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March  2020, 13(3): 503-518. doi: 10.3934/dcdss.2020028

New approximate solutions to the nonlinear Klein-Gordon equations using perturbation iteration techniques

Department of Mathematics, Faculty of Art and Sciences, Manisa Celal Bayar University, Manisa, 45140, Turkey

* Corresponding author

Received  April 2018 Revised  May 2018 Published  March 2019

In this study, we present the new approximate solutions of the nonlinear Klein-Gordon equations via perturbation iteration technique and newly developed optimal perturbation iteration method. Some specific examples are given and obtained solutions are compared with other methods and analytical results to confirm the good accuracy of the proposed methods.We also discuss the convergence of the optimal perturbation iteration method for partial differential equations. The results reveal that perturbation iteration techniques, unlike many other techniques in literature, converge rapidly to exact solutions of the given problems at lower order of approximations.

Citation: Necdet Bildik, Sinan Deniz. New approximate solutions to the nonlinear Klein-Gordon equations using perturbation iteration techniques. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 503-518. doi: 10.3934/dcdss.2020028
References:
[1]

S. Abbasbandy and F. Samadian Zakaria, Soliton solutions for the fifth-order KdV equation with the homotopy analysis method, Nonlinear Dynamics, 51 (2008), 83-87.  doi: 10.1007/s11071-006-9193-y.  Google Scholar

[2]

G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method Kluwer, Boston, MA, 1994. doi: 10.1007/978-94-015-8289-6.  Google Scholar

[3]

Y. Aksoy, et al. New perturbation-iteration solutions for nonlinear heat transfer equations, International Journal of Numerical Methods for Heat & Fluid Flow, 22 (2012), 814-828. Google Scholar

[4]

Y. Aksoy and M. Pakdemirli, New perturbation–iteration solutions for Bratu-type equations, Computers & Mathematics with Applications, 59 (2010), 2802-2808.  doi: 10.1016/j.camwa.2010.01.050.  Google Scholar

[5]

M. Alquran, Solitons and periodic solutions to nonlinear partial differential equations by the Sine-Cosine method, Appl. Math. Inf. Sci., 6 (2012), 85-88.   Google Scholar

[6]

A. Atangana and A. Secer, The time-fractional coupled-Korteweg-de-Vries equations, Abstract and Applied Analysis, 2013 (2013), Art. ID 947986, 8 pp. doi: 10.1155/2013/947986.  Google Scholar

[7]

A. Bekir, New solitons and periodic wave solutions for some nonlinear physical models by using the sine-cosine method, Physica Scripta, 77 (2008), 045008.   Google Scholar

[8]

N. Bildik and S. Deniz, Implementation of Taylor collocation and Adomian decomposition method for systems of ordinary differential equations, AIP Conference Proceedings. Vol. 1648. No. 1. AIP Publishing, 2015. Google Scholar

[9]

N. Bildik and S. Deniz, Comparative Study between Optimal Homotopy Asymptotic Method and Perturbation-Iteration Technique for Different Types of Nonlinear Equations, Iranian Journal of Science and Technology, 42 (2018), 647-654.  doi: 10.1007/s40995-016-0039-2.  Google Scholar

[10]

N. Bildik and S. Deniz, A new efficient method for solving delay differential equations and a comparison with other methods, The European Physical Journal Plus, 132 (2017), 51.   Google Scholar

[11]

N. Bildik and S. Deniz, A Practical Method for Analytical Evaluation of Approximate Solutions of Fisher's Equations, ITM Web of Conferences. Vol. 13. EDP Sciences, 2017. Google Scholar

[12]

S. T. DemirayY. Pandir and H. Bulut, New solitary wave solutions of Maccari system, Ocean Engineering, 103 (2015), 153-159.   Google Scholar

[13]

S. Deniz, Optimal perturbation iteration method for solving nonlinear heat transfer equations, Journal of Heat Transfer, 139 (2017), 074503.   Google Scholar

[14]

S. Deniz and N. Bildik, A new analytical technique for solving Lane-Emden type equations arising in astrophysics, Bulletin of the Belgian Mathematical Society-Simon Stevin, 24 (2017), 305-320.   Google Scholar

[15]

S. Deniz and N. Bildik, Applications of optimal perturbation iteration method for solving nonlinear differential equations., AIP Conference Proceedings. Vol. 1798. No. 1. AIP Publishing, 2017. Google Scholar

[16]

S. Deniz and N. Bildik, Optimal perturbation iteration method for Bratu-type problems, Journal of King Saud University Science, 30 (2018), 91-99.   Google Scholar

[17]

Z. Fu, et al., New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations, Physics Letters A, 290 (2001), 72-76. doi: 10.1016/S0375-9601(01)00644-2.  Google Scholar

[18]

Y. GurefeA. Sonmezoglu and E. Misirli, Application of the trial equation method for solving some nonlinear evolution equations arising in mathematical physics, Pramana, 77 (2011), 1023-1029.   Google Scholar

[19]

J.-H. He and X.-H. Wu, Construction of solitary solution and compacton-like solution by variational iteration method, Chaos, Solitons & Fractals, 29 (2006), 108-113.  doi: 10.1016/j.chaos.2005.10.100.  Google Scholar

[20]

N. Herisanu and V. Marinca, Accurate analytical solutions to oscillators with discontinuities and fractional-power restoring force by means of the optimal homotopy asymptotic method, Computers & Mathematics with Applications, 60 (2010), 1607-1615.  doi: 10.1016/j.camwa.2010.06.042.  Google Scholar

[21]

S. Iqbal, et al., Some solutions of the linear and nonlinear Klein-Gordon equations using the optimal homotopy asymptotic method, Applied Mathematics and Computation, 216 (2010), 2898-2909. doi: 10.1016/j.amc.2010.04.001.  Google Scholar

[22]

H. Jafari and V. Daftardar-Gejji, Solving linear and nonlinear fractional diffusion and wave equations by Adomian decomposition, Applied Mathematics and Computation, 180 (2006), 488-497.  doi: 10.1016/j.amc.2005.12.031.  Google Scholar

