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# The method of finding solutions of partial dynamic equations on time scales

- Hsuan-Ku Liu
^{1}Email author

**2013**:141

https://doi.org/10.1186/1687-1847-2013-141

© Liu; licensee Springer 2013

**Received: **14 November 2012

**Accepted: **30 April 2013

**Published: **17 May 2013

## Abstract

On time scales, one area lacking of development is the method of finding solutions on partial dynamic equations. This paper proposes a method for finding the exact solution of linear partial dynamic equations on arbitrage time scales. We modify the variational iteration method on ℝ to find an approximation of the nonlinear partial dynamic equation on $\overline{{q}^{\mathbb{N}}}$. As an example, the modified variational iteration method is applied to *q*-Berger equations and to *q*-Fisher equations. Their numerical results reveal that the proposed method is very effective.

## Keywords

- partial dynamic equations on time scales
- nonlinear
*q*-difference equation - variational iterative method
- approximate solutions

## 1 Introduction

A time scale is a nonempty closed subset of real numbers. On time scale calculus, notations and theorems have been well established for the univariate case [1]. Solutions of ordinary differential equations, such as initial value problems and boundary value problems, have been studied and published during the past two decades on time scales. In recent years, Hoffacker [2] and Ahlbrandt and Morian [3] demonstrated the related ideas to the multivariate case and studied partial dynamic equations on time scales. Notations and definitions on multivariate time scales calculus can be found in Bohner and Guseinov [4, 5]. Jackson [6] extended the existing ideas of the time scales calculus [3] to the multivariate case. The method of generalized Laplace transform on time scales is applied to find solutions of the homogeneous and nonhomogeneous heat and wave equations. Recent developments in the method of finding solutions have aroused further interest in the discussion of partial dynamic equations on time scales.

For the nonlinear cases, methods of finding solutions are not mentioned for partial dynamic equations on time scales. One of the difficulties for developing a theory of series solutions to nonlinear equations on time scales is that the formula for multiplications of two generalized polynomials is not easily found. If a time scale has constant graininess, Haile and Hall [7] provided an exact formula for the multiplication of two generalized polynomials. Using the obtained results, the series solutions for linear dynamic equations are proposed on the time scales ℝ and $\mathbb{T}=h\mathbb{Z}$ (difference equations with step size *h*). On generalized time scales, Mozyrska and Pawtuszewicz [8] presented a formula for the multiplication of generalized polynomials of degree one and degree $n\in \mathbb{N}$. Liu [9] provided a product rule of two generalized polynomials on the time scale $\overline{{q}^{\mathbb{Z}}}=\{{q}^{n}\mid n\in \mathbb{N}\}\cup \{0\}$.

The variational iteration method proposed by He [10] is a powerful mathematical tool in analyzing the nonlinear problems on ℝ (the set of real numbers). Over the last few years, the variational iteration method (VIM) has been widely applied to analyze the nonlinear boundary value problems [11], the nonlinear heat diffusion equations [12] and the nonlinear reaction-diffusion equations [13]. An advantage of the VIM is that there is no need to make the assumption of the small parameters. On nonlinear partial dynamic equations, approximate solutions obtained by the variational iteration method are not found yet.

In this paper, we first explore a simple method to find the exact solution of linear partial dynamic equations on time scales. For the nonlinear cases, we derive a product rule of two generalized polynomials on $\overline{{q}^{\mathbb{Z}}}$, which provides an idea for developing a series solutions on *q*-calculus. Applying the product rules, we extend the variational iteration method from the set of real numbers ℝ to the time scales $\overline{{q}^{\mathbb{Z}}}$. The extension provides a method to find an approximate solution on the nonlinear partial dynamic equation on $\overline{{q}^{\mathbb{Z}}}$. Moreover, the VIM is applied to find an approximation of the *q*-Berger equation and the *q*-Fisher equation. By the numerical results, we found that the modified VIM is very effective. The VIM can be applied to other time scales when the multiplication rule of two generalized polynomials on these time scales is obtained.

