Show that the function u(x, y, t)=t^-1 e^-(x^2+y^2) / 4 t satisfies the two-dimensional heat equ

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

Show that the function $u(x, y, t)=t^{-1} e^{-\left(x^{2}+y^{2}\right) / 4 t}$ satisfies the two-dimensional heat equation $$\frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}$$

Key Concepts

Solution Verification

Solution verification is the process of substituting a candidate function into a differential equation to check if it satisfies the equation under given conditions. This involves calculating and equating the relevant partial derivatives (with respect to both time and space) in the context of the heat equation. It is a fundamental method in the study of differential equations to confirm the correctness of proposed solutions.

Heat Equation

The heat equation is a partial differential equation that models the distribution of heat (or variations in temperature) in a given region over time. In multiple dimensions, it relates the time derivative of the temperature distribution to the spatial second-order derivatives, representing the diffusion of heat. This equation is not only fundamental in physics but also serves as a model for various diffusion processes in different fields.

Laplacian Operator

The Laplacian operator is a second-order differential operator defined as the sum of the second partial derivatives with respect to each spatial coordinate. In two dimensions, it takes the form of the sum of the second derivatives with respect to x and y, and it quantifies the rate at which the average value of a function around a point diverges from the function’s value at that point. This operator is central in formulating and solving the heat equation.

Gaussian Function

A Gaussian function, often characterized by its bell-shaped curve, appears frequently in the study of diffusion processes. Its particular properties, including symmetry and rapid decay at infinity, make it an excellent candidate for demonstrating solutions to the heat equation. Gaussian functions naturally arise as fundamental solutions (or heat kernels) in the theory of the heat equation, providing insight into how initial heat distributions evolve over time.

Differentiation Techniques (Chain Rule and Product Rule)

Verifying that a function is a solution to a differential equation involves computing its partial derivatives. In this context, the chain rule and the product rule are essential differentiation techniques used to handle composite functions and products of functions, respectively. These rules ensure correct computation of derivatives when the function has exponential and power-law factors, as seen in the Gaussian-related candidate solution.

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