[23]

Si rendaoreji and S. Jiong, Auxiliary equation method for solving nonlinear partial differential equations, Physics Letters A, 309 (2003), 387-396.  doi: 10.1016/S0375-9601(03)00196-8.  Google Scholar

[24]

A. S. V. R. Kanth and K. Aruna, Differential transform method for solving the linear and nonlinear Klein-Gordon equation, Computer Physics Communications, 180 (2009), 708-711.  doi: 10.1016/j.cpc.2008.11.012.  Google Scholar

[25]

M. M. Khader and K. M. Saad, A numerical approach for solving the fractional Fisher equation using Chebyshev spectral collocation method, Chaos, Solitons & Fractals, 110 (2018), 169-177.  doi: 10.1016/j.chaos.2018.03.018.  Google Scholar

[26]

L. Kong, et al. Semi-explicit symplectic partitioned Runge–Kutta Fourier pseudo-spectral scheme for Klein–Gordon–Schrödinger equations, Computer Physics Communications, 181 (2010), 1369-1377. doi: 10.1016/j.cpc.2010.04.003.  Google Scholar

[27]

C. S. Liu, Trial equation method and its applications to nonlinear evolution equations, Acta Physica Sinica, 54 (2005), 2505-2509.   Google Scholar

[28]

W. Malfliet and W. Hereman, The tanh method: Ⅰ. Exact solutions of nonlinear evolution and wave equations, Physica Scripta, 54 (1996), 563-568.  doi: 10.1088/0031-8949/54/6/003.  Google Scholar

[29]

V. Marinca and N. Herisanu, Application of optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer, International Communications in Heat and Mass Transfer, 35 (2008), 710-715.   Google Scholar

[30]

V. Marinca and et al., An optimal homotopy asymptotic method applied to the steady flow of a fourth-grade fluid past a porous plate, Applied Mathematics Letters, 22 (2009), 245-251.  doi: 10.1016/j.aml.2008.03.019.  Google Scholar

[31]

V. MarincaN. Herisanu and I. Nemes, Optimal homotopy asymptotic method with application to thin film flow, Open Physics, 6 (2008), 648-653.   Google Scholar

[32]

V. Marinca and N. Herisanu, The optimal homotopy asymptotic method for solving Blasius equation, Applied Mathematics and Computation, 231 (2014), 134-139.  doi: 10.1016/j.amc.2013.12.121.  Google Scholar

[33]

Y. MolliqM. Salmi Md Noorani and I. Hashim, Variational iteration method for fractional heat-and wave-like equations, Nonlinear Analysis: Real World Applications, 10 (2009), 1854-1869.  doi: 10.1016/j.nonrwa.2008.02.026.  Google Scholar

[34]

M. M. RashidiG. Domairry and S. Dinarvand, Approximate solutions for the Burger and regularized long wave equations by means of the homotopy analysis method, Communications in Nonlinear Science and Numerical Simulation, 14 (2009), 708-717.   Google Scholar

[35]

K. M. Saad, et al., Optimal q-homotopy analysis method for time-space fractional gas dynamics equation, The European Physical Journal Plus, 132 (2017), 23. Google Scholar

[36]

K. M. Saad and E. H..F Al-Sharif, Analytical study for time and time-space fractional Burgersequation, Advances in Difference Equations, 2017 (2017), Paper No. 300, 15 pp. doi: 10.1186/s13662-017-1358-0.  Google Scholar

[37]

J. J. Sakurai, Advanced Quantum Mechanics, AddisonWesley, New York, 1967. Google Scholar

[38]

F. Shakeri and M. Dehghan, Numerical solution of the Klein-Gordon equation via He variational iteration method, Nonlinear Dynamics, 51 (2008), 89-97.  doi: 10.1007/s11071-006-9194-x.  Google Scholar

[39]

H. TariD. D. Ganji and M. Rostamian, Approximate solutions of K(2, 2), KdV and modified KdV equations by variational iteration method, homotopy perturbation method and homotopy analysis method, International Journal of Nonlinear Sciences and Numerical Simulation, 8 (2007), 203-210.   Google Scholar

[40]

A.-M.. Wazwaz, The modified decomposition method for analytic treatment of differential equations, Applied Mathematics and Computation, 173 (2006), 165-176.  doi: 10.1016/j.amc.2005.02.048.  Google Scholar

[41]

E. Yusufo lu, The variational iteration method for studying the Klein-Gordon equation, Applied Mathematics Letters, 21 (2008), 669-674.  doi: 10.1016/j.aml.2007.07.023.  Google Scholar

[42]

X. Zhao, et al. A new Riccati equation expansion method with symbolic computation to construct new travelling wave solution of nonlinear differential equations, Applied Mathematics and Computation, 172 (2006), 24-39. doi: 10.1016/j.amc.2005.01.145.  Google Scholar

show all references

References:
[1]

S. Abbasbandy and F. Samadian Zakaria, Soliton solutions for the fifth-order KdV equation with the homotopy analysis method, Nonlinear Dynamics, 51 (2008), 83-87.  doi: 10.1007/s11071-006-9193-y.  Google Scholar

[2]

G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method Kluwer, Boston, MA, 1994. doi: 10.1007/978-94-015-8289-6.  Google Scholar

[3]

Y. Aksoy, et al. New perturbation-iteration solutions for nonlinear heat transfer equations, International Journal of Numerical Methods for Heat & Fluid Flow, 22 (2012), 814-828. Google Scholar

[4]

Y. Aksoy and M. Pakdemirli, New perturbation–iteration solutions for Bratu-type equations, Computers & Mathematics with Applications, 59 (2010), 2802-2808.  doi: 10.1016/j.camwa.2010.01.050.  Google Scholar

[5]

M. Alquran, Solitons and periodic solutions to nonlinear partial differential equations by the Sine-Cosine method, Appl. Math. Inf. Sci., 6 (2012), 85-88.   Google Scholar