This paper is organized as follows. In Section 2, the basic ideas of partial dynamic equations on time scales are introduced. In Section 3, a method is explored to find an exact solution of linear initial value problems on time scales. In Section 4, a product rule of two generalized polynomials at 0 is derived on $\overline{{q}^{\mathbb{Z}}}$ and the variational iteration method is applied to find an approximate solution of the Burger equation and the Fisher equation. In Section 5, numerical examples reveal that the proposed method is very effective. Finally, a concise conclusion and future directions are provided in Section 6.

## 2 Basic concepts on time scales

A time scale is an arbitrary nonempty closed subset of the real numbers. The calculus of time scales was introduced by Hilger [14] in order to create a theory that can unify discrete and continuous analysis.

### 2.1 An introduction to time scales

In this subsection, we first define the forward and backward jump operators on time scales and then introduce the delta derivative and the integration.

**Definition 1**Let $\mathbb{T}$ be a time scale. For $t\in \mathbb{T}$ the forward jump operator $\sigma :\mathbb{T}\to \mathbb{T}$ is defined by

According to the forward jump operator and the gain function, the delta derivative on the time scale $\mathbb{T}$ is given as follows.

**Definition 2**Assume that $f:\mathbb{T}\to \mathbb{R}$ is a function and let $t\in \mathbb{T}$. If $\sigma (t)>t$, the delta derivative of $f(t)$ at

*t*on the time scale $\mathbb{T}$ is defined as

A function $f(t)$ on $\mathbb{T}$ is said to be differentiable at *t* if its derivative exists at *t*, $\mathrm{\forall}t\in \mathbb{T}$.

Integration on a time scale can be viewed as an anti-derivative.

**Definition 3**If we have delta derivative $g(t)={f}^{\Delta}(t)$ on the time scale $\mathbb{T}$, then the anti-derivative is

Following the delta derivative and integration, we define the generalized polynomials as follows.

**Definition 4**On the time scale $\mathbb{T}$, the generalized polynomials ${h}_{k}(\cdot ,{t}_{0}):\mathbb{T}\to \mathbb{R}$ are defined recursively as follows:

*s*, the delta derivative of ${h}_{k}$ with respect to

*t*satisfies

### 2.2 An introduction to *q*-calculus

where ℕ denotes the set of positive integers.

*a*and

*q*are real numbers such that $0<q<1$, then the

*q*-shift factorial [15] is defined by

Following Definition 2, the *q*-derivative is given as follows.

**Definition 5**Assume that $f:\overline{{q}^{\mathbb{N}}}\to \mathbb{R}$ is a function and $t\in \overline{{q}^{\mathbb{N}}}$. The

*q*-derivative [1] at

*t*is defined by

*q*-polynomials are represented as

on $\overline{{q}^{\mathbb{N}}}$ [1].

A *q*-difference equation is an equation that contains *q*-derivatives of a function defined on $\overline{{q}^{\mathbb{N}}}$.

### 2.3 Multivariable calculus on time scales

The differentiation and integrations are introduced for functions of two variables on time scales [3]. Definitions on multivariate calculus on time scales can be found in Bohner and Guseinov [4, 5]. Following the line of ideas, the dynamic equations on time scales are extended to partial dynamic equations on time scales [2, 3, 14].

- 1.
The forward jump operators $\sigma :\mathbb{T}\to \mathbb{T}$ by $\sigma (\mathbf{t})=(\sigma ({t}_{1}),\sigma ({t}_{2}))$ are defined as $\sigma ({t}_{1})=inf(s\in {\mathbb{T}}_{1}\mid s>{t}_{1})$ and $\sigma ({t}_{2})=inf(s\in {\mathbb{T}}_{2}\mid s>{t}_{2})$.

- 2.
The backward jump operators $\tau :\mathbb{T}\to \mathbb{T}$ by $\tau (\mathbf{t})=(\tau ({t}_{1}),\tau ({t}_{2}))$ are defined as $\tau ({t}_{1})=sup(s\in {\mathbb{T}}_{1}\mid s<{t}_{1})$ and $\tau ({t}_{2})=sup(s\in {\mathbb{T}}_{2}\mid s<{t}_{2})$.

