boussinesq-Burgers’ equations when c0.5,k1,b2 and
. 5 . 0 t
Figure 3.3 One soliton approximate solution of u( tx, ), obtained by OHAM for Boussinesq- Burgers’ equations with parameters c0.5,k1 and b2.
89
Figure 3.4 One soliton approximate solution of v( tx, ), obtained by OHAM for Boussinesq- Burgers’ equations with parameters ,
2 1
c k1andb2.
From the above Figures, one can see a very good agreement between the exact solutions and the solutions obtained by HPM and OHAM. Tables 3.1-3.4 depict the performance of OHAM in comparison with HPM and clearly witness the reliability and efficiency of OHAM for the solutions of Boussinesq-Burgers’ equations.