On symmetry group classification of a fin equation
- Özlem Orhan^{1},
- Gülden Gün^{1} and
- Teoman Özer^{2}Email author
https://doi.org/10.1186/1029-242X-2013-147
© Orhan et al.; licensee Springer. 2013
Received: 12 December 2012
Accepted: 12 March 2013
Published: 3 April 2013
Abstract
We deal with the Noether symmetry classification of a nonlinear fin equation, in which thermal conductivity and heat transfer coefficient are assumed to be functions of the temperature. In this study Noether symmetries of the fin equation are investigated using the partial Lagrangian approach. This classification includes Noether symmetries, first integrals and some invariant solutions with respect to different choices of thermal conductivity and heat transfer coefficient functions.
Keywords
Noether symmetries partial Lagrangian approach classification first integrals nonlinear fin equation1 Introduction
The aim of the present study is to classify the Noether point symmetries of a fin equation. In the literature, symmetry classifications of differential equations with respect to Lie point symmetries and Noether symmetries have an important role for understanding possible solutions of differential equations [1–11]. Noether symmetries can also be used in finding the first integrals (conserved forms) of the nonlinear problems. The earliest studies on Noether symmetries based on the Noether theorem are due to German mathematician Emily Noether [1]. Applications of the Noether theorem to differential equations can provide some important information about the problems in mechanics, physics, and engineering sciences [12–15]. In order to apply the Noether theorem, the differential equations should have a standard Lagrangian. On the other hand, one can apply the partial Lagrangian method to differential equations to investigate Noether symmetries and first integrals by using Euler-Lagrange equations [12]. Here, we determine the partial Lagrangian and Noether symmetries of the fin equation by applying partial Noether approach to a nonlinear fin equation.
This study is organized as follows. In Section 2, we present some fundamental definitions of the Euler-Lagrange operator, partial Lagrangian and partial Noether operators. In Section 3 we discuss the nonlinear fin equation and the corresponding determining equations. This section also includes different cases corresponding to different choices of thermal conductivity and heat transfer coefficient. Furthermore, Noether point symmetries and first integrals for each different case are presented. Section 4 presents some invariant solutions, and the last section summarizes some important results in the study.
2 Preliminaries
Here, the vector space of all differential functions of all finite orders is represented by $\mathcal{A}$ that is a universal space. Also, operators apart from the total derivative operator (2.2) are defined on the space $\mathcal{A}$.
Definition 1
is called the Euler operator or Euler-Lagrange operator.
Definition 2
Then the expression (2.9) is called the local conservation law for system (2.8). Furthermore, ${D}_{x}I={\text{O\u0327}}^{\alpha}{E}_{\alpha}$ is called the characteristic form of conservation law (2.9) where the functions ${\text{O\u0327}}^{\alpha}=({\text{O\u0327}}^{1},\dots ,{\text{O\u0327}}^{m})$ are the associated characteristics of the conservation law (2.9).
Definition 4 [12]
which are called Euler-Lagrange equations and the Lagrangian L is called a standard Lagrangian. However, if $\frac{\delta L}{\delta {u}^{\alpha}}\ne 0$, the Lagrangian L is called as a partial Lagrangian and the corresponding differential equations are called partial Euler-Lagrange equations.
3 Noether symmetries of a fin equation
where K and H are thermal conductivity and heat transfer coefficient, respectively, which are considered as functions of temperature, and $y=y(x)$ is the temperature function and x is a dimensional spatial variable. The Lie point symmetries equation (3.1) is investigated in the reference [18]. In this study, we consider the partial Noether approach to analyze Noether symmetries of equation (3.1).
which is a differential equation including unknown functions $K(y)$, $H(y)$, $a(x)$ and $b(x)$. Using equations (3.16)-(3.18), one can classify Noether symmetries and the corresponding first integrals of nonlinear fin equation (3.1) based on different forms of the thermal conductivity $K(y)$ and the heat transfer coefficient $H(y)$ and differential relations for $a(x)$ and $b(x)$.
Case 1: $K(y)=k(\mathit{constant})$.
