Question

Consider the equation $$ a u_{x x}-b u_{t}+c u=0 $$ where \(a, b,\) and \(c\) are constants. (a) Let \(u(x, t)=e^{\delta t} w(x, t),\) where \(\delta\) is constant, and find the corresponding partial differential equation for \(w\). (b) If \(b \neq 0\), show that \(\delta\) can be chosen so that the partial differential equation found in part (a) has no term in \(w\). Thus, by a change of dependent variable, it is possible to reduce Eq. (i) to the heat conduction equation.

Short answer

Question: Find a new partial differential equation for the function w(x, t) after substituting u(x, t) = e^{\delta t} w(x, t) into the equation a u_{x x}-b u_{t}+c u=0. Show that if b ≠ 0, a value for δ can be chosen to remove the term w from the equation, reducing it to the heat conduction equation. Answer: After substituting u(x, t) = e^{\delta t} w(x, t) into the given equation and simplifying, we found the new partial differential equation for w(x, t): a w_{xx}(x, t) -b\delta w(x, t) -b w_t(x, t) + c w(x, t)= 0. By choosing δ such that c = b δ, the term containing w can be removed, and the equation reduces to the heat conduction equation: w_{t} = \frac{a}{b} w_{xx}.

Step-by-step solution

Setup of the problem

We are given the equation: $$ a u_{x x}-b u_{t}+c u=0 $$ where u(x, t) = e^{\delta t} w(x, t). We need to find the corresponding partial differential equation for w(x,t).

Calculate the second derivative of u with respect to x and the first derivative with respect to t

To find the second derivative of u with respect to x and the first derivative with respect to t, we compute: $$ u(x, t) = e^{\delta t} w(x, t) \\ u_t(x, t) = \delta e^{\delta t} w(x, t) + e^{\delta t} w_t(x, t) \\ u_{xx}(x, t) = e^{\delta t} w_{xx}(x, t) $$

Substitute the derivatives into the original equation

We will now substitute the derivatives of u that we found in Step 2 back into the original equation: $$ a u_{x x}-b u_{t}+c u=0 \\ a(e^{\delta t} w_{xx}(x, t)) - b(\delta e^{\delta t} w(x, t) + e^{\delta t} w_t(x, t)) + c(e^{\delta t} w(x, t))= 0 $$

Simplify the equation to find the partial differential equation for w

Simplify the equation by pulling out a factor of \(e^{\delta t}\): $$ e^{\delta t}(a w_{xx}(x, t) - b\delta w(x, t) - bw_t(x, t) + c w(x, t)) = 0 $$ Since \(e^{\delta t} \neq 0\), we can divide both sides of the equation by \(e^{\delta t}\), giving us our partial differential equation for w: $$ a w_{xx}(x, t) -b\delta w(x, t) -b w_t(x, t) + c w(x, t)= 0 \quad (a) $$

Show that if b ≠ 0, δ can be chosen so that the term containing w can be removed

If \(b \neq 0\), we can choose a value for δ such that the term with w vanishes. Let us choose \(\delta\) such that \(c = b \delta\). This allows us to rewrite the equation (a) as: $$ a w_{xx}(x, t) - b\delta w(x, t) - b w_t(x, t) + b\delta w(x, t) = 0 \\ a w_{xx}(x, t) - b w_t(x, t) = 0 $$ Since \(b \neq 0\), we can divide by b to get the heat conduction equation: $$ w_{t} = \frac{a}{b} w_{xx} $$ Thus, by choosing δ such that \(c = b \delta\), the partial differential equation found in part (a) reduces to the heat conduction equation.

Key concepts

Heat Conduction Equation

The heat conduction equation is a standard partial differential equation (PDE) that describes how heat propagates through a given medium over time. It is also known as the diffusion equation. This equation is vital in understanding thermal conductivity and various heat transfer processes. The mathematical form is typically given by:\[ u_t = rac{a}{b} u_{xx} \]Here, the term \(u(x, t)\) represents the temperature distribution in the medium as a function of space \(x\) and time \(t\). The constants \(a\) and \(b\) are related to the physical properties of the material, such as thermal conductivity and specific heat capacity. This equation is characterized by its time derivative \(u_t\) and spatial second derivative \(u_{xx}\), signifying the rate of change of temperature and spatial distribution of heat, respectively. The presence of the second spatial derivative indicates it is a second-order PDE, focusing on the curvature of the temperature profile. This particular form of the heat equation, derived in Step 5 of the original solution, shows how we can manipulate and simplify PDEs by choosing appropriate transformations or parameter values, such as \(\delta\) in this case, to reveal fundamental behaviors of physical systems.

Dependent Variable Transformation

A transformation of dependent variables is a crucial technique when working with partial differential equations. This method involves changing the form of the dependent variable to simplify the equation or show it in a form that's easier to solve or interpret.In the exercise, the dependent variable transformation used was:\[ u(x, t) = e^{\delta t} w(x, t) \]This transformation introduces a new variable, \(w(x, t)\), into the equation, helping to simplify it by handling specific terms more easily. By changing \(u\) to a product of an exponential function and a new function, the problem becomes more accessible to solve under certain conditions, especially when dealing with terms involving time-dependent exponential growth or decay.Transformations like these offer flexibility in mathematical modeling. They help isolate specific patterns or behaviors (e.g., exponential growth with respect to \(t\)) from the broader problem, allowing for clearer analysis. In this solved exercise, such a transformation is leverage to create a PDE for \(w\) that can ultimately be converted to a more well-known form, such as the heat conduction equation.

Constant Coefficient Equation

In mathematics and physics, constant coefficient equations involve equations where the coefficients (such as \(a\), \(b\), and \(c\) present in the original exercise) are constants, not dependent on the independent variables (here, \(x\) and \(t\)).These equations are significant due to their relative simplicity and well-documented solution methods. The PDE we start with:\[a u_{xx} - b u_{t} + c u = 0\]is a constant coefficient equation, meaning terms multiplying derivatives or functions are constant. This property is advantageous because it allows the use of well-established mathematical techniques and transforms, like Fourier or Laplace transforms, for solving the equation. Furthermore, it regularly appears in uniform media, where properties such as thermal and electrical conductivity do not vary with position or time.Such equations are pivotal in theoretical and applied scenarios across scientific disciplines, providing insight into behaviors like wave propagation, heat conduction, and quantum mechanics, by reducing the complexity associated with variable coefficients.