In this section, we extend the variational iteration method to find an approximate solution of nonlinear initial value problems on the time scale \overline{{q}^{\mathbb{N}}}.

To extend the variational iteration method, we first display a production rule [9] of two *q*-polynomials at 0 which will be used to derive an approximate solution in the following discussion.

**Theorem 1** *Let* {h}_{i}(t,0) *and* {h}_{j}(t,0) *be two* *q*-*polynomials at zero*. *We have*

{h}_{i}(t,0){h}_{j}(t,0)=\frac{{({q}^{i+1};q)}_{j}}{{(q;q)}_{j}}{h}_{i+j}(t,0).

*Proof* Since

{h}_{i+j}(t,0)=\prod _{\nu =0}^{i+j-1}\frac{t}{{\sum}_{\mu =0}^{\nu}{q}^{\mu}},

we have

\begin{array}{rl}{h}_{i+j}(t,0)& =\left(\prod _{\nu =0}^{i-1}\frac{t}{{\sum}_{\mu =0}^{\nu}{q}^{\mu}}\right)\left(\prod _{\nu =i}^{i+j-1}\frac{t}{{\sum}_{\mu =0}^{\nu}{q}^{\mu}}\right)\\ ={h}_{i}(t,0)\left(\frac{{\prod}_{\nu =0}^{j-1}{\sum}_{\mu =0}^{\nu}{q}^{\mu}}{{\prod}_{\nu =0}^{j-1}{\sum}_{\mu =0}^{\nu}{q}^{\mu}}\right){t}^{j}\left(\prod _{\nu =i}^{i+j-1}\frac{1}{{\sum}_{\mu =0}^{\nu}{q}^{\mu}}\right)\\ ={h}_{i}(t,0)\left(\prod _{\nu =0}^{j-1}\frac{t}{{\sum}_{\mu =0}^{\nu}{q}^{\mu}}\right)\left(\prod _{\nu =0}^{j-1}\sum _{\mu =0}^{\nu}{q}^{\mu}\right)\left(\prod _{\nu =i}^{i+j-1}\frac{1}{{\sum}_{\mu =0}^{\nu}{q}^{\mu}}\right)\\ ={h}_{i}(t,0){h}_{j}(t,0)\left(\prod _{\nu =0}^{j-1}\frac{{\sum}_{\mu =0}^{\nu}{q}^{\mu}}{{\sum}_{\mu =0}^{\nu +i}{q}^{\mu}}\right).\end{array}

This implies that

\begin{array}{rl}{h}_{i}(t,0){h}_{j}(t,0)& =\left(\prod _{\nu =0}^{j-1}\frac{{\sum}_{\mu =0}^{\nu +i}{q}^{\mu}}{{\sum}_{\mu =0}^{\nu}{q}^{\mu}}\right){h}_{i+j}(t,0)=\prod _{\nu =0}^{j-1}\frac{(1-{q}^{\upsilon +i+1})}{(1-{q}^{\upsilon +1})}{h}_{i+j}(t)\\ =\frac{{({q}^{i+1};q)}_{j}}{{(q;q)}_{j}}{h}_{i+j}(t,0).\end{array}

□

**Proposition 1** *Let* {h}_{i}(t,0) *and* {h}_{j}(t,0) *be any two* *q*-*polynomials*. *We have*

{h}_{i}(t,0){h}_{j}(t,0)={h}_{j}(t,0){h}_{i}(t,0).

*Proof* It suffices to show that

\frac{{({q}^{i+1};q)}_{j}}{{(q,q)}_{j}}=\frac{{({q}^{j+1};q)}_{i}}{{(q,q)}_{i}}.