[6]

A. Atangana and A. Secer, The time-fractional coupled-Korteweg-de-Vries equations, Abstract and Applied Analysis, 2013 (2013), Art. ID 947986, 8 pp. doi: 10.1155/2013/947986.  Google Scholar

[7]

A. Bekir, New solitons and periodic wave solutions for some nonlinear physical models by using the sine-cosine method, Physica Scripta, 77 (2008), 045008.   Google Scholar

[8]

N. Bildik and S. Deniz, Implementation of Taylor collocation and Adomian decomposition method for systems of ordinary differential equations, AIP Conference Proceedings. Vol. 1648. No. 1. AIP Publishing, 2015. Google Scholar

[9]

N. Bildik and S. Deniz, Comparative Study between Optimal Homotopy Asymptotic Method and Perturbation-Iteration Technique for Different Types of Nonlinear Equations, Iranian Journal of Science and Technology, 42 (2018), 647-654.  doi: 10.1007/s40995-016-0039-2.  Google Scholar

[10]

N. Bildik and S. Deniz, A new efficient method for solving delay differential equations and a comparison with other methods, The European Physical Journal Plus, 132 (2017), 51.   Google Scholar

[11]

N. Bildik and S. Deniz, A Practical Method for Analytical Evaluation of Approximate Solutions of Fisher's Equations, ITM Web of Conferences. Vol. 13. EDP Sciences, 2017. Google Scholar

[12]

S. T. DemirayY. Pandir and H. Bulut, New solitary wave solutions of Maccari system, Ocean Engineering, 103 (2015), 153-159.   Google Scholar

[13]

S. Deniz, Optimal perturbation iteration method for solving nonlinear heat transfer equations, Journal of Heat Transfer, 139 (2017), 074503.   Google Scholar

[14]

S. Deniz and N. Bildik, A new analytical technique for solving Lane-Emden type equations arising in astrophysics, Bulletin of the Belgian Mathematical Society-Simon Stevin, 24 (2017), 305-320.   Google Scholar

[15]

S. Deniz and N. Bildik, Applications of optimal perturbation iteration method for solving nonlinear differential equations., AIP Conference Proceedings. Vol. 1798. No. 1. AIP Publishing, 2017. Google Scholar

[16]

S. Deniz and N. Bildik, Optimal perturbation iteration method for Bratu-type problems, Journal of King Saud University Science, 30 (2018), 91-99.   Google Scholar

[17]

Z. Fu, et al., New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations, Physics Letters A, 290 (2001), 72-76. doi: 10.1016/S0375-9601(01)00644-2.  Google Scholar

[18]

Y. GurefeA. Sonmezoglu and E. Misirli, Application of the trial equation method for solving some nonlinear evolution equations arising in mathematical physics, Pramana, 77 (2011), 1023-1029.   Google Scholar

[19]

J.-H. He and X.-H. Wu, Construction of solitary solution and compacton-like solution by variational iteration method, Chaos, Solitons & Fractals, 29 (2006), 108-113.  doi: 10.1016/j.chaos.2005.10.100.  Google Scholar

[20]

N. Herisanu and V. Marinca, Accurate analytical solutions to oscillators with discontinuities and fractional-power restoring force by means of the optimal homotopy asymptotic method, Computers & Mathematics with Applications, 60 (2010), 1607-1615.  doi: 10.1016/j.camwa.2010.06.042.  Google Scholar

[21]

S. Iqbal, et al., Some solutions of the linear and nonlinear Klein-Gordon equations using the optimal homotopy asymptotic method, Applied Mathematics and Computation, 216 (2010), 2898-2909. doi: 10.1016/j.amc.2010.04.001.  Google Scholar

[22]

H. Jafari and V. Daftardar-Gejji, Solving linear and nonlinear fractional diffusion and wave equations by Adomian decomposition, Applied Mathematics and Computation, 180 (2006), 488-497.  doi: 10.1016/j.amc.2005.12.031.  Google Scholar

[23]

Si rendaoreji and S. Jiong, Auxiliary equation method for solving nonlinear partial differential equations, Physics Letters A, 309 (2003), 387-396.  doi: 10.1016/S0375-9601(03)00196-8.  Google Scholar

[24]

A. S. V. R. Kanth and K. Aruna, Differential transform method for solving the linear and nonlinear Klein-Gordon equation, Computer Physics Communications, 180 (2009), 708-711.  doi: 10.1016/j.cpc.2008.11.012.  Google Scholar

[25]

M. M. Khader and K. M. Saad, A numerical approach for solving the fractional Fisher equation using Chebyshev spectral collocation method, Chaos, Solitons & Fractals, 110 (2018), 169-177.  doi: 10.1016/j.chaos.2018.03.018.  Google Scholar

[26]

L. Kong, et al. Semi-explicit symplectic partitioned Runge–Kutta Fourier pseudo-spectral scheme for Klein–Gordon–Schrödinger equations, Computer Physics Communications, 181 (2010), 1369-1377. doi: 10.1016/j.cpc.2010.04.003.  Google Scholar

[27]

C. S. Liu, Trial equation method and its applications to nonlinear evolution equations, Acta Physica Sinica, 54 (2005), 2505-2509.   Google Scholar

[28]

W. Malfliet and W. Hereman, The tanh method: Ⅰ. Exact solutions of nonlinear evolution and wave equations, Physica Scripta, 54 (1996), 563-568.  doi: 10.1088/0031-8949/54/6/003.  Google Scholar

[29]

V. Marinca and N. Herisanu, Application of optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer, International Communications in Heat and Mass Transfer, 35 (2008), 710-715.   Google Scholar

[30]

V. Marinca and et al., An optimal homotopy asymptotic method applied to the steady flow of a fourth-grade fluid past a porous plate, Applied Mathematics Letters, 22 (2009), 245-251.  doi: 10.1016/j.aml.2008.03.019.  Google Scholar

