---
title: "Solve one dimensional heat equation"
subject: "Mathematics III"
chapter: 4
key: "semester_3/mathematics_iii:solve-one-dimensional-heat-equation-two"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/solve-one-dimensional-heat-equation-two
site: BCT Engineering
---

# Solve one dimensional heat equation

Mathematics III · Chapter 4: Fourier Series

- Asked in: 2082 Kartik (new course), 4 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/solve-one-dimensional-heat-equation-two
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-4

## Question

Derive the two dimensional heat equation for steady state heat flow.

## Short answer

Heat balance on a thin insulated plate gives ∂u/∂t = (K/(ρs))(∂^2u/∂x^2 + ∂^2u/∂y^2). In steady state ∂u/∂t = 0, so temperature satisfies Laplace's equation: ∂^2u/∂x^2 + ∂^2u/∂y^2 = 0.

## Answer

The steady-state heat equation for a plate is Laplace's equation: ∂²u/∂x² + ∂²u/∂y² = 0.

**Derivation.** Take a thin rectangular plate of thickness δz, density ρ, specific heat s and thermal conductivity K. The faces are insulated, so heat flows only in the xy-plane. Consider a small element with sides δx and δy, and let u(x, y, t) be the temperature.

Heat flow rate across a face is −K × (area) × (temperature gradient normal to the face).

```text
        ↓ heat flow in y
     ┌────────────┐
 →   │  δx × δy   │  →   heat flow in x
     └────────────┘
        ↓
```

- **Along x.** Heat entering through the left face: −K δy δz (∂u/∂x)ₓ. Heat leaving through the right face: −K δy δz (∂u/∂x)ₓ₊δₓ. Net gain = K δy δz [(∂u/∂x)ₓ₊δₓ − (∂u/∂x)ₓ] ≈ K δx δy δz ∂²u/∂x².
- **Along y.** In the same way the net gain = K δx δy δz ∂²u/∂y².

Total net gain per unit time = K δx δy δz (u_xx + u_yy).

This equals the rate at which the heat content increases, ρ s δx δy δz ∂u/∂t. Hence

∂u/∂t = c² (∂²u/∂x² + ∂²u/∂y²),  c² = K/(ρs)

**Steady state.** In the steady state the temperature does not change with time, so ∂u/∂t = 0:

**∂²u/∂x² + ∂²u/∂y² = 0**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The plate faces are insulated so heat flows:
   A. in z too
   B. only in the xy-plane
   C. only along x
   D. radially
   Answer: B. Stated. (Key point: Assumptions)

2. Heat flow rate across a face is:
   A. K × area
   B. −K × volume
   C. −K × area × normal temperature gradient
   D. ρ s × area
   Answer: C. Fourier's law. (Key point: Flow)

3. The unsteady equation is:
   A. ∂u/∂t = K u_xx
   B. ∂u/∂t = c(u_x + u_y)
   C. c^2 = ρs/K
   D. ∂u/∂t = c^2(u_xx + u_yy), c^2 = K/(ρs)
   Answer: D. Net gain equals ρsδxδyδz u_t. (Key point: Equation)

4. Steady state gives:
   A. u_xx + u_yy = 0
   B. u_t = u_xx
   C. u = 0
   D. u_xx = u_yy
   Answer: A. u_t = 0. (Key point: Steady)

## Related in Mathematics III

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