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## 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, 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. [2] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method Kluwer, Boston, MA, 1994. doi: 10.1007/978-94-015-8289-6. [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. [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. [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. [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. [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. [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. [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. [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. [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. [12] S. T. Demiray, Y. Pandir and H. Bulut, New solitary wave solutions of Maccari system, Ocean Engineering, 103 (2015), 153-159. [13] S. Deniz, Optimal perturbation iteration method for solving nonlinear heat transfer equations, Journal of Heat Transfer, 139 (2017), 074503. [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. [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. [16] S. Deniz and N. Bildik, Optimal perturbation iteration method for Bratu-type problems, Journal of King Saud University Science, 30 (2018), 91-99. [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. [18] Y. Gurefe, A. 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. [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. [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. [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. [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. [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. [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. [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. [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. [27] C. S. Liu, Trial equation method and its applications to nonlinear evolution equations, Acta Physica Sinica, 54 (2005), 2505-2509. [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. [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. [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. [31] V. Marinca, N. Herisanu and I. Nemes, Optimal homotopy asymptotic method with application to thin film flow, Open Physics, 6 (2008), 648-653. [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. [33] Y. Molliq, M. 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. [34] M. M. Rashidi, G. 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. [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. [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. [37] J. J. Sakurai, Advanced Quantum Mechanics, AddisonWesley, New York, 1967. [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. [39] H. Tari, D. 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. [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. [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. [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.

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. [2] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method Kluwer, Boston, MA, 1994. doi: 10.1007/978-94-015-8289-6. [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. [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. [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. [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. [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. [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. [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. [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. [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. [12] S. T. Demiray, Y. Pandir and H. Bulut, New solitary wave solutions of Maccari system, Ocean Engineering, 103 (2015), 153-159. [13] S. Deniz, Optimal perturbation iteration method for solving nonlinear heat transfer equations, Journal of Heat Transfer, 139 (2017), 074503. [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. [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. [16] S. Deniz and N. Bildik, Optimal perturbation iteration method for Bratu-type problems, Journal of King Saud University Science, 30 (2018), 91-99. [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. [18] Y. Gurefe, A. 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. [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. [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. [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. [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. [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. [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. [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. [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. [27] C. S. Liu, Trial equation method and its applications to nonlinear evolution equations, Acta Physica Sinica, 54 (2005), 2505-2509. [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. [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. [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. [31] V. Marinca, N. Herisanu and I. Nemes, Optimal homotopy asymptotic method with application to thin film flow, Open Physics, 6 (2008), 648-653. [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. [33] Y. Molliq, M. 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. [34] M. M. Rashidi, G. 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. [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. [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. [37] J. J. Sakurai, Advanced Quantum Mechanics, AddisonWesley, New York, 1967. [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. [39] H. Tari, D. 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. [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. [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. [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.
Absolute errors obtained by ADM-DTM and PIM for Example 1
Absolute errors obtained by OPIM for Example 1
Absolute errors of third order PIM and OPIM solutions for Example 2
Comparison between the third order approximate solutions obtained by OPIM($\blacksquare$) and by VIM($\bullet$) and the exact solution (–) for Example 2
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. $2.015 \times {{10}^{ -2 }}$ $4.696 \times {{10}^{ -3 }}$ $4.689 \times {{10}^{ -3 }}$
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. $6.822 \times {{10}^{ -4 }}$ $2.979 \times {{10}^{ -4 }}$ $1.825 \times {{10}^{ -8 }}$
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. $4.225 \times {{10}^{ -9 }}$ $6.376 \times {{10}^{ -8 }}$ $2.428 \times {{10}^{ -7 }}$ $1.01 \times {{10}^{ -7 }}$ $3.129 \times {{10}^{ -6 }}$
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. $8.322 \times {{10}^{ -12 }}$ $1.326 \times {{10}^{ -10 }}$ $6.67 \times {{10}^{ -10 }}$ $2.088 \times {{10}^{ -9 }}$ $5.038 \times {{10}^{ -9 }}$
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 $1.108 \times {10}^{-14}$ $8.743 \times {10}^{-20}$ $1.055 \times {10}^{-20}$
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 $4.353 \times {10}^{-14}$ $5.674 \times {10}^{-16}$ $8.777 \times {10}^{-16}$
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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