[31]

V. MarincaN. Herisanu and I. Nemes, Optimal homotopy asymptotic method with application to thin film flow, Open Physics, 6 (2008), 648-653.   Google Scholar

[32]

V. Marinca and N. Herisanu, The optimal homotopy asymptotic method for solving Blasius equation, Applied Mathematics and Computation, 231 (2014), 134-139.  doi: 10.1016/j.amc.2013.12.121.  Google Scholar

[33]

Y. MolliqM. Salmi Md Noorani and I. Hashim, Variational iteration method for fractional heat-and wave-like equations, Nonlinear Analysis: Real World Applications, 10 (2009), 1854-1869.  doi: 10.1016/j.nonrwa.2008.02.026.  Google Scholar

[34]

M. M. RashidiG. Domairry and S. Dinarvand, Approximate solutions for the Burger and regularized long wave equations by means of the homotopy analysis method, Communications in Nonlinear Science and Numerical Simulation, 14 (2009), 708-717.   Google Scholar

[35]

K. M. Saad, et al., Optimal q-homotopy analysis method for time-space fractional gas dynamics equation, The European Physical Journal Plus, 132 (2017), 23. Google Scholar

[36]

K. M. Saad and E. H..F Al-Sharif, Analytical study for time and time-space fractional Burgersequation, Advances in Difference Equations, 2017 (2017), Paper No. 300, 15 pp. doi: 10.1186/s13662-017-1358-0.  Google Scholar

[37]

J. J. Sakurai, Advanced Quantum Mechanics, AddisonWesley, New York, 1967. Google Scholar

[38]

F. Shakeri and M. Dehghan, Numerical solution of the Klein-Gordon equation via He variational iteration method, Nonlinear Dynamics, 51 (2008), 89-97.  doi: 10.1007/s11071-006-9194-x.  Google Scholar

[39]

H. TariD. D. Ganji and M. Rostamian, Approximate solutions of K(2, 2), KdV and modified KdV equations by variational iteration method, homotopy perturbation method and homotopy analysis method, International Journal of Nonlinear Sciences and Numerical Simulation, 8 (2007), 203-210.   Google Scholar

[40]

A.-M.. Wazwaz, The modified decomposition method for analytic treatment of differential equations, Applied Mathematics and Computation, 173 (2006), 165-176.  doi: 10.1016/j.amc.2005.02.048.  Google Scholar

[41]

E. Yusufo lu, The variational iteration method for studying the Klein-Gordon equation, Applied Mathematics Letters, 21 (2008), 669-674.  doi: 10.1016/j.aml.2007.07.023.  Google Scholar

[42]

X. Zhao, et al. A new Riccati equation expansion method with symbolic computation to construct new travelling wave solution of nonlinear differential equations, Applied Mathematics and Computation, 172 (2006), 24-39. doi: 10.1016/j.amc.2005.01.145.  Google Scholar