Now we analyze differential equation (3.19) for different $H(y)$ functions corresponding to different solutions of (3.18), and we get differential relations between functions $a(x)$ and $b(x)$, which give Noether symmetries and the corresponding first integral for each case.
Case 1.1: $H(y)=h(\mathit{constant})$.
Case 1.2: $H(y)=y$.
Case 1.3: $H(y)={y}^{n}$, $n>1$.
Case 1.4: $H(y)=Exp(y)$.
Case 1.5: $H(y)=\frac{1}{my+n}$.
Case 1.6: Arbitrary function $H(y)$.
Case 2: $K(y)=kExp(\alpha y)$, k and β are constants.
and consider the following cases as the solutions of (3.42), and we get the mathematical relations between functions $a(x)$ and $b(x)$.
Case 2.1: $H(y)=h(\mathit{constant})$.
Case 2.2: Arbitrary $H(y)$.
Case 2.3: $H(y)=\frac{h}{{(\beta y+\gamma )}^{2}}$, β and γ are arbitrary constants.
Case 3: $K(y)=k{y}^{\beta}$, $\beta \ne -1$.
Case 3.1: $K(y)=k{y}^{\beta}$, $\beta =-1$.
4 Invariant solutions
Some group invariant solutions of nonlinear fin equation (3.1) can be constructed from the Noether symmetries and the first integrals. In this section, we consider some different special cases to present invariant solutions of (3.1).
where c, ${c}_{1}$ are constants.
where c, ${c}_{2}$ are constants. This solution (4.4) is the group invariant solution that satisfies the original fin equation (3.1).
where ${c}_{4}$ is constant, which satisfies fin equation (3.1).
5 Concluding remarks
Noether symmetry classification table of fin equation
Thermal conductivity | Heat transfer coefficient | Infinitesimals and first integrals |
---|---|---|
k(constant) | H(y) | $\xi ={k}^{2}{c}_{1}$, η = 0, $I=k\int H(y)\phantom{\rule{0.2em}{0ex}}dy-\frac{1}{2}{k}^{2}{y}^{\mathrm{\prime}2}$ |
k(constant) | h | $\xi ={k}^{2}({c}_{1}+x{c}_{2}+{x}^{2}{c}_{3})$, $\eta =\frac{1}{2}{k}^{2}y({c}_{2}+2x{c}_{3})+k(\frac{3}{4}h{x}^{2}{c}_{2}+\frac{1}{2}h{x}^{3}{c}_{3}+{c}_{4}+x{c}_{5})$, ${I}_{1}=hky-\frac{1}{2}{k}^{2}{y}^{\mathrm{\prime}2}$, ${I}_{2}=\frac{1}{8}(-2{h}^{2}{x}^{3}-4hkxy+2k(3h{x}^{2}+2ky){y}^{\prime}-4{k}^{2}x{y}^{\mathrm{\prime}2})$, ${I}_{3}=\frac{1}{8}(-{h}^{2}{x}^{4}-4hk{x}^{2}y-4{k}^{2}{y}^{2}+2k(2h{x}^{3}+4kxy){y}^{\prime}-4{k}^{2}{x}^{2}{y}^{\mathrm{\prime}2})$, ${I}_{4}=-hx+k{y}^{\prime}$, ${I}_{5}=-\frac{1}{2}h{x}^{2}-ky+kx{y}^{\prime}$ |
