The function \(u(x, \tau)\) satisfies the following free boundary problem with free boundary \(x=x_{f}(\tau)\), where \(x_{f}(0)=0\) : $$\begin{gathered} \frac{\partial u}{\partial \tau}=\frac{\partial^{2} u}{\partial x^{2}}, \quad 00 \end{gathered}$$ Show that there is a similarity solution with \(u(x, \tau)=u^{*}(x / \sqrt{\tau})\) \(x_{f}(\tau)=\xi_{0} \sqrt{\tau}\), where \(\xi_{0}\) satisfies the transcendental equation $$\frac{1}{2} \xi_{0} e^{\xi_{0}^{2} / 4} \int_{0}^{\xi_{0}} e^{-s^{2} / 4} d s=1$$ (This is a solution to the Stefan problem, in which \(u(x, \tau)\) models the temperature in a pure material that melts at temperature \(u=0\); here a semi- infinite bar of the material is initially solid at the melting temperature, and melting is initiated by raising the temperature at \(x=0\) to 1 and holding it there, so that the region \(0

Step by step solution

01

Assume a Similarity Solution

To find a similarity solution, we assume that the solution can be represented in the form of a single variable function, where the physical variables x and t are combined into a single dimensionless variable. The form of the solution is given as \(u(x, \tau)=u^{*}(\frac{x}{\sqrt{\tau}})\), where \(u^{*}\) is a function to be determined.

02

Derive Conditions for \(u^{*}\)

We apply the given boundary conditions to the assumed solution form to derive the conditions that \(u^{*}\) must satisfy. The condition at \(x=0\), \(u(0, \tau)=1\), implies that \(u^{*}(0) = 1\). For the condition at \(x=x_{f}(\tau)\), where \(x_{f}(\tau)=\xi_{0} \sqrt{\tau}\), the assumed solution becomes \(u(x_{f}(\tau), \tau) = u^{*}(\xi_{0}) = 0\). The condition for the heat flux can be also expressed in terms of \(u^{*}\).

03

Transform the Partial Differential Equation

Using the change of variables and applying the Chain Rule, transform the partial differential equation (PDE) into an ordinary differential equation (ODE) in terms of \(u^{*}\) and \(\xi\).

04

Solve the ODE for \(u^{*}\)

Solve the obtained ODE to find the function \(u^{*}\) that satisfies the transformed boundary conditions.

05

Apply the Free Boundary Condition

Use the condition that the heat flux into the free boundary is balanced by the latent heat uptake to find the relation between \(\xi_0\) and \(\tau\). This gives the transcendental equation that \(\xi_{0}\) must satisfy.

06

Validate the Transcendental Equation

Verify that the transcendental equation \(\frac{1}{2} \xi_{0} e^{\xi_{0}^{2} / 4} \int_{0}^{\xi_{0}} e^{-s^{2} / 4} d s=1\) is indeed satisfied by the obtained solution for \(\xi_{0}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Similarity Solution

In the study of differential equations, a similarity solution drastically simplifies complex problems. It is a technique that reduces the number of independent variables by combining them into a single dimensionless variable. This method is particularly useful in problems that exhibit self-similar behavior, where the shape of the solution remains unchanged over time, merely scaling in size. For the Stefan problem, to arrive at a similarity solution, one assumes that the temperature profile can be described by a function that only depends on a ratio involving the spatial coordinate and the square root of time, often denoted as \( u^*(\frac{x}{\sqrt{\tau}}) \). Such a solution reduces a partial differential equation (PDE) to an ordinary differential equation (ODE), which is typically easier to solve.

Free Boundary Problem

The Stefan problem is an example of a free boundary problem, where the boundary of the domain is not fixed but instead moves with time, and is part of the solution. In our particular scenario, the free boundary \( x_f(\tau) \) represents the moving interface between the solid and liquid phases of the material. The movement of this boundary depends on the heat absorbed by the melting process and must satisfy the condition that the temperature at the boundary stays at the melting point, here given as zero. Solving free boundary problems usually requires determining both the solution to the differential equation and the equation governing the boundary's motion, making them more complex than fixed boundary problems.

Partial Differential Equation

A partial differential equation (PDE) is an equation involving functions of several variables and their partial derivatives. PDEs are used to formulate problems involving functions of several variables and are either solved by hand, or used to create a relevant computer model. They play a pivotal role in several disciplines including physics, engineering, and finance. In the Stefan problem under discussion, the heat equation \( \frac{\partial u}{\partial \tau} = \frac{\partial^2 u}{\partial x^2} \) is a second-order PDE that describes the diffusion of heat in the material.

Ordinary Differential Equation

Upon employing a similarity solution in a PDE, we can often convert it into an ordinary differential equation (ODE). An ODE contains derivatives with respect to only one variable, as opposed to the multiple variables present in a PDE, making it simpler to tackle. In context, the Stefan problem's PDE is transformed into an ODE in terms of the similarity variable and the function \( u^* \). This simplifies the problem and allows us to isolate \( u^*(\xi) \) which describes the temperature profile in terms of the similarity variable, \( \xi \) being \( \frac{x}{\sqrt{\tau}} \).

Transcendental Equation

A transcendental equation contains functions that are not algebraic, such as exponential, trigonometric, or logarithmic functions. These equations often arise in the context of free boundary problems, representing a condition that must be met by certain parameters of the system. In our exercise, the transcendental equation involving the similarity variable \( \xi_0 \) takes the form of an integral equation. This particular equation dictates the movement of the free boundary in the Stefan problem and does not admit a simple closed-form solution. Its verification and solution are typically approached via numerical methods, which allow us to predict the evolution of the interface in the physical phenomena modeled by the equation.