Figure 1.  Absolute errors obtained by ADM-DTM and PIM for Example 1
Figure 2.  Absolute errors obtained by OPIM for Example 1
Figure 3.  Absolute errors of third order PIM and OPIM solutions for Example 2
Figure 4.  Comparison between the third order approximate solutions obtained by OPIM($ \blacksquare $) and by VIM($ \bullet $) and the exact solution (–) for Example 2
Table 1.  Absolute errors of the second order ADM-DTM (ADM-DTM-2nd), PIM (PIM-2nd), OPIM(OPIM-2nd) approximate solutions at $ x = 0.5 $ for Example 1
t ADM-DTM-2nd PIM-2nd OPIM-2nd
0.1 $2.083 \times {{10}^{ -6 }}$ $4.859 \times {{10}^{ -9 }}$ $3.802 \times {{10}^{ -9 }}$
0.2 $3.329 \times {{10}^{ -5 }}$ $3.107 \times {{10}^{ -7 }}$ $2.94 \times {{10}^{ -7 }}$
0.3 $1.682 \times {{10}^{ -4 }}$ $3.533 \times {{10}^{ -6 }}$ $3.45 \times {{10}^{ -6 }}$
0.4 $5.305 \times {{10}^{ -4 }}$ $1.98 \times {{10}^{ -5 }}$ $1.955 \times {{10}^{ -5 }}$
0.5 $1.291 \times {{10}^{ -3 }}$ $7.532 \times {{10}^{ -5 }}$ $7.472 \times {{10}^{ -5 }}$
0.6 $2.668 \times {{10}^{ -3 }}$ $2.241 \times {{10}^{ -4 }}$ $2.229 \times {{10}^{ -4 }}$
0.7 $4.921 \times {{10}^{ -3 }}$ $5.625 \times {{10}^{ -4 }}$ $5.604 \times {{10}^{ -4 }}$
0.8 $8.353 \times {{10}^{ -3 }}$ $1.247 \times {{10}^{ -3 }}$ $1.243 \times {{10}^{ -3 }}$
0.9 $1.33 \times {{10}^{ -2 }}$ $2.512 \times {{10}^{ -3 }}$ $2.507 \times {{10}^{ -3 }}$
1. $2.015 \times {{10}^{ -2 }}$ $4.696 \times {{10}^{ -3 }}$ $4.689 \times {{10}^{ -3 }}$
t ADM-DTM-2nd PIM-2nd OPIM-2nd
0.1 $2.083 \times {{10}^{ -6 }}$ $4.859 \times {{10}^{ -9 }}$ $3.802 \times {{10}^{ -9 }}$
0.2 $3.329 \times {{10}^{ -5 }}$ $3.107 \times {{10}^{ -7 }}$ $2.94 \times {{10}^{ -7 }}$
0.3 $1.682 \times {{10}^{ -4 }}$ $3.533 \times {{10}^{ -6 }}$ $3.45 \times {{10}^{ -6 }}$
0.4 $5.305 \times {{10}^{ -4 }}$ $1.98 \times {{10}^{ -5 }}$ $1.955 \times {{10}^{ -5 }}$
0.5 $1.291 \times {{10}^{ -3 }}$ $7.532 \times {{10}^{ -5 }}$ $7.472 \times {{10}^{ -5 }}$
0.6 $2.668 \times {{10}^{ -3 }}$ $2.241 \times {{10}^{ -4 }}$ $2.229 \times {{10}^{ -4 }}$
0.7 $4.921 \times {{10}^{ -3 }}$ $5.625 \times {{10}^{ -4 }}$ $5.604 \times {{10}^{ -4 }}$
0.8 $8.353 \times {{10}^{ -3 }}$ $1.247 \times {{10}^{ -3 }}$ $1.243 \times {{10}^{ -3 }}$
0.9 $1.33 \times {{10}^{ -2 }}$ $2.512 \times {{10}^{ -3 }}$ $2.507 \times {{10}^{ -3 }}$
1. $2.015 \times {{10}^{ -2 }}$ $4.696 \times {{10}^{ -3 }}$ $4.689 \times {{10}^{ -3 }}$
Table 2.  Absolute errors of the third order ADM-DTM (ADM-DTM-3rd), PIM (PIM-3rd), OPIM(OPIM-3rd) approximate solutions at $ x = 0.5 $ for Example 1
t ADM-DTM-3rd PIM-3rd OPIM-3rd
0.1 $6.943 \times {{10}^{ -10 }}$ $3.831 \times {{10}^{ -12 }}$ $2.081 \times {{10}^{ -12 }}$
0.2 $4.441 \times {{10}^{ -8 }}$ $9.748 \times {{10}^{ -10 }}$ $3.316 \times {{10}^{ -11 }}$
0.3 $5.054 \times {{10}^{ -7 }}$ $2.468 \times {{10}^{ -8 }}$ $1.667 \times {{10}^{ -10 }}$
0.4 $2.836 \times {{10}^{ -6 }}$ $2.424 \times {{10}^{ -7 }}$ $5.221 \times {{10}^{ -10 }}$
0.5 $1.08 \times {{10}^{ -5 }}$ $1.413 \times {{10}^{ -6 }}$ $1.259 \times {{10}^{ -9 }}$
0.6 $3.219 \times {{10}^{ -5 }}$ $5.914 \times {{10}^{ -6 }}$ $2.574 \times {{10}^{ -9 }}$
0.7 $8.099 \times {{10}^{ -5 }}$ $1.965 \times {{10}^{ -5 }}$ $4.686 \times {{10}^{ -9 }}$
0.8 $1.8 \times {{10}^{ -4 }}$ $5.501 \times {{10}^{ -5 }}$ $7.838 \times {{10}^{ -9 }}$
0.9 $3.638 \times {{10}^{ -4 }}$ $1.35 \times {{10}^{ -4 }}$ $1.227 \times {{10}^{ -8 }}$
1. $6.822 \times {{10}^{ -4 }}$ $2.979 \times {{10}^{ -4 }}$ $1.825 \times {{10}^{ -8 }}$
t ADM-DTM-3rd PIM-3rd OPIM-3rd
0.1 $6.943 \times {{10}^{ -10 }}$ $3.831 \times {{10}^{ -12 }}$ $2.081 \times {{10}^{ -12 }}$
0.2 $4.441 \times {{10}^{ -8 }}$ $9.748 \times {{10}^{ -10 }}$ $3.316 \times {{10}^{ -11 }}$
0.3 $5.054 \times {{10}^{ -7 }}$ $2.468 \times {{10}^{ -8 }}$ $1.667 \times {{10}^{ -10 }}$
0.4 $2.836 \times {{10}^{ -6 }}$ $2.424 \times {{10}^{ -7 }}$ $5.221 \times {{10}^{ -10 }}$
