© Hindawi Publishing Corp. GROUP METHOD ANALYSIS OF TWO-DIMENSIONAL PLATE IN HEAT FLUX

The group transformation theoretic approach is applied to present an analytic study of the temperature distribution in a triangular plate, Ω, placed in the field of heat flux, along one boundary, in a form of polynomial functions of any degree “n.” The Laplace’s equation has been reduced to second-order linear ordinary differential equation with an appropriate boundary conditions. Exact solution has been obtained for general shape of Ω and different boundary conditions. 2000 Mathematics Subject Classification: 58J35, 54H15. 1. Introduction. The Laplace’s equation arises in many branches of physics, from which it attracts a wide band of researchers. Electrostatic potential, temperature in the case of steady state heat conduction, velocity potential in the case of steady irrotational flow of an ideal fluid, concentration of a substance that is diffusing through a solid, and the displacements of a two-dimensional membrane in equilibrium state are counter examples in which the Laplace’s equation is satisfied. The mathematical technique used in the present analysis is the parametergroup transformation. The group methods, as a class of methods lead to the reduction of the number of independent variables, were first introduced by Birkhoff [6] in 1948, where he made use of one-parameter transformation groups. In 1952, Morgan [9] presented a theory which has led to improvements over earlier similarity methods. The method has been applied intensively by Abd-el-Malek et al. [1, 2, 3, 4, 5, 7]. In this paper, we present a general procedure for applying one-parameter group transformation to the Laplace’s equation in a triangular domain. Under the transformation, the partial differential equation with boundary conditions in polynomial form, of any degree, is reduced to an ordinary differential equation with the appropriate corresponding conditions. The equation is then solved analytically for the general form of the triangular domain and boundary conditions.

1. Introduction.The Laplace's equation arises in many branches of physics, from which it attracts a wide band of researchers.Electrostatic potential, temperature in the case of steady state heat conduction, velocity potential in the case of steady irrotational flow of an ideal fluid, concentration of a substance that is diffusing through a solid, and the displacements of a two-dimensional membrane in equilibrium state are counter examples in which the Laplace's equation is satisfied.
The mathematical technique used in the present analysis is the parametergroup transformation.The group methods, as a class of methods lead to the reduction of the number of independent variables, were first introduced by Birkhoff [6] in 1948, where he made use of one-parameter transformation groups.In 1952, Morgan [9] presented a theory which has led to improvements over earlier similarity methods.The method has been applied intensively by Abd-el-Malek et al. [1,2,3,4,5,7].
In this paper, we present a general procedure for applying one-parameter group transformation to the Laplace's equation in a triangular domain.Under the transformation, the partial differential equation with boundary conditions in polynomial form, of any degree, is reduced to an ordinary differential equation with the appropriate corresponding conditions.The equation is then solved analytically for the general form of the triangular domain and boundary conditions.

Mathematical formulation.
The governing equation for the distribution of temperature T (x,y) is given by with the following boundary conditions: It is required to find the distribution of the temperature T (x,y) inside the domain Ω, defined in Figure 2.1, and the heat flux across L 3 , where (

Solution of the problem.
The method of solution depends on the application of a one-parameter group transformation to the partial differential equation (2.1).Under this transformation, the two independent variables will be reduced by one and the differential equation (2.1) transforms into an ordinary differential equation in only one independent variable, which is the similarity variable.

The group systematic formulation.
The procedure is initiated with the group G, a class of transformation of one-parameter "a" of the form G : S = C S (a)S + K S (a), (3.1)where S stands for x, y, w, and q and the C's and K's are real valued and at least differentiable in the real argument "a."

The invariance analysis.
To transform the differential equation, transformations of the derivatives of w and q are obtained from G via chain-rule operations where S stands for w and q.Equation (2.5) is said to be invariantly transformed, for some function H 1 (a), whenever q wx x + w ȳ ȳ + 2 wx qx + w qx x = H 1 (a) q w xx + w yy + 2w x q x + wq xx . ( Substitution from (3.1) into (3.3)yields = H 1 (a) q w xx + w yy + 2w x q x + wq xx , (3.4) where The invariance of (3.4) implies that ζ 1 (a) ≡ 0. This is satisfied by putting which yields Moreover, the boundary conditions (2.6) are also invariant in form, implying that Finally, we get the one-parameter group G which transforms invariantly the differential equation (2.5) and the boundary conditions (2.6).The group G is of the form (3.9)