Suppose i>j, we have

\begin{array}{r}\frac{{({q}^{i+1};q)}_{j}}{{(q,q)}_{j}}-\frac{{({q}^{j+1};q)}_{i}}{{(q,q)}_{i}}\\ \phantom{\rule{1em}{0ex}}=\frac{(1-{q}^{j+1})\cdots (1-{q}^{i+j})}{(1-q)\cdots (1-{q}^{i})}-\frac{(1-{q}^{i+1})\cdots (1-{q}^{i+j})}{(1-q)\cdots (1-{q}^{j})}\\ \phantom{\rule{1em}{0ex}}=\frac{(1-{q}^{j+1})\cdots (1-{q}^{i+j})}{(1-q)\cdots (1-{q}^{i})}-\frac{(1-{q}^{i+1})\cdots (1-{q}^{i+j})(1-{q}^{j+1})\cdots (1-{q}^{i})}{(1-q)\cdots (1-{q}^{j})(1-{q}^{j+1})\cdots (1-{q}^{i})}=0.\end{array}

□

Let {h}_{k} and {g}_{k} be generalized polynomials of \overline{{q}_{1}^{\mathbb{N}}} and \overline{{q}_{2}^{\mathbb{N}}}. The variational iteration method is now applied to find an approximate solution of the nonlinear partial dynamic equations as the form

\{\begin{array}{l}{u}^{{\Delta}_{1}}=Nu\phantom{\rule{1em}{0ex}}\text{on}\overline{{q}_{1}^{\mathbb{N}}}\times \overline{{q}_{2}^{\mathbb{N}}},\\ u(0,{t}_{2})={g}_{k}({t}_{2},0)\phantom{\rule{1em}{0ex}}\text{on}\overline{{q}_{2}^{\mathbb{N}}}.\end{array}

When the linear operator *L* is selected as

and the other operator *N* is selected as -Nu, the variational iteration formula is obtained as

{u}_{n+1}({t}_{1},{t}_{2})={u}_{n}({t}_{1},{t}_{2})-{\int}_{0}^{{t}_{1}}\{{u}_{n}^{{\Delta}_{1}}(s,{t}_{2})-N{u}_{n}(s,{t}_{2})\}\Delta s

with the initial approximation

{u}_{0}={g}_{k}({t}_{2},0).

**Example 3** Consider the partial dynamic equations as the form

\{\begin{array}{l}{u}^{{\Delta}_{1}}=u{u}^{{\Delta}_{2}}\phantom{\rule{1em}{0ex}}\text{on}{\mathbb{T}}_{1}\times {\mathbb{T}}_{2},\\ u(0,{t}_{2})={g}_{k}({t}_{2},0)\phantom{\rule{1em}{0ex}}\text{on}{\mathbb{T}}_{2}.\end{array}

With the variational iteration formula, we obtain the first few components of {u}_{n}({t}_{1},{t}_{2}):

\begin{array}{c}\begin{array}{rl}{u}_{1}({t}_{1},{t}_{2})& ={u}_{0}({t}_{1},{t}_{2})-{\int}_{0}^{{t}_{1}}\{{u}_{0}^{{\Delta}_{1}}(s,{t}_{2})-{u}_{0}(s,{t}_{2}){u}_{0}^{{\Delta}_{2}}(s,{t}_{2})\}\Delta s\\ ={g}_{k}({t}_{2},0)+{\int}_{0}^{{t}_{1}}[{g}_{k}({t}_{2},0){g}_{k-1}({t}_{2},0)]\Delta s\\ ={g}_{k}({t}_{2},0)+{H}_{2}(k,k-1){g}_{2k-1}({t}_{2},0){h}_{1}({t}_{1},0),\end{array}\hfill \\ \begin{array}{rl}{u}_{2}({t}_{1},{t}_{2})=& {u}_{1}({t}_{1},{t}_{2})-{\int}_{0}^{{t}_{1}}\{{u}_{1}^{{\Delta}_{1}}(s,{t}_{2})-{u}_{1}(s,{t}_{2}){u}_{1}^{{\Delta}_{2}}(s,{t}_{2})\}\Delta s\\ =& {g}_{k}({t}_{2},0)+{H}_{2}(k,k-1){g}_{2k-1}({t}_{2},0){h}_{1}({t}_{1},0)\\ -{\int}_{0}^{{t}_{1}}[{H}_{2}(k,k-1){g}_{2k-1}({t}_{2},0)-({g}_{k}({t}_{2},0)+{H}_{2}(k,k-1){g}_{2k-1}({t}_{2},0){h}_{1}(s,0))\\ \times ({g}_{k-1}({t}_{2},0)+{H}_{2}(k,k-1){g}_{2k-2}({t}_{2},0){h}_{1}(s,0))]\Delta s\\ =& {g}_{k}({t}_{2},0)+{H}_{2}(k,k-1){g}_{2k-1}({t}_{2},0){h}_{1}({t}_{1},0)\\ +{H}_{2}(k,k-1){H}_{2}(k,2k-2){g}_{3k-2}({t}_{2},0){h}_{2}({t}_{1},0)\\ +{H}_{2}(k,k-1){H}_{2}(k-1,2k-1){g}_{3k-2}({t}_{2},0){h}_{2}({t}_{1},0)\\ +{H}_{2}(k,k-1){H}_{2}(k,k-1){H}_{2}(2k-1,2k-2){g}_{4k-3}({t}_{2},0){H}_{1}(1,1){h}_{3}({t}_{1},0),\end{array}\hfill \end{array}

