Application of modified wavelet and homotopy perturbation methods to nonlinear oscillation problems

The shifted second kind Chebyshev polynomials are defined on the interval [0, 1] by Un∗<!-- * --> $\begin{array}{} \displaystyle U_n^* \end{array}$(x) = U_n(2x – 1). All the results of second kind Chebyshev polynomials can be easily transformed to give the corresponding results for their shifted forms. The orthogonality relation with respect to the weight function x−<!-- - -->x2 $\begin{array}{} \displaystyle \sqrt {x - {x^2}} \end{array}$ is given by

∫<!-- ? -->01x−<!-- - -->x2Un∗<!-- * -->(x)Um∗<!-- * -->(x)dx=0,form≠<!-- ? -->n,π<!-- p -->8,form=n. $$\begin{array}{} \displaystyle \int\limits_0^1 {\sqrt {x - {x^2}} } U_n^*(x)U_m^*(x)dx = \left\{ {\begin{array}{*{20}{c}} {0,}&{for\,\,\,m \ne n,}\\ {\frac{\pi }{8},}&{for\,\,\,m = n.} \end{array}} \right. \end{array}$$(8)

The first derivative Un∗<!-- * --> $\begin{array}{} \displaystyle U_n^* \end{array}$(x) is given in the following corollary.

Wavelets constitute a family of functions constructed from dilation and translation of a single function called the mother wavelet. When the dilation parameter a and the translation parameter b vary continuously, we have the following family of continuous wavelet:

ψ<!-- ? -->a,b(t)=a−<!-- - -->12ψ<!-- ? -->t−<!-- - -->ba,a,b∈<!-- ? -->ℜ<!-- R -->,a≠<!-- ? -->0 $$\begin{array}{} \displaystyle {\psi _{a,b}}(t) = {\left| a \right|^{\frac{{ - 1}}{2}}}\psi \left( {\frac{{t - b}}{a}} \right)\,,\,\,\,a,b \in \Re, \,\,a \ne 0 \end{array}$$(10)

Shifted second kind Chebyshev wavelets ψ_nm(t) = ψ(k, n, m, t) have four arguments: k, n can assume any positive integer, m is the order of second kind Chebyshev polynomials and tis the normalized time. They are defined on the interval [0, 1] by

ψ<!-- ? -->nmt=2k+32π<!-- p -->Um∗<!-- * -->2kt−<!-- - -->nfort∈<!-- ? -->n2k,n+12km=0,1,.....M,n=0,1,.....,2k−<!-- - -->10,forotherwise $$\begin{array}{} \displaystyle {\psi _{nm}}\left( t \right) = \left\{ \begin{array}{} {\frac{{{2^{\frac{{\left( {k + 3} \right)}}{2}}}}}{{\sqrt \pi }}U_m^*\left( {{2^k}t - n} \right)}~for~{t \in \left[ {\frac{n}{{{2^k}}},\,\,\frac{{n + 1}}{{{2^k}}}} \right]} \quad m = 0,1,.....M,\,\,\,\,n = {0,1,.....,2^k} - 1 \\ \qquad\qquad\quad~ 0,~for~~otherwise \end{array}\right. \end{array}$$(11)

A function f(t) defined on the interval [0, 1] may be expanded in terms of second kind Chebyshev wavelets as

f(t)=∑<!-- ? -->n=0∞<!-- 8 -->∑<!-- ? -->m=0∞<!-- 8 -->cnmψ<!-- ? -->nm(t), $$\begin{array}{} \displaystyle f(t) = \sum\limits_{n = 0}^\infty {\sum\limits_{m = 0}^\infty {{c_{nm}}} } {\psi _{nm}}(t), \end{array}$$(12)

where

cnm=f(t),ψ<!-- ? -->nm(t)ω<!-- ? -->=∫<!-- ? -->01ω<!-- ? -->(t)f(t)ψ<!-- ? -->nm(t)dt, $$\begin{array}{} \displaystyle {c_{nm}} = {\left\langle {f(t),{\psi _{nm}}(t)} \right\rangle _\omega } = \int\limits_0^1 {\omega (t)f(t){\psi _{nm}}} (t)dt, \end{array}$$(13)

and ω<!-- ? -->(t)=t−<!-- - -->t2. $\begin{array}{} \displaystyle \omega (t) = \sqrt {t - {t^2}} . \end{array}$ If the infinite series is truncated, then f(t) can be approximated as