0.5 $1.08 \times {{10}^{ -5 }}$ $1.413 \times {{10}^{ -6 }}$ $1.259 \times {{10}^{ -9 }}$
0.6 $3.219 \times {{10}^{ -5 }}$ $5.914 \times {{10}^{ -6 }}$ $2.574 \times {{10}^{ -9 }}$
0.7 $8.099 \times {{10}^{ -5 }}$ $1.965 \times {{10}^{ -5 }}$ $4.686 \times {{10}^{ -9 }}$
0.8 $1.8 \times {{10}^{ -4 }}$ $5.501 \times {{10}^{ -5 }}$ $7.838 \times {{10}^{ -9 }}$
0.9 $3.638 \times {{10}^{ -4 }}$ $1.35 \times {{10}^{ -4 }}$ $1.227 \times {{10}^{ -8 }}$
1. $6.822 \times {{10}^{ -4 }}$ $2.979 \times {{10}^{ -4 }}$ $1.825 \times {{10}^{ -8 }}$
Table 3.  The absolute errors of second order approximation by OPIM with the exact solution of Example 1
x t=0.1 t=0.2 t=0.3 t=0.4 t=0.5
0.1 $5.506 \times {{10}^{ -9 }}$ $3.537 \times {{10}^{ -7 }}$ $4.019 \times {{10}^{ -6 }}$ $2.248 \times {{10}^{ -5 }}$ $8.521 \times {{10}^{ -5 }}$
0.2 $5.341 \times {{10}^{ -9 }}$ $3.492 \times {{10}^{ -7 }}$ $3.981 \times {{10}^{ -6 }}$ $2.229 \times {{10}^{ -5 }}$ $8.458 \times {{10}^{ -5 }}$
0.3 $5.023 \times {{10}^{ -9 }}$ $3.392 \times {{10}^{ -7 }}$ $3.889 \times {{10}^{ -6 }}$ $2.183 \times {{10}^{ -5 }}$ $8.293 \times {{10}^{ -5 }}$
0.4 $4.522 \times {{10}^{ -9 }}$ $3.215 \times {{10}^{ -7 }}$ $3.72 \times {{10}^{ -6 }}$ $2.096 \times {{10}^{ -5 }}$ $7.981 \times {{10}^{ -5 }}$
0.5 $3.802 \times {{10}^{ -9 }}$ $2.94 \times {{10}^{ -7 }}$ $3.45 \times {{10}^{ -6 }}$ $1.955 \times {{10}^{ -5 }}$ $7.472 \times {{10}^{ -5 }}$
0.6 $2.832 \times {{10}^{ -9 }}$ $2.546 \times {{10}^{ -7 }}$ $3.055 \times {{10}^{ -6 }}$ $1.747 \times {{10}^{ -5 }}$ $6.719 \times {{10}^{ -5 }}$
0.7 $1.577 \times {{10}^{ -9 }}$ $2.012 \times {{10}^{ -7 }}$ $2.512 \times {{10}^{ -6 }}$ $1.46 \times {{10}^{ -5 }}$ $5.674 \times {{10}^{ -5 }}$
0.8 $4.911 \times {{10}^{ -12 }}$ $1.318 \times {{10}^{ -7 }}$ $1.797 \times {{10}^{ -6 }}$ $1.08 \times {{10}^{ -5 }}$ $4.29 \times {{10}^{ -5 }}$
0.9 $1.918 \times {{10}^{ -9 }}$ $4.417 \times {{10}^{ -8 }}$ $8.865 \times {{10}^{ -7 }}$ $5.939 \times {{10}^{ -6 }}$ $2.519 \times {{10}^{ -5 }}$
1. $4.225 \times {{10}^{ -9 }}$ $6.376 \times {{10}^{ -8 }}$ $2.428 \times {{10}^{ -7 }}$ $1.01 \times {{10}^{ -7 }}$ $3.129 \times {{10}^{ -6 }}$
x t=0.1 t=0.2 t=0.3 t=0.4 t=0.5
0.1 $5.506 \times {{10}^{ -9 }}$ $3.537 \times {{10}^{ -7 }}$ $4.019 \times {{10}^{ -6 }}$ $2.248 \times {{10}^{ -5 }}$ $8.521 \times {{10}^{ -5 }}$
0.2 $5.341 \times {{10}^{ -9 }}$ $3.492 \times {{10}^{ -7 }}$ $3.981 \times {{10}^{ -6 }}$ $2.229 \times {{10}^{ -5 }}$ $8.458 \times {{10}^{ -5 }}$
0.3 $5.023 \times {{10}^{ -9 }}$ $3.392 \times {{10}^{ -7 }}$ $3.889 \times {{10}^{ -6 }}$ $2.183 \times {{10}^{ -5 }}$ $8.293 \times {{10}^{ -5 }}$
0.4 $4.522 \times {{10}^{ -9 }}$ $3.215 \times {{10}^{ -7 }}$ $3.72 \times {{10}^{ -6 }}$ $2.096 \times {{10}^{ -5 }}$ $7.981 \times {{10}^{ -5 }}$
0.5 $3.802 \times {{10}^{ -9 }}$ $2.94 \times {{10}^{ -7 }}$ $3.45 \times {{10}^{ -6 }}$ $1.955 \times {{10}^{ -5 }}$ $7.472 \times {{10}^{ -5 }}$
0.6 $2.832 \times {{10}^{ -9 }}$ $2.546 \times {{10}^{ -7 }}$ $3.055 \times {{10}^{ -6 }}$ $1.747 \times {{10}^{ -5 }}$ $6.719 \times {{10}^{ -5 }}$
0.7 $1.577 \times {{10}^{ -9 }}$ $2.012 \times {{10}^{ -7 }}$ $2.512 \times {{10}^{ -6 }}$ $1.46 \times {{10}^{ -5 }}$ $5.674 \times {{10}^{ -5 }}$
0.8 $4.911 \times {{10}^{ -12 }}$ $1.318 \times {{10}^{ -7 }}$ $1.797 \times {{10}^{ -6 }}$ $1.08 \times {{10}^{ -5 }}$ $4.29 \times {{10}^{ -5 }}$
0.9 $1.918 \times {{10}^{ -9 }}$ $4.417 \times {{10}^{ -8 }}$ $8.865 \times {{10}^{ -7 }}$ $5.939 \times {{10}^{ -6 }}$ $2.519 \times {{10}^{ -5 }}$
1. $4.225 \times {{10}^{ -9 }}$ $6.376 \times {{10}^{ -8 }}$ $2.428 \times {{10}^{ -7 }}$ $1.01 \times {{10}^{ -7 }}$ $3.129 \times {{10}^{ -6 }}$
Table 4.  The absolute errors of third order approximation by OPIM with the exact solution of Example 1
x t=0.1 t=0.2 t=0.3 t=0.4 t=0.5
