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Article Contents

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

• * Corresponding author
• 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.

Mathematics Subject Classification: Primary: 70K60, 81Q05, 35Q40; Secondary: 65N1.

 Citation:

• 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. $2.015 \times {{10}^{ -2 }}$ $4.696 \times {{10}^{ -3 }}$ $4.689 \times {{10}^{ -3 }}$

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. $6.822 \times {{10}^{ -4 }}$ $2.979 \times {{10}^{ -4 }}$ $1.825 \times {{10}^{ -8 }}$

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. $4.225 \times {{10}^{ -9 }}$ $6.376 \times {{10}^{ -8 }}$ $2.428 \times {{10}^{ -7 }}$ $1.01 \times {{10}^{ -7 }}$ $3.129 \times {{10}^{ -6 }}$

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. $8.322 \times {{10}^{ -12 }}$ $1.326 \times {{10}^{ -10 }}$ $6.67 \times {{10}^{ -10 }}$ $2.088 \times {{10}^{ -9 }}$ $5.038 \times {{10}^{ -9 }}$

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 $1.108 \times {10}^{-14}$ $8.743 \times {10}^{-20}$ $1.055 \times {10}^{-20}$

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 $4.353 \times {10}^{-14}$ $5.674 \times {10}^{-16}$ $8.777 \times {10}^{-16}$

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
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