To use the notations for partial derivatives with respect to time scale variables ${t}_{1}$ and ${t}_{2}$, respectively, we employ lexicographic ordering for consistency. Let ${f}^{{\Delta}_{1}}$ denote the time scale partial derivative with respect to ${t}_{1}$ and let ${f}^{{\Delta}_{2}}$ denote the time scale partial derivative with respect to ${t}_{2}$. Definitions of these partial derivatives are given as below [2, 3].

**Definition 6**Let

*f*be a real-valued function on $\mathbb{T}$. At $({t}_{1},{t}_{2})\in \mathbb{T}={\mathbb{T}}_{1}\times {\mathbb{T}}_{2}$ we say

*f*has a ${\Delta}_{1}$-partial derivative ${f}^{{\Delta}_{1}}({t}_{1},{t}_{2})$ if for each $\epsilon >0$, there exists a neighborhood

*U*of ${t}_{1}$, with $U=({t}_{1}-\delta ,{t}_{1}+\delta )\cap {\mathbb{T}}_{1}$ for $\delta >0$, such that

*f*has a ${\Delta}_{2}$-partial derivative ${f}^{{\Delta}_{2}}({t}_{1},{t}_{2})$ if for each $\epsilon >0$, there exists a neighborhood

*V*of ${t}_{2}$, with $V=({t}_{2}-\delta ,{t}_{2}+\delta )\cap {\mathbb{T}}_{2}$ for $\delta >0$, such that

for all $s\in V$.

- 1.
${f}^{{\Delta}_{ij}}(\mathbf{t})$ (if this value exists) denotes first taking the partial derivative with respect to ${t}_{i}$ and then taking the partial derivative with respect to ${t}_{j}$, so that ${f}^{{\Delta}_{ij}}={({f}^{{\Delta}_{i}})}^{{\Delta}_{j}}$, $i,j=1,2$.

- 2.
${f}^{{\Delta}_{i}^{n}}(\mathbf{t})$ (if this value exists) denotes taking the partial derivative of $f(\mathbf{t})$ with respect to ${t}_{i}$

*n*times.

## 3 The exact solution of linear initial value problems on time scales

Let ${h}_{k}({t}_{1},0)$ and ${g}_{k}({t}_{2},0)$ be the generalized polynomials on ${\mathbb{T}}_{1}$ and ${\mathbb{T}}_{2}$, respectively. In this section, the variational iteration method on ℝ is extended to provide a method of finding the exact solution of linear partial dynamic equations on time scales. The introduction and the details of the variational iteration method can be found in the Appendix and in [10].

### 3.1 The exact solution of the first-order linear partial dynamic equations

where $f({t}_{2})={\sum}_{i=0}^{K}{a}_{i}{g}_{i}({t}_{2},0)$ on ${\mathbb{T}}_{2}$ and ${a}_{i}$, $i=1,\dots ,K$ are real numbers.

where *L* is a linear operator on ${\mathbb{T}}_{1}$, *N* is a linear (or nonlinear) operator on ${\mathbb{T}}_{2}$ (or ${\mathbb{T}}_{1}\times {\mathbb{T}}_{2}$), *λ* is a Lagrange multiplier which can be identified optimally by variational theory, ${u}_{n}$ is the *n* th approximation, and ${\tilde{u}}_{n}$ denotes a restricted variation, that is, $\delta {\tilde{u}}_{n}=0$.

*L*is selected as

*N*is selected as

As *k* is large enough such that ${f}^{{\Delta}_{2}^{k}}$ equals to zero, the series solution ${u}_{k}$ is the exact solution of (1).

**Example 1**Consider the initial value problem

is the exact solution of (3).

*Proof*We now verify that the obtained function ${u}_{k}$ actually solves the initial value problem (3). First, we show that the obtained function satisfies the initial condition. Since ${h}_{0}(t,s)\equiv 1$ for all

*t*,

*s*and ${h}_{j}(0,0)\equiv 1$ for $j>0$, we have

This implies that ${u}_{k}^{{\Delta}_{1}}({t}_{1},{t}_{2})-{c}^{2}{u}_{k}^{{\Delta}_{2}}({t}_{1},{t}_{2})=0$ on ${\mathbb{T}}_{1}\times {\mathbb{T}}_{2}$. □

### 3.2 The exact solution of the second-order linear partial dynamic equations

where $f({t}_{2})={\sum}_{i=0}^{K}{a}_{i}{g}_{i}({t}_{2},0)$ on ${\mathbb{T}}_{2}$ and ${a}_{i}$, $i=1,\dots ,K$ are real numbers.