0.1 $8.322 \times {{10}^{ -14 }}$ $1.326 \times {{10}^{ -12 }}$ $6.67 \times {{10}^{ -12 }}$ $2.088 \times {{10}^{ -11 }}$ $5.038 \times {{10}^{ -11 }}$
0.2 $3.329 \times {{10}^{ -13 }}$ $5.305 \times {{10}^{ -12 }}$ $2.668 \times {{10}^{ -11 }}$ $8.353 \times {{10}^{ -11 }}$ $2.015 \times {{10}^{ -10 }}$
0.3 $7.49 \times {{10}^{ -13 }}$ $1.194 \times {{10}^{ -11 }}$ $6.003 \times {{10}^{ -11 }}$ $1.88 \times {{10}^{ -10 }}$ $4.534 \times {{10}^{ -10 }}$
0.4 $1.332 \times {{10}^{ -12 }}$ $2.122 \times {{10}^{ -11 }}$ $1.067 \times {{10}^{ -10 }}$ $3.341 \times {{10}^{ -10 }}$ $8.06 \times {{10}^{ -10 }}$
0.5 $2.081 \times {{10}^{ -12 }}$ $3.316 \times {{10}^{ -11 }}$ $1.667 \times {{10}^{ -10 }}$ $5.221 \times {{10}^{ -10 }}$ $1.259 \times {{10}^{ -9 }}$
0.6 $2.996 \times {{10}^{ -12 }}$ $4.774 \times {{10}^{ -11 }}$ $2.401 \times {{10}^{ -10 }}$ $7.518 \times {{10}^{ -10 }}$ $1.814 \times {{10}^{ -9 }}$
0.7 $4.078 \times {{10}^{ -12 }}$ $6.499 \times {{10}^{ -11 }}$ $3.268 \times {{10}^{ -10 }}$ $1.023 \times {{10}^{ -9 }}$ $2.469 \times {{10}^{ -9 }}$
0.8 $5.326 \times {{10}^{ -12 }}$ $8.488 \times {{10}^{ -11 }}$ $4.268 \times {{10}^{ -10 }}$ $1.337 \times {{10}^{ -9 }}$ $3.224 \times {{10}^{ -9 }}$
0.9 $6.741 \times {{10}^{ -12 }}$ $1.074 \times {{10}^{ -10 }}$ $5.402 \times {{10}^{ -10 }}$ $1.692 \times {{10}^{ -9 }}$ $4.081 \times {{10}^{ -9 }}$
1. $8.322 \times {{10}^{ -12 }}$ $1.326 \times {{10}^{ -10 }}$ $6.67 \times {{10}^{ -10 }}$ $2.088 \times {{10}^{ -9 }}$ $5.038 \times {{10}^{ -9 }}$
x t=0.1 t=0.2 t=0.3 t=0.4 t=0.5
0.1 $8.322 \times {{10}^{ -14 }}$ $1.326 \times {{10}^{ -12 }}$ $6.67 \times {{10}^{ -12 }}$ $2.088 \times {{10}^{ -11 }}$ $5.038 \times {{10}^{ -11 }}$
0.2 $3.329 \times {{10}^{ -13 }}$ $5.305 \times {{10}^{ -12 }}$ $2.668 \times {{10}^{ -11 }}$ $8.353 \times {{10}^{ -11 }}$ $2.015 \times {{10}^{ -10 }}$
0.3 $7.49 \times {{10}^{ -13 }}$ $1.194 \times {{10}^{ -11 }}$ $6.003 \times {{10}^{ -11 }}$ $1.88 \times {{10}^{ -10 }}$ $4.534 \times {{10}^{ -10 }}$
0.4 $1.332 \times {{10}^{ -12 }}$ $2.122 \times {{10}^{ -11 }}$ $1.067 \times {{10}^{ -10 }}$ $3.341 \times {{10}^{ -10 }}$ $8.06 \times {{10}^{ -10 }}$
0.5 $2.081 \times {{10}^{ -12 }}$ $3.316 \times {{10}^{ -11 }}$ $1.667 \times {{10}^{ -10 }}$ $5.221 \times {{10}^{ -10 }}$ $1.259 \times {{10}^{ -9 }}$
0.6 $2.996 \times {{10}^{ -12 }}$ $4.774 \times {{10}^{ -11 }}$ $2.401 \times {{10}^{ -10 }}$ $7.518 \times {{10}^{ -10 }}$ $1.814 \times {{10}^{ -9 }}$
0.7 $4.078 \times {{10}^{ -12 }}$ $6.499 \times {{10}^{ -11 }}$ $3.268 \times {{10}^{ -10 }}$ $1.023 \times {{10}^{ -9 }}$ $2.469 \times {{10}^{ -9 }}$
0.8 $5.326 \times {{10}^{ -12 }}$ $8.488 \times {{10}^{ -11 }}$ $4.268 \times {{10}^{ -10 }}$ $1.337 \times {{10}^{ -9 }}$ $3.224 \times {{10}^{ -9 }}$
0.9 $6.741 \times {{10}^{ -12 }}$ $1.074 \times {{10}^{ -10 }}$ $5.402 \times {{10}^{ -10 }}$ $1.692 \times {{10}^{ -9 }}$ $4.081 \times {{10}^{ -9 }}$
1. $8.322 \times {{10}^{ -12 }}$ $1.326 \times {{10}^{ -10 }}$ $6.67 \times {{10}^{ -10 }}$ $2.088 \times {{10}^{ -9 }}$ $5.038 \times {{10}^{ -9 }}$
Table 5.  Absolute errors of third order OPIM, third order VIM and fourth order ADM solutions at t = 0.1 for Example 2
x (ADM) (VIM) (OPIM)
1 $5.201 \times {10}^{-11}$ $4.809 \times {10}^{-12}$ $3.334 \times {10}^{-19}$
2 $3.362 \times {10}^{-11}$ $2.607 \times {10}^{-13}$ $3.108 \times {10}^{-19}$
3 $2.379 \times {10}^{-11}$ $4.985 \times {10}^{-14}$ $5.274 \times {10}^{-20}$
4 $1.509 \times {10}^{-11}$ $2.774 \times {10}^{-15}$ $6.118 \times {10}^{-21}$
5 $1.496 \times {10}^{-11}$ $1.292 \times {10}^{-16}$ $8.936 \times {10}^{-20}$
6 $2.471 \times {10}^{-12}$ $2.315 \times {10}^{-18}$ $4.014 \times {10}^{-21}$
7 $2.250 \times {10}^{-12}$ $1.403 \times {10}^{-18}$ $1.124 \times {10}^{-21}$
8 $1.613 \times {10}^{-13}$ $6.288 \times {10}^{-19}$ $7.052 \times {10}^{-21}$
9 $1.541 \times {10}^{-13}$ $2.369 \times {10}^{-19}$ $8.017 \times {10}^{-20}$