The complete set of absolute invariants.
Our aim is to make use of group methods to represent the problem in the form of an ordinary differential equation.Then, we have to proceed in our analysis to obtain a complete set of absolute invariants.
If η ≡ η(x, y) is the absolute invariant of the independent variables, then are the two absolute invariants corresponding to w and q.The application of a basic theorem in group theory, see [8], states that a function g(x, y; w, q) is an absolute invariant of a one-parameter group if it satisfies the following first-order linear differential equation: where S i stands for x, y, w, and q, respectively, and and a 0 denotes the value of "a" which yields the identity element of the group.From (3.6), (3.7), (3.8), and (3.12), we get To obtain the absolute invariant of the dependent variables w and q, apply (3.11), we get Since q(x) and R(x) are independent of y, while η is a function of x and y, then θ(η) must be a constant, say θ(η) = 1, hence (3.17) and the second absolute invariant is 4. The reduction to an ordinary differential equation.As the general analysis proceeds, the established forms of the dependent and the independent absolute invariant are used to obtain an ordinary differential equation.Generally, the absolute invariant η(x, y) has the form given in (3.15).
Substituting from (3.15) and (3.18) into (2.5)yields For (4.1) to be reduced to an expression in the single independent invariant η, the coefficients should be constants or functions of η alone.Thus, Also, from (4.2) and (4.3), we can show that Take C 3 = 1 and C 1 = n + 1, we get C 2 = n(n + 1), hence we have Under the similarity variable η, the boundary conditions are such that the boundary L 1 or L 2 does not coincide with the vertical axis.

Analytical solution.
The solution corresponds to n ≥ 0 is hence we get ( The heat flux across L 3 is hence we get (5.4) Applying the boundary conditions (4.7), we get where (5.6) Solving (5.5) for a given value of "n," we get b 0 and b 1 .
(5.10) Similarly, we get (5.11) The system of (5.5) has a unique solution if the determinant (5.12) That is, which is satisfied if the difference between "Φ 1 /n" and the vertex angle between L 1 and L 2 is not an odd multiple of "π/(2n)."

Special cases
Case 6.1.Boundary conditions are combinations of two different degrees of polynomials.
The governing equation for the distribution of temperature T (x,y) is given by with the following boundary conditions: From the principle of superposition, write where the boundary conditions for T 1 (x, y) are and the boundary conditions for T 2 (x, y) are where q(x) = α 1 q 1 (x) + α 2 q 2 (x).(6.6) Setting n = 1 in the general solution (5.2) and (5.4), we get (6.7) Setting n = 5 in the general solution (5.2) and (5.4), we get Geometrical configuration of the problem of Case 6.2.
Hence, the analytic solutions have the form (6.9) Case 6.2.One of the boundaries is vertical.The governing equation for the distribution of temperature T (x,y) is given by with the following boundary conditions: (6.11) Write T (x,y) = w(x,y)q(y), q(y) ≡ 0 in Ω, (6.12) by which differential equation (6.10) takes the form and the boundary conditions (6.11) take the form Applying the invariant analysis, we get and the absolute invariant η(x, y) is The complete set of absolute invariants corresponding to "w" and "q" are q(y) = R(y), w(x, y) = F(η).(6.17) Substituting (6.16) and (6.17) in (6.13), with R(y) = y n+1 , we get Under the similarity variable η, the boundary conditions take the form Solution of (6.18) with the boundary conditions (6.19) is where 3. Geometrical configuration of the problem of Case 6.3.
Solving (6.21) for the given value of "n," we get both b 0 and b 1 .
The heat flux across L 3 is where (6.23) Case 6.3.The two boundary conditions are identical.The governing equation for the distribution of temperature T (x,y) is given by with the following boundary conditions: Applying the invariant analysis, we get and the absolute invariant η(x, y) is η(x, y) = y x .(6.30) The complete set of absolute invariants corresponding to "w" and "q" are q(x) = R(x), w(x, y) = F(η).(6.31) Substituting (6.30) and (6.31) in (6.27), with R(x) = x 3 , we get Under the similarity variable η, the boundary conditions take the form

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: 1, 2, 3,... }, α and β are constants.