where {H}_{1}(k,l)=\frac{{({q}_{1}^{k+1};{q}_{1})}_{l}}{{({q}_{1};{q}_{1})}_{l}} and {H}_{2}(k,l)=\frac{{({q}_{2}^{k+1};{q}_{2})}_{l}}{{({q}_{2};{q}_{2})}_{l}}.

In the same manner, the rest of components of the iteration formula are obtained iteratively.

### 4.1 Applications to the *q*-Burger equation and the Fisher equation

*q*-Burger equation

First of all, we consider the *q*-Burger equation as the form

\{\begin{array}{l}{u}^{{\Delta}_{1}}-u{u}^{{\Delta}_{2}}-\alpha {u}^{{\Delta}_{2}^{2}}=0\phantom{\rule{1em}{0ex}}\text{on}\overline{{q}_{1}^{\mathbb{N}}}\times \overline{{q}_{2}^{\mathbb{N}}},\\ u(0,{t}_{2})={g}_{k}({t}_{2},0)\phantom{\rule{1em}{0ex}}\text{on}\overline{{q}_{2}^{\mathbb{N}}}.\end{array}

When the linear operator and the nonlinear operator are selected as Lu={u}^{{\Delta}_{1}} and -Nu=-u{u}^{{\Delta}_{2}}-\alpha {u}^{{\Delta}_{2}^{2}}, respectively, the variational iteration formula is obtained as

{u}_{n+1}({t}_{1},{t}_{2})={u}_{n}({t}_{1},{t}_{2})-{\int}_{0}^{{t}_{1}}\{{u}_{n}^{{\Delta}_{1}}(s,{t}_{2})-{u}_{n}(s,{t}_{2}){u}_{n}^{{\Delta}_{2}}(s,{t}_{2})-\alpha {u}_{n}^{{\Delta}_{2}^{2}}(s,{t}_{2})\}\Delta s.

(6)

Let G({t}_{2})={H}_{2}(k,k-1){g}_{2k-1}({t}_{2},0)+\alpha {g}_{k-2}({t}_{2},0) and {H}_{2}(k,l)=\frac{{({q}_{2}^{k+1};{q}_{2})}_{l}}{{({q}_{2};{q}_{2})}_{l}}. With the initial condition {u}_{0}({t}_{1},{t}_{2})\equiv u(0,{t}_{2})={g}_{k}({t}_{2},0), we have