10 $1.108 \times {10}^{-14}$ $8.743 \times {10}^{-20}$ $1.055 \times {10}^{-20}$
x (ADM) (VIM) (OPIM)
1 $5.201 \times {10}^{-11}$ $4.809 \times {10}^{-12}$ $3.334 \times {10}^{-19}$
2 $3.362 \times {10}^{-11}$ $2.607 \times {10}^{-13}$ $3.108 \times {10}^{-19}$
3 $2.379 \times {10}^{-11}$ $4.985 \times {10}^{-14}$ $5.274 \times {10}^{-20}$
4 $1.509 \times {10}^{-11}$ $2.774 \times {10}^{-15}$ $6.118 \times {10}^{-21}$
5 $1.496 \times {10}^{-11}$ $1.292 \times {10}^{-16}$ $8.936 \times {10}^{-20}$
6 $2.471 \times {10}^{-12}$ $2.315 \times {10}^{-18}$ $4.014 \times {10}^{-21}$
7 $2.250 \times {10}^{-12}$ $1.403 \times {10}^{-18}$ $1.124 \times {10}^{-21}$
8 $1.613 \times {10}^{-13}$ $6.288 \times {10}^{-19}$ $7.052 \times {10}^{-21}$
9 $1.541 \times {10}^{-13}$ $2.369 \times {10}^{-19}$ $8.017 \times {10}^{-20}$
10 $1.108 \times {10}^{-14}$ $8.743 \times {10}^{-20}$ $1.055 \times {10}^{-20}$
Table 6.  Absolute errors of third order OPIM, third order VIM and fourth order ADM solutions at t = 0.3 for Example 2
x (ADM) (VIM) (OPIM)
1 $4.427 \times {10}^{-8}$ $3.177 \times {10}^{-8}$ $5.036 \times {10}^{-17}$
2 $6.142 \times {10}^{-9}$ $1.651 \times {10}^{-9}$ $6.018 \times {10}^{-17}$
3 $3.528 \times {10}^{-10}$ $3.211 \times {10}^{-10}$ $3.305 \times {10}^{-16}$
4 $2.774 \times {10}^{-11}$ $1.788 \times {10}^{-11}$ $9.012 \times {10}^{-15}$
5 $8.682 \times {10}^{-12}$ $8.318 \times {10}^{-13}$ $7.047 \times {10}^{-15}$
6 $1.430 \times {10}^{-13}$ $1.741 \times {10}^{-14}$ $2.512 \times {10}^{-16}$
7 $5.498 \times {10}^{-13}$ $9.126 \times {10}^{-15}$ $6.369 \times {10}^{-16}$
8 $1.514 \times {10}^{-13}$ $4.081 \times {10}^{-15}$ $8.169 \times {10}^{-17}$
9 $4.975 \times {10}^{-14}$ $1.537 \times {10}^{-15}$ $9.142 \times {10}^{-16}$
10 $4.353 \times {10}^{-14}$ $5.674 \times {10}^{-16}$ $8.777 \times {10}^{-16}$
x (ADM) (VIM) (OPIM)
1 $4.427 \times {10}^{-8}$ $3.177 \times {10}^{-8}$ $5.036 \times {10}^{-17}$
2 $6.142 \times {10}^{-9}$ $1.651 \times {10}^{-9}$ $6.018 \times {10}^{-17}$
3 $3.528 \times {10}^{-10}$ $3.211 \times {10}^{-10}$ $3.305 \times {10}^{-16}$
4 $2.774 \times {10}^{-11}$ $1.788 \times {10}^{-11}$ $9.012 \times {10}^{-15}$
5 $8.682 \times {10}^{-12}$ $8.318 \times {10}^{-13}$ $7.047 \times {10}^{-15}$
6 $1.430 \times {10}^{-13}$ $1.741 \times {10}^{-14}$ $2.512 \times {10}^{-16}$
7 $5.498 \times {10}^{-13}$ $9.126 \times {10}^{-15}$ $6.369 \times {10}^{-16}$
8 $1.514 \times {10}^{-13}$ $4.081 \times {10}^{-15}$ $8.169 \times {10}^{-17}$
9 $4.975 \times {10}^{-14}$ $1.537 \times {10}^{-15}$ $9.142 \times {10}^{-16}$
10 $4.353 \times {10}^{-14}$ $5.674 \times {10}^{-16}$ $8.777 \times {10}^{-16}$
Table 7.  ADM, VIM-PIM, DTM and OPIM solutions at t = 0.1 for Example 3
x (ADM) (VIM-PIM) (DTM) (OPIM-$u_1$) (OPIM-$u_2$)
0.0 0.994999 0.995000 0.995000 1.00235 1.00436
0.1 1.093291 1.093291 1.093336 1.10291 1.10548
0.2 1.190502 1.190503 1.190602 1.20252 1.20566
0.3 1.285668 1.285668 1.285829 1.30016 1.3039
0.4 1.377844 1.377844 1.378073 1.39487 1.39921
0.5 1.466118 1.466119 1.466420 1.4857 1.49062
0.6 1.549620 1.549621 1.550000 1.57173 1.57723
0.7 1.627529 1.627531 1.627994 1.65209 1.65815
0.8 1.699081 1.699084 1.699640 1.72598 1.73257
0.9 1.763575 1.763579 1.764245 1.79265 1.79972
1.0 1.820382 1.820387 1.821201 1.85142 1.85893
x (ADM) (VIM-PIM) (DTM) (OPIM-$u_1$) (OPIM-$u_2$)
0.0 0.994999 0.995000 0.995000 1.00235 1.00436
0.1 1.093291 1.093291 1.093336 1.10291 1.10548
0.2 1.190502 1.190503 1.190602 1.20252 1.20566
0.3 1.285668 1.285668 1.285829 1.30016 1.3039
0.4 1.377844 1.377844 1.378073 1.39487 1.39921
0.5 1.466118 1.466119 1.466420 1.4857 1.49062
0.6 1.549620 1.549621 1.550000 1.57173 1.57723
0.7 1.627529 1.627531 1.627994 1.65209 1.65815
0.8 1.699081 1.699084 1.699640 1.72598 1.73257
0.9 1.763575 1.763579 1.764245 1.79265 1.79972
1.0 1.820382 1.820387 1.821201 1.85142 1.85893
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