Question

Lassify the following PDEs into linear/nonlinear, homogeneous/inhomogeneous:

a) One dimensional heat equation,
∂u/∂t=∂²ₓ u
b) One dimensional heat equation on [0,1] with varying diffusion,

∂u/∂t=x(x−1)∂²ₓ u

Answer 1

The one-dimensional heat equation is a linear, homogeneous PDE, while the one-dimensional heat equation with varying diffusion is a nonlinear, homogeneous PDE.

Step-by-step explanation:

The student asked to classify the following partial differential equations (PDEs):

1. The one-dimensional heat equation ∂u/∂t=∂²ₓ u, which is a linear, homogeneous PDE.
2. The one-dimensional heat equation on [0,1] with varying diffusion ∂u/∂t=x(x−1)∂²ₓ u, which is a nonlinear, homogeneous PDE.

A linear PDE is an equation where the unknown function and its derivatives appear to the first power and are not multiplied by each other. A nonlinear PDE has terms that are of higher power or are products of the unknown function or its derivatives. Homogeneity in a PDE means that if the function u is a solution, then a constant multiple of the function is also a solution, and there are no terms free of the function or its derivatives.

The first equation does not have any coefficients that depend on x or u, and therefore it is linear. Since there are no non-zero terms on the right-hand side that are independent of the function u, it is also homogeneous. In contrast, the second equation includes a term x(x-1) multiplying the second derivative which introduces nonlinearity, as the coefficient depends on the spatial variable x. Nonetheless, it remains homogeneous since no additional terms independent of u are added to the equation.

Answer 2

a) The one-dimensional heat equation, ∂u/∂t=∂²ₓu, is a linear and homogeneous partial differential equation (PDE).

b) The one-dimensional heat equation on [0,1] with varying diffusion, ∂u/∂t=x(x−1)∂²ₓu, is a linear and inhomogeneous partial differential equation (PDE).

Step-by-step explanation:

In the one-dimensional heat equation (∂u/∂t=∂²ₓu), the presence of the first and second derivatives with respect to time (t) and space (x) implies a linear relationship between the dependent variable u and its derivatives. Linearity is a fundamental property that allows for the superposition of solutions, making it easier to analyze and solve. The absence of any additional terms involving u or its derivatives confirms homogeneity, indicating that the equation is not influenced by external forces or sources.

Now, considering the one-dimensional heat equation on [0,1] with varying diffusion (∂u/∂t=x(x−1)∂²ₓu), the linear nature persists due to the first and second derivatives, but the presence of the term x(x−1) introduces a dependency on the spatial variable x. This term, dependent on x, makes the equation inhomogeneous. The inhomogeneity indicates the influence of external factors, represented by the variable x(x−1), affecting the evolution of the temperature distribution. Despite this inhomogeneity, linearity is maintained because the equation remains linear with respect to the unknown function u and its derivatives.

In summary, the classification of these partial differential equations as linear/nonlinear and homogeneous/inhomogeneous is essential for understanding their behavior and facilitating the appropriate mathematical methods for their solution. The linear and homogeneous nature of the standard one-dimensional heat equation contrasts with the linear and inhomogeneous characteristics of the modified version with varying diffusion.