k(constant) | $\frac{1}{my+n}$ | $\xi ={k}^{2}{c}_{1}$, η = 0, $I=\frac{klog(k(n+my))}{m}-\frac{1}{2}{k}^{2}{y}^{\mathrm{\prime}2}$ |
k(constant) | ${e}^{y}$ | $\xi ={k}^{2}{c}_{1}$, η = 0, $I={e}^{y}k-\frac{1}{2}{k}^{2}{y}^{\mathrm{\prime}2}$ |
k(constant) | y | $\xi ={k}^{2}(\frac{1}{2}{e}^{\frac{-2x}{\sqrt{k}}}\sqrt{k}({e}^{\frac{4x}{\sqrt{k}}}{c}_{1}-{c}_{2})+{c}_{3})$, $\eta =\frac{1}{2}{k}^{2}y(2{e}^{\frac{2x}{\sqrt{k}}}{c}_{1}-{e}^{\frac{-2x}{\sqrt{k}}}({e}^{\frac{4x}{\sqrt{k}}}{c}_{1}-{c}_{2}))+k({e}^{\frac{x}{\sqrt{k}}}{c}_{4}+{e}^{\frac{-x}{\sqrt{k}+{c}_{5}}})$, ${I}_{1}=\frac{1}{4}{e}^{\frac{2x}{\sqrt{k}}}(-{k}^{\frac{3}{2}}{y}^{2}+2{k}^{2}y{y}^{\prime}-{k}^{\frac{5}{2}}{({y}^{\prime})}^{2})$, ${I}_{2}=\frac{1}{4}{e}^{\frac{-2x}{\sqrt{k}}}({k}^{\frac{3}{2}}{y}^{2}+2{k}^{2}y{y}^{\prime}+{k}^{\frac{5}{2}}{({y}^{\prime})}^{2})$, ${I}_{3}=\frac{1}{2}(k{y}^{2}-{k}^{2}{({y}^{\prime})}^{2})$, ${I}_{4}={e}^{\frac{x}{\sqrt{k}}}(k{y}^{\prime}-\sqrt{k}y)$, ${I}_{5}={e}^{\frac{-x}{\sqrt{k}}}(\sqrt{k}y+k{y}^{\prime})$ |
k(constant) | ${y}^{n}$ | $\xi ={e}^{2y\alpha}{k}^{2}{c}_{1}$, η = 0, $I=\frac{2k{y}^{1+n}-{k}^{2}(1+n){({y}^{\prime})}^{2}}{2(1+n)}$ |
k Exp(αy) | H(y) | $\xi ={e}^{2y\alpha}{k}^{2}{c}_{1}$, η = 0, $I=k({e}^{2\alpha y}\int {e}^{-\alpha y}H(y)\phantom{\rule{0.2em}{0ex}}dy-2\alpha \int {e}^{2\alpha y}(\int {e}^{-\alpha y}H(y)\phantom{\rule{0.2em}{0ex}}dy)\phantom{\rule{0.2em}{0ex}}dy-\frac{1}{2}{e}^{2\alpha y}k{y}^{\mathrm{\prime}2})$ |
k Exp(αy) | h(constant) | $\xi ={e}^{2y\alpha}{k}^{2}({c}_{1}+x{c}_{2}+{x}^{2}{c}_{3})$, $\eta =\frac{{e}^{2y\alpha}{k}^{2}({c}_{2}+2x{c}_{3})}{2\alpha}+{e}^{y\alpha}k(\frac{3}{4}h{x}^{2}{c}_{2}+\frac{1}{2}h{x}^{3}{c}_{3}+{c}_{4}+x{c}_{5})$, ${I}_{1}=\frac{{e}^{\alpha y}hk}{\alpha}-\frac{1}{2}{e}^{2\alpha y}{k}^{2}{\alpha}^{2}{({y}^{\prime})}^{2}$, ${I}_{2}=-\frac{2{e}^{y}hkx\alpha +{h}^{2}{x}^{3}{\alpha}^{2}-{e}^{\alpha y}k\alpha (2{e}^{\alpha y}k+3h{x}^{2}\alpha ){y}^{\prime}+2{e}^{2\alpha y}{k}^{2}x{\alpha}^{2}{({y}^{\prime})}^{2}}{4{\alpha}^{2}}$, ${I}_{3}=-\frac{1}{8{\alpha}^{2}}(4{e}^{2\alpha