f(t)≈<!-- ? -->∑<!-- ? -->n=02k−<!-- - -->1∑<!-- ? -->m=0Mcnmψ<!-- ? -->nm(t)=CTψ<!-- ? -->(t), $$\begin{array}{} \displaystyle f(t) \approx \sum\limits_{n = 0}^{{2^k} - 1} {\sum\limits_{m = 0}^M {{c_{nm}}} } {\psi _{nm}}(t) = {C^T}\psi (t), \end{array}$$(14)

where C and ψ(t) are 2^k(M + 1) × 1 matrices defined by

C=c0,0,c0,1,...c0,M,....c2k−<!-- - -->1,M,....c2k−<!-- - -->1,1,...........c2k−<!-- - -->1,MT $$\begin{array}{} \displaystyle C{ = ^{^{^{}}}}{\left[ {{c_{0,0}},{c_{0,1}},...{c_{0,M}},....{c_{{2^k} - 1,M}},....{c_{{2^k} - 1,1}},...........{c_{{2^k} - 1,M}}} \right]^T}^{^{}} \end{array}$$(15)

ψ<!-- ? -->(t)=ψ<!-- ? -->0,0(t),ψ<!-- ? -->0,1(t),..........ψ<!-- ? -->0,M(t),.....ψ<!-- ? -->2k−<!-- - -->1,M(t),...............ψ<!-- ? -->2k−<!-- - -->1,1(t),.....ψ<!-- ? -->2k−<!-- - -->1,M(t)T $$\begin{array}{} \displaystyle \psi (t){ = ^{^{}}}{\left[ {{\psi _{0,0}}(t),{\psi _{0,1}}(t),..........{\psi _{0,M}}(t),.....{\psi _{{2^k} - 1,M}}(t),...............{\psi _{{2^k} - 1,1}}(t),.....{\psi _{{2^k} - 1,M}}(t)} \right]^T}^{^{}} \end{array}$$(16)

In this section, we give some numerical results obtained using the algorithms presented in the previous sections. We consider the following examples.

Wavelet method:

Eq. (17) is solved using the wavelet scheme,

CTD2ψ<!-- ? -->(t)+CTψ<!-- ? -->(t)1+μ<!-- ? -->CTψ<!-- ? -->(t)2=0 $$\begin{array}{} \displaystyle {C^T}{D^2}\psi (t) + \frac{{{C^T}\psi (t)}}{{1 + \mu {{\left( {{C^T}\psi (t)} \right)}^2}\,}} = 0 \end{array}$$(19)

Moreover, the initial conditions are

CTψ<!-- ? -->(t)=l,CTDψ<!-- ? -->(t)=0,att=0 $$\begin{array}{} \displaystyle {C^T}\psi (t) = l,\,\,{C^T}D\psi (t) = 0,\,\, \mathrm{at}\;\; t = 0 \end{array}$$(20)

Using Eq. (19) and Eq. (20), the following system of algebraic equations can be obtained

64c2+2c0+8t−<!-- - -->4c1+32t2−<!-- - -->32t+6c21+μ<!-- ? -->2c0+8t−<!-- - -->4c1+32t2−<!-- - -->32t+6c22=0 $$\begin{array}{} \displaystyle 64{c_2} + \frac{{2{c_0} + \left( {8t - 4} \right){c_1} + \left( {32{t^2} - 32t + 6} \right)\,{c_2}}}{{1 + \mu {{\left( {2{c_0} + \left( {8t - 4} \right)\,{c_1} + \left( {32{t^2} - 32t + 6} \right)\,{c_2}} \right)}^2}}} = 0 \end{array}$$(21)

2c0−<!-- - -->4c1+6c2=l $$\begin{array}{} \displaystyle 2{c_0} - 4{c_1} + 6{c_2} = l \end{array}$$(22)

8c1−<!-- - -->32c2=0 $$\begin{array}{} \displaystyle 8{c_1} - 32{c_2} = 0 \end{array}$$(23)

The suggested method of solution x(t) is approximated as follows (Appendix B):

x(t)=CTψ<!-- ? -->(t) $$\begin{array}{} \displaystyle x(t) = {C^T}\psi (t) \end{array}$$(24)

Homotopy perturbation method:

Solving the nonlinear Eq. (17) using a new approach of homotopy perturbation method (Appendix A)

x(t)=cos⁡<!-- ? -->tl+l3μ<!-- ? -->64−<!-- - -->50l3μ<!-- ? -->218432+cos⁡<!-- ? -->3t13l4μ<!-- ? -->26144−<!-- - -->l3μ<!-- ? -->64+cos⁡<!-- ? -->5t11l4μ<!-- ? -->218432 $$\begin{array}{} \displaystyle x(t) = \cos t\,\,\left( {l + \frac{{{l^3}\mu }}{{64}} - \frac{{50{l^3}{\mu ^2}}}{{18432}}} \right) + \cos 3t\,\,\left( {\frac{{13{l^4}{\mu ^2}}}{{6144}} - \frac{{{l^3}\mu }}{{64}}} \right) + \cos 5t\left( {\frac{{11{l^4}{\mu ^2}}}{{18432}}} \right) \end{array}$$(25)

The velocity becomes

x˙<!-- ? -->(t)=sin⁡<!-- ? -->t−<!-- - -->l−<!-- - -->l3μ<!-- ? -->64+50l3μ<!-- ? -->218432+3sin⁡<!-- ? -->3t−<!-- - -->13l4μ<!-- ? -->26144+l3μ<!-- ? -->64−<!-- - -->5sin⁡<!-- ? -->5t11l4μ<!-- ? -->218432 $$\begin{array}{} \displaystyle \dot x(t) = \sin t\,\,\left( { - l - \frac{{{l^3}\mu }}{{64}} + \frac{{50{l^3}{\mu ^2}}}{{18432}}} \right) + 3\sin 3t\,\,\left( { - \frac{{13{l^4}{\mu ^2}}}{{6144}} + \frac{{{l^3}\mu }}{{64}}} \right) - 5\sin 5t\left( {\frac{{11{l^4}{\mu ^2}}}{{18432}}} \right) \end{array}$$(26)

Wavelet method:

The wavelet scheme of Eq. (27) is

CTD2ψ<!-- ? -->(t)+CTψ<!-- ? -->(t)1+α<!-- a -->CTψ<!-- ? -->(t)+β<!-- ? -->CTψ<!-- ? -->(t)2=0 $$\begin{array}{} \displaystyle {C^T}{D^2}\psi (t) + \frac{{{C^T}\psi (t)}}{{1 + \alpha \,\left( {{C^T}\psi (t)} \right) + \beta {{\left( {{C^T}\psi (t)} \right)}^2}\,}} = 0 \end{array}$$(29)

The initial conditions are

CTψ<!-- ? -->(t)=l,CTDψ<!-- ? -->(t)=0,att=0 $$\begin{array}{} \displaystyle {C^T}\psi (t) = l,\,\,{C^T}D\psi (t) = 0,\,\, \mathrm{at}\;\; t = 0 \end{array}$$(30)

Using Eq. (29) and Eq. (30), the following system of algebraic equations can be obtained

64c2+2c0+8t−<!-- - -->4c1+32t2−<!-- - -->32t+6c21+α<!-- a -->2c0+8t−<!-- - -->4c1+32t2−<!-- - -->32t+6c2+β<!-- ? -->2c0+8t−<!-- - -->4c1+32t2−<!-- - -->32t+6c22=0 $$\begin{array}{} \displaystyle 64{c_2} + \frac{{2{c_0} + \left( {8t - 4} \right){c_1} + \left( {32{t^2} - 32t + 6} \right)\,{c_2}}}{{1 + \alpha \left( {2{c_0} + \left( {8t - 4} \right)\,{c_1} + \left( {32{t^2} - 32t + 6} \right)\,{c_2}} \right) + \beta {{\left( {2{c_0} + \left( {8t - 4} \right)\,{c_1} + \left( {32{t^2} - 32t + 6} \right)\,{c_2}} \right)}^2}}} = 0 \end{array}$$(31)

2c0−<!-- - -->4c1+6c2=l $$\begin{array}{} \displaystyle 2{c_0} - 4{c_1} + 6{c_2} = l \end{array}$$(32)

8c1−<!-- - -->32c2=0 $$\begin{array}{} \displaystyle 8{c_1} - 32{c_2} = 0 \end{array}$$(33)

The solution of x(t) is approximated as follows (Appendix B):

x(t)=CTψ<!-- ? -->(t) $$\begin{array}{} \displaystyle x(t) = {C^T}\psi (t) \end{array}$$(34)

Homotopy perturbation method:

Solving the nonlinear Eq. (27) using a new approach of homotopy perturbation method, we get

x(t)=lcost+sin2t22α<!-- a -->l23+β<!-- ? -->l34cos⁡<!-- ? -->t+β<!-- ? -->l38cos2t+4α<!-- a -->l23+β<!-- ? -->l38 $$\begin{array}{} \displaystyle x(t) = l\,\cos \,t + {\sin ^2}\left( {\frac{t}{2}} \right)\,\left[ {\,\left( {\frac{{2\alpha \,{l^2}}}{3} + \frac{{\beta {l^3}}}{4}} \right)\cos t + \frac{{\beta \,{l^3}}}{8}\cos \,2t + \frac{{4\alpha {l^2}}}{3} + \frac{{\beta \,{l^3}}}{8}} \right] \end{array}$$(35)

The velocity becomes

x˙<!-- ? -->(t)=−<!-- - -->lsint+2sint2cost2−<!-- - -->2α<!-- a -->l23+β<!-- ? -->l34sin⁡<!-- ? -->t−<!-- - -->β<!-- ? -->l34sin2t $$\begin{array}{} \displaystyle \dot x(t) = - l\sin \,t + 2\sin \left( {\frac{t}{2}} \right)\cos \left( {\frac{t}{2}} \right)\,\left[ { - \,\left( {\frac{{2\alpha \,{l^2}}}{3} + \frac{{\beta {l^3}}}{4}} \right)\sin t - \frac{{\beta \,{l^3}}}{4}\sin \,2t} \right] \end{array}$$(36)

Wavelet method:

Eq. (37) is solved using the wavelet scheme,

CTD2ψ<!-- ? -->(t)+CTψ<!-- ? -->(t)+ς<!-- ? -->CTψ<!-- ? -->(t)3=0 $$\begin{array}{} \displaystyle {C^T}{D^2}\psi (t) + \,\left( {{C^T}\psi (t)} \right) + \varsigma {\left( {{C^T}\psi (t)} \right)^3} = 0 \end{array}$$(39)

Moreover, the initial conditions are

CTψ<!-- ? -->(t)=l,CTDψ<!-- ? -->(t)=0,att=0 $$\begin{array}{} \displaystyle {C^T}\psi (t) = l,\,\,{C^T}D\psi (t) = 0,\,\,\mathrm{at} \;\;t = 0 \end{array}$$(40)

Using Eq. (39) and Eq. (40), the following system of algebraic equations can be obtained

64c2+2c0+8t−<!-- - -->4c1+32t2−<!-- - -->32t+6c2+ς<!-- ? -->2c0+8t−<!-- - -->4c1+32t2−<!-- - -->32t+6c23=0 $$\begin{array}{} \displaystyle 64{c_2} + \left( {2{c_0} + \left( {8t - 4} \right){c_1} + \left( {32{t^2} - 32t + 6} \right)\,{c_2}} \right)\, + \varsigma \,\,{\left( {2{c_0} + \left( {8t - 4} \right){c_1} + \left( {32{t^2} - 32t + 6} \right)\,{c_2}} \right)^3} = 0 \end{array}$$(41)

2c0−<!-- - -->4c1+6c2=l $$\begin{array}{} \displaystyle 2{c_0} - 4{c_1} + 6{c_2} = l \end{array}$$(42)

8c1−<!-- - -->32c2=0 $$\begin{array}{} \displaystyle 8{c_1} - 32{c_2} = 0 \end{array}$$(43)

The solution of u(t) is approximated as follows (Appendix B):

u(t)=CTψ<!-- ? -->(t) $$\begin{array}{} \displaystyle u(t) = {C^T}\psi (t) \end{array}$$(44)

Homotopy perturbation method:

Solving the nonlinear Eq. (37) using a new approach of homotopy perturbation method, we get

u(t)=lcos⁡<!-- ? -->t−<!-- - -->ς<!-- ? -->l332cos⁡<!-- ? -->t−<!-- - -->cos⁡<!-- ? -->3t $$\begin{array}{} \displaystyle u(t) = l\,\cos t - \frac{{\varsigma \,{l^3}}}{{32}}\left( {\cos t - \cos 3t} \right) \end{array}$$(45)

The velocity becomes

u˙<!-- ? -->(t)=−<!-- - -->lsint+ς<!-- ? -->l332sin⁡<!-- ? -->t−<!-- - -->3sin⁡<!-- ? -->3t $$\begin{array}{} \displaystyle \dot u(t) = - l\sin \,t + \frac{{\varsigma \,{l^3}}}{{32}}\left( {\sin t - 3\sin 3t} \right) \end{array}$$(46)