A thin rectangular plate, 0 ≤x ≤a, 0 ≤y ≤b, with negligible heat loss from its sides, has a line

step 1 In the problem, for example, for c equals 75, we get 100 minus 50x minus 25 y equal 75. So this

A thin rectangular plate, 0 ≤x ≤a, 0 ≤y ≤b, with negligible...

A thin rectangular plate, $0 \leq x \leq a, 0 \leq y \leq b$, with negligible heat loss from its sides, has a linear temperature variation along the edge at $y=b$ given by $T=20+100(x / a)^{\circ} \mathrm{C}$. The other three edges are maintained at $20^{\circ} \mathrm{C}$. Determine the temperature distribution $T(x, y)$.

Key Concepts

Laplace's Equation

In steady-state heat conduction in a homogeneous medium, the temperature distribution in a region without internal heat sources is governed by Laplace's equation. This partial differential equation states that the sum of the second spatial derivatives of the temperature is zero, reflecting the balance of heat flow in all directions within the material.

Boundary Conditions

Boundary conditions specify the temperature values along the edges of the domain and are essential in uniquely determining the solution to the differential equation. In problems like this, fixed temperature values (Dirichlet conditions) on the boundaries constrain the solution, and any non-homogeneous edge conditions often require special treatment, such as expanding in a Fourier series.

Separation of Variables

Separation of variables is a mathematical method used to solve partial differential equations by assuming that the solution can be written as a product of functions, each depending on a single coordinate. This technique simplifies Laplace's equation into ordinary differential equations that can be solved individually and then recombined to satisfy the overall boundary conditions.

Fourier Series Method

The Fourier series method is employed to represent non-homogeneous boundary conditions as an infinite sum of sinusoidal functions. This expansion is particularly useful in solving PDEs with complex boundary conditions, enabling the matching of a linear or other variable temperature distribution along one edge by decomposing it into its harmonic components.

Steady-State Heat Conduction

Steady-state heat conduction refers to a condition where the temperature distribution within a body does not change over time because the rate of heat entering any region equals the rate of heat leaving it. This concept justifies the use of Laplace's equation, as the time derivative of temperature is zero in these equilibrium conditions.

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