y}{k}^{2}+4{e}^{\alpha y}hk{x}^{2}\alpha +{h}^{2}{x}^{4}{\alpha}^{2}-2{e}^{\alpha y}k\alpha (4{e}^{\alpha y}kx+2h{x}^{3}\alpha )({y}^{\prime})+4{e}^{2\alpha y}{k}^{2}{x}^{2}{\alpha}^{2}{({y}^{\prime})}^{2})$, ${I}_{4}={e}^{\alpha y}k{\alpha}^{2}{y}^{\prime}-hx$, ${I}_{5}={e}^{\alpha y}kx{y}^{\prime}-\frac{1}{2}h{x}^{2}-\frac{{e}^{\alpha y}k}{\alpha}$ |
k Exp(αy) | $\frac{h}{{(\beta y+\gamma )}^{2}}$ | $\xi ={e}^{2y\alpha}{k}^{2}{c}_{1}$, η = 0, $I=\frac{1}{2{\beta}^{2}(\gamma +\beta y)}{e}^{-\frac{\alpha \gamma}{\beta}}(-2({e}^{\frac{\alpha \gamma}{\beta}+\alpha y}hk\beta -hk\alpha ExpIntegralEi(\alpha (\frac{\gamma}{\beta}+y))(\gamma +\beta y)){e}^{-\frac{\alpha \gamma}{\beta}+2\alpha y}{k}^{2}{\beta}^{2}(\gamma +\beta y){({y}^{\prime})}^{2})$ |
$k{y}^{\beta}$ | h | $\xi ={k}^{2}{y}^{2}\beta ({c}_{1}+x{c}_{2}+{x}^{2}{c}_{3})$, $\eta =\frac{{k}^{2}{y}^{1+2\beta}({c}_{2}+2x{c}_{3})}{2(1+\beta )}+k{y}^{\beta}(\frac{3}{4}h{x}^{2}{c}_{2}+\frac{1}{2}{x}^{3}{c}_{3}+{c}_{4}+x{c}_{5})$, ${I}_{1}=-\frac{k{y}^{\beta}(-2hy+k(1+\beta ){y}^{\beta}{y}^{\mathrm{\prime}2})}{2(1+\beta )}$, ${I}_{2}=-\frac{1}{4(1+\beta )}((hx-k{y}^{\beta}{y}^{\prime})(h{x}^{2}(1+\beta )+2k{y}^{1+\beta}-2kx(1+\beta ){y}^{\beta}{y}^{\prime}))$, ${I}_{3}=-\frac{1}{8{(1+\beta )}^{2}}{(h{x}^{2}(1+\beta )+2k{y}^{1+\beta}-2kx(1+\beta ){y}^{\beta}{y}^{\prime})}^{2}$, ${I}_{4}=-hx+k{y}^{\beta}{y}^{\prime}$, ${I}_{5}=\frac{h{x}^{2}}{2}-\frac{k{y}^{(1+\beta )}}{1+\beta}+kx{y}^{\beta}{y}^{\prime}$ |
$k{y}^{\beta}$, β = −1 | h | $\xi =\frac{{k}^{2}}{{y}^{2}}({c}_{1}+x{c}_{2}+{x}^{2}{c}_{3})$, $\eta =\frac{1}{y}(k(\frac{3}{4}h{x}^{2}{c}_{2}+\frac{1}{2}h{x}^{3}{c}_{3}+{c}_{4}+x{c}_{5})+{k}^{2}({c}_{2}+2x{c}_{3})lny)$, ${I}_{1}=\frac{k}{2}(h+2hlny-\frac{k{y}^{\mathrm{\prime}2}}{{y}^{2}}),{I}_{2}=\frac{k{y}^{\prime}-hxy}{4{y}^{2}}((h{x}^{2}+2klny)y-2kx{y}^{\prime})$, ${I}_{3}=-\frac{1}{8{y}^{2}}{((h{x}^{2}+2klny)y-2kx{y}^{\prime})}^{2}$, ${I}_{4}=\frac{k{y}^{\prime}}{y}-hx$, ${I}_{5}=kx\frac{{y}^{\prime}}{y}-klny-\frac{h{x}^{2}}{2}$ |
Declarations
Acknowledgements
Dedicated to Professor Hari M Srivastava.
Authors’ Affiliations
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