\begin{array}{c}\begin{array}{rl}{u}_{1}({t}_{1},{t}_{2})& ={u}_{0}({t}_{1},{t}_{2})-{\int}_{0}^{{t}_{1}}\{{u}_{0}^{{\Delta}_{1}}(s,{t}_{2})-{u}_{0}(s,{t}_{2}){u}_{0}^{{\Delta}_{2}}(s,{t}_{2})-\alpha {u}_{0}^{{\Delta}_{2}^{2}}(s,{t}_{2})\}\Delta s\\ ={g}_{k}({t}_{2},0)-{\int}_{0}^{{t}_{1}}\{-{g}_{k}({t}_{2},0){g}_{k-1}({t}_{2},0)-\alpha {g}_{k-2}({t}_{2},0)\}\Delta s\\ ={g}_{k}({t}_{2},0)+({H}_{2}(k,k-1){g}_{2k-1}({t}_{2},0)+\alpha {g}_{k-2}({t}_{2},0)){h}_{1}({t}_{1},0)\\ ={g}_{k}({t}_{2},0)+G({t}_{2}){h}_{1}({t}_{1},0),\end{array}\hfill \\ \begin{array}{rl}{u}_{2}({t}_{1},{t}_{2})=& {u}_{1}({t}_{1},{t}_{2})-{\int}_{0}^{{t}_{1}}\{{u}_{1}^{{\Delta}_{1}}(s,{t}_{2})-{u}_{1}(s,{t}_{2}){u}_{1}^{{\Delta}_{2}}(s,{t}_{2})-\alpha {u}_{1}^{{\Delta}_{2}^{2}}(s,{t}_{2})\}\Delta s\\ =& {g}_{k}({t}_{2},0)+G({t}_{2}){h}_{1}({t}_{1},0)\\ -{\int}_{0}^{{t}_{1}}\{G({t}_{2})-({g}_{k}({t}_{2},0)+G({t}_{2}){h}_{1}(s,0))\\ \times ({g}_{k-1}({t}_{2},0)+{g}^{{\Delta}_{2}}({t}_{2}){h}_{1}(s,0))\\ -\alpha ({g}_{k-2}({t}_{2},0)+{g}^{{\Delta}_{2}^{2}}({t}_{2}){h}_{1}(s,0))\}\Delta s\\ =& {g}_{k}({t}_{2},0)+{H}_{2}(k,k-1){g}_{2k-1}({t}_{2},0){h}_{1}({t}_{1},0)\\ +G({t}_{2}){g}_{k-1}({t}_{2},0){h}_{2}({t}_{1},0)+{g}^{{\Delta}_{2}}({t}_{2}){g}_{k}({t}_{2},0){h}_{2}({t}_{1},0)\\ +G({t}_{2}){g}^{{\Delta}_{2}}({t}_{2}){H}_{1}(1,1){h}_{3}({t}_{1},0)\\ +\alpha {g}_{k-2}({t}_{2},0){h}_{1}({t}_{1},0)+\alpha {g}^{{\Delta}_{2}^{2}}({t}_{2}){h}_{2}({t}_{1},0)\\ =& {g}_{k}({t}_{2},0)+[{H}_{2}(k,k-1){g}_{2k-1}({t}_{2},0)+\alpha {g}_{k-2}({t}_{2},0)]{h}_{1}({t}_{1},0)\\ +[G({t}_{2}){g}_{k-1}({t}_{2},0)+{g}^{{\Delta}_{2}}({t}_{2}){g}_{k}({t}_{2},0)+\alpha {g}^{{\Delta}_{2}^{2}}({t}_{2})]{h}_{2}({t}_{1},0)\\ +G({t}_{2}){g}^{{\Delta}_{2}}({t}_{2}){H}_{1}(1,1){h}_{3}({t}_{1},0).\end{array}\hfill \end{array}

In the same manner, the rest of components of the iteration formula are obtained iteratively.

*q*-Fisher equation

Secondly, we consider the *q*-Fisher equation, which is a nonlinear reaction diffusion equation, as the form

\{\begin{array}{l}{u}^{{\Delta}_{1}}-\alpha {u}^{{\Delta}_{2}^{2}}-\beta u(1-u)=0\phantom{\rule{1em}{0ex}}\text{on}\overline{{q}_{1}^{\mathbb{N}}}\times \overline{{q}_{2}^{\mathbb{N}}},\\ u(0,{t}_{2})={g}_{k}({t}_{2},0)\phantom{\rule{1em}{0ex}}\text{on}\overline{{q}_{2}^{\mathbb{N}}}.\end{array}

The variational iteration formula is obtained as

\begin{array}{rl}{u}_{n+1}({t}_{1},{t}_{2})=& {u}_{n}({t}_{1},{t}_{2})-{\int}_{0}^{{t}_{1}}\{{u}_{n}^{{\Delta}_{1}}(s,{t}_{2})-\alpha {u}_{n}^{{\Delta}_{2}^{2}}(s,{t}_{2})\\ -\beta {u}_{n}(s,{t}_{2})+\beta {u}_{n}(s,{t}_{2}){u}_{n}(s,{t}_{2})\}\Delta s.\end{array}

(7)

