Home | | **Transforms and Partial Differential Equations** | Solution of Laplaceâ€™s equation (Two dimensional heat equation)

Let u = X(x) . Y(y) be the solution of (1), where â€žXâ€Ÿ is a function of â€žxâ€Ÿ alone and â€žYâ€Ÿ is a function of â€žyâ€Ÿ alone.

**Solution of Laplaceâ€™s equation (Two dimensional heat equation)**

The Laplace equation is

Let u = X(x) . Y(y) be the solution of (1), where â€žXâ€Ÿ is a function of â€žxâ€Ÿ alone and â€žYâ€Ÿ is a function of â€žyâ€Ÿ alone.

Now the left side of (2) is a function of â€žxâ€Ÿ alone and the right side is a function of â€žtâ€Ÿ alone. Since â€žxâ€Ÿ and â€žtâ€Ÿ are independent variables, (2) can be true only if each side is equal to a constant.

Solving equations (3), we get

(i) when â€žkâ€Ÿ is positive and k = l2, say