Show that the function u(x, t)=t^-1 / 2 e^-x^2 / 4 t...

Show that the function $u(x, t)=t^{-1 / 2} e^{-x^{2} / 4 t}$ satisfies the partial differential equation $$\frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial x^{2}}$$ This equation is called the one-dimensional heat equation because it models heat diffusion in an insulated rod (with $u(x, t)$ representing the temperature at position $x$ at time $t$ ) and other similar phenomena.

Key Concepts

Partial Differential Equations (PDEs)

PDEs are equations that involve partial derivatives of an unknown function with respect to two or more independent variables. They are widely used to model a variety of physical phenomena, including heat conduction, fluid flow, and wave propagation, by relating the rates of change in different directions.

Heat Equation

The heat equation is a specific partial differential equation that models the distribution of temperature (or other diffusive substances) in a given region over time. It expresses how heat diffuses through a medium and is fundamental in both theoretical studies and practical applications in physics and engineering.

Solution Verification for PDEs

Verifying that a function is a solution to a PDE involves substituting the function into the equation to check if the equality holds. This process requires calculating the necessary partial derivatives and ensuring that the expression derived from the candidate solution meets the requirements of the differential equation.

Partial Derivatives and the Chain Rule

Partial derivatives measure how a multivariate function changes with respect to one variable while keeping the others constant. The chain rule is a key tool in this context, allowing the differentiation of composite functions by accounting for the dependencies between variables, which is essential when verifying solutions to PDEs.

Fundamental Solutions and the Gaussian Kernel

A fundamental solution to a differential equation is a solution that represents the response of the system to an initial impulse or point source. In the context of the heat equation, the Gaussian kernel is a common fundamental solution, illustrating how an initial point heat source evolves over time and spreads out due to diffusion.

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