*L*is selected as

*N*is selected as

As *k* is large enough such that ${f}^{{\Delta}_{2}^{(2k)}}$ equals to zero, the series solution ${u}_{k}$ is the exact solution of (4).

**Example 2**Consider the IVP

is the exact solution of (5).

where $\lfloor k/2\rfloor $ denotes the floor of $k/2$.

When the initial condition can be represented as a finite series of generalized polynomials, we have proposed a useful method of finding the exact solution of partial dynamic equations on time scales. When the initial condition is represented as an infinite series of generalized polynomials, the approximate solution can be obtained by the same manner. In the following section, we consider the nonlinear partial dynamic equation on the specific time scales.

## 4 Approximation solutions of nonlinear *q*-partial dynamic equations

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*

*Proof*Since

□

**Proposition 1**

*Let*${h}_{i}(t,0)$

*and*${h}_{j}(t,0)$

*be any two*

*q*-

*polynomials*.

*We have*

*Proof*It suffices to show that

□

*L*is selected as

*N*is selected as $-Nu$, the variational iteration formula is obtained as

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

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

*q*-Burger equation as the form

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

*q*-Fisher equation

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

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

## 5 Numerical results

The approximate solutions introduced in the previous sections will be illustrated with some examples.

*q*-shift factorial with $q=0.9$ is given as

*q*-polynomials are represented as

where $H(k,l)=\frac{{({0.9}^{k+1};0.9)}_{l}}{{(0.9;0.9)}_{l}}$. For example, $H(1,1)=\frac{{({0.9}^{2};0.9)}_{1}}{{(0.9;0.9)}_{1}}=\frac{(1-{0.9}^{2})}{(1-0.9)}=\frac{1.9}{0.1}=1.9$.

**Example 4**Consider the nonlinear partial dynamic equation as the form

**Example 5**Consider the Fisher equation as the form

Now, we have demonstrated a method for finding an approximate solution of nonlinear partial dynamic equations on $\overline{{q}_{1}^{\mathbb{N}}}\times \overline{{q}_{2}^{\mathbb{N}}}$. The proposed tool could also be applied to other nonlinear *q*-partial dynamic equations.

In future studies, we intend to derive the multiplication rule of two generalized polynomials and extend the application of the variational iteration method to nonlinear partial dynamic equations on other time scales.

## 6 Conclusion and future direction

In this paper, we have propose a method to find the exact solution of the linear partial dynamic equation on time scales and to find an approximate solution of the nonlinear *q*-partial dynamic equations. Moreover, this method is applied to provide an approximate solution of the *q*-Berger equations and the *q*-Fisher equations.

To extend the method to other time scales, it is important to derive a multiplication rule of two generalized polynomials on the other time scales. On the other hand, approximate solutions as well as their properties of the nonlinear partial dynamic equations, such as Benjamin-Ono equations and the Benjamin-Bona-Mahony equations, are not found on $\overline{{q}^{\mathbb{N}}}$ yet. In the future studies, we would intend to derive the multiplication rule of two generalized polynomials or to provide an approximation of other nonlinear *q*-partial dynamic equations by using the proposing method.

## Appendix: Basic ideas of the variational iteration method

*L*is a linear operator,

*N*is a nonlinear operator and

*g*is an inhomogeneous term. According to the variational iteration method, we can construct a correction functional as follows:

*λ*is a general Lagrange multiplier, ${u}_{0}$ is an initial approximation which must be chosen suitably and ${\tilde{u}}_{n}$ is considered as a restricted variation, that is, $\delta {\tilde{u}}_{n}=0$. To find the optimal value of

*λ*, we make the above correction functional stationary with respect to ${u}_{n}$, noticing that $\delta {u}_{n}(0)=0$, and have

Having obtained the optimal Lagrange multiplier, the successive approximations ${u}_{n}$, $n\ge 0$, of the solution *u* are determined upon the initial function ${u}_{0}$. Therefore, the exact solution is obtained at the limit of the resulting successive approximations.

## Declarations

## Authors’ Affiliations

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