Let G({t}_{2})=\alpha {h}_{k-2}({t}_{2})+\beta {g}_{k}({t}_{2},0)-\beta {H}_{2}(k,k){g}_{2k}({t}_{2},0) and {H}_{2}(k,l)=\frac{{({q}_{2}^{k+1};{q}_{2})}_{l}}{{({q}_{2};{q}_{2})}_{l}}. With the initial condition {u}_{0}({t}_{1},{t}_{2})\equiv u(0,{t}_{2})={g}_{k}({t}_{2},0), we have

\begin{array}{c}\begin{array}{rl}{u}_{1}({t}_{1},{t}_{2})& ={u}_{0}({t}_{1},{t}_{2})-{\int}_{0}^{{t}_{1}}\{{u}_{0}^{{\Delta}_{1}}(s,{t}_{2})-\alpha {u}_{0}^{{\Delta}_{2}^{2}}(s,{t}_{2})-\beta {u}_{0}(s,{t}_{2})+\beta {u}_{0}(s,{t}_{2}){u}_{0}(s,{t}_{2})\}\Delta s\\ ={g}_{k}({t}_{2},0)+{\int}_{0}^{{t}_{1}}\alpha {g}_{k-2}({t}_{2},0)+\beta {g}_{k}({t}_{2},0)-\beta {H}_{2}(k,k){g}_{2k}({t}_{2},0)\Delta s\\ ={g}_{k}({t}_{2},0)+(\alpha {h}_{k-2}({t}_{2})+\beta {g}_{k}({t}_{2},0)-\beta {H}_{2}(k,k){g}_{2k}({t}_{2},0)){h}_{1}({t}_{1},0)\\ ={g}_{k}({t}_{2},0)+G({t}_{2}){h}_{1}({t}_{1},0),\end{array}\hfill \\ \begin{array}{rl}{u}_{2}({t}_{1},{t}_{2})=& {u}_{1}({t}_{1},{t}_{2})-{\int}_{0}^{{t}_{1}}\{{u}_{1}^{{\Delta}_{1}}(s,{t}_{2})-\alpha {u}_{1}^{{\Delta}_{2}^{2}}(s,{t}_{2})-\beta {u}_{1}(s,{t}_{2})+\beta {u}_{1}(s,{t}_{2}){u}_{1}(s,{t}_{2})\}\Delta s\\ =& {g}_{k}({t}_{2},0)+G({t}_{2}){h}_{1}({t}_{1},0)\\ -{\int}_{0}^{{t}_{1}}\{G({t}_{2})-\alpha ({g}_{k-2}({t}_{2},0)+{g}^{{\Delta}_{2}^{2}}({t}_{2}){h}_{1}(s,0))-\beta ({g}_{k}({t}_{2},0)+G({t}_{2}){h}_{1}(s,0))\\ +\beta ({g}_{k}({t}_{2},0)+G({t}_{2}){h}_{1}(s,0))({g}_{k}({t}_{2},0)+G({t}_{2}){h}_{1}(s,0))\}\Delta s\\ =& {g}_{k}({t}_{2},0)+\alpha {g}_{k-2}({t}_{2},0){h}_{1}({t}_{1},0)+\alpha {g}^{{\Delta}_{2}^{2}}({t}_{2}){h}_{2}({t}_{1},0)\\ +\beta {g}_{k}({t}_{2},0){h}_{1}({t}_{1},0)+\beta G({t}_{2}){h}_{2}({t}_{1},0)\\ -{\int}_{0}^{{t}_{1}}\beta ({H}_{2}(k,k){g}_{2k}({t}_{2},0)+2G({t}_{2}){g}_{k}({t}_{2},0){h}_{1}(s,0)\\ +{g}^{2}({t}_{2}){H}_{1}(1,1){h}_{2}(s,0))\Delta s\\ =& {g}_{k}({t}_{2},0)+[\alpha {g}_{k-2}({t}_{2},0)+\beta {g}_{k}({t}_{2},0)-\beta {H}_{2}(k,k){g}_{2k}({t}_{2},0)]{h}_{1}({t}_{1},0)\\ +(\alpha {g}^{{\Delta}_{2}^{2}}({t}_{2})+\beta G({t}_{2})-2\beta G({t}_{2}){g}_{k}({t}_{2},0)){h}_{2}({t}_{1},0)\\ -\beta {g}^{2}({t}_{2}){H}_{1}(1,1){h}_{3}({t}_{1},0).\end{array}\hfill \end{array}

In the same manner, the rest of components of the iteration formula are obtained iteratively.