# American Institute of Mathematical Sciences

• Previous Article
Analysis of a Lymphatic filariasis-schistosomiasis coinfection with public health dynamics: Model obtained through Mittag-Leffler function
• DCDS-S Home
• This Issue
• Next Article
Generalized variational calculus in terms of multi-parameters involving Atangana-Baleanu's derivatives and application
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:

show all references

##### References:
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
 [1] Chi-Kun Lin, Kung-Chien Wu. On the fluid dynamical approximation to the nonlinear Klein-Gordon equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2233-2251. doi: 10.3934/dcds.2012.32.2233 [2] Stefano Pasquali. A Nekhoroshev type theorem for the nonlinear Klein-Gordon equation with potential. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3573-3594. doi: 10.3934/dcdsb.2017215 [3] Zhong-Zhi Bai. On convergence of the inner-outer iteration method for computing PageRank. Numerical Algebra, Control & Optimization, 2012, 2 (4) : 855-862. doi: 10.3934/naco.2012.2.855 [4] Masahito Ohta, Grozdena Todorova. Strong instability of standing waves for nonlinear Klein-Gordon equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 315-322. doi: 10.3934/dcds.2005.12.315 [5] Michinori Ishiwata, Makoto Nakamura, Hidemitsu Wadade. Remarks on the Cauchy problem of Klein-Gordon equations with weighted nonlinear terms. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4889-4903. doi: 10.3934/dcds.2015.35.4889 [6] Hironobu Sasaki. Remark on the scattering problem for the Klein-Gordon equation with power nonlinearity. Conference Publications, 2007, 2007 (Special) : 903-911. doi: 10.3934/proc.2007.2007.903 [7] Karen Yagdjian. The semilinear Klein-Gordon equation in de Sitter spacetime. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 679-696. doi: 10.3934/dcdss.2009.2.679 [8] Satoshi Masaki, Jun-ichi Segata. Modified scattering for the Klein-Gordon equation with the critical nonlinearity in three dimensions. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1595-1611. doi: 10.3934/cpaa.2018076 [9] Aslihan Demirkaya, Panayotis G. Kevrekidis, Milena Stanislavova, Atanas Stefanov. Spectral stability analysis for standing waves of a perturbed Klein-Gordon equation. Conference Publications, 2015, 2015 (special) : 359-368. doi: 10.3934/proc.2015.0359 [10] Hironobu Sasaki. Small data scattering for the Klein-Gordon equation with cubic convolution nonlinearity. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 973-981. doi: 10.3934/dcds.2006.15.973 [11] Jun Yang. Vortex structures for Klein-Gordon equation with Ginzburg-Landau nonlinearity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2359-2388. doi: 10.3934/dcds.2014.34.2359 [12] Changxing Miao, Jiqiang Zheng. Scattering theory for energy-supercritical Klein-Gordon equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2073-2094. doi: 10.3934/dcdss.2016085 [13] Elena Kopylova. On dispersion decay for 3D Klein-Gordon equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5765-5780. doi: 10.3934/dcds.2018251 [14] Weizhu Bao, Chunmei Su. Uniform error estimates of a finite difference method for the Klein-Gordon-Schrödinger system in the nonrelativistic and massless limit regimes. Kinetic & Related Models, 2018, 11 (4) : 1037-1062. doi: 10.3934/krm.2018040 [15] Miguel Escobedo, Minh-Binh Tran. Convergence to equilibrium of a linearized quantum Boltzmann equation for bosons at very low temperature. Kinetic & Related Models, 2015, 8 (3) : 493-531. doi: 10.3934/krm.2015.8.493 [16] Soichiro Katayama. Global existence for systems of nonlinear wave and klein-gordon equations with compactly supported initial data. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1479-1497. doi: 10.3934/cpaa.2018071 [17] Andrew Comech. Weak attractor of the Klein-Gordon field in discrete space-time interacting with a nonlinear oscillator. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2711-2755. doi: 10.3934/dcds.2013.33.2711 [18] Baoxiang Wang. Scattering of solutions for critical and subcritical nonlinear Klein-Gordon equations in $H^s$. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 753-763. doi: 10.3934/dcds.1999.5.753 [19] Zaihui Gan. Cross-constrained variational methods for the nonlinear Klein-Gordon equations with an inverse square potential. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1541-1554. doi: 10.3934/cpaa.2009.8.1541 [20] Olivier Goubet, Marilena N. Poulou. Semi discrete weakly damped nonlinear Klein-Gordon Schrödinger system. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1525-1539. doi: 10.3934/cpaa.2014.13.1525

2018 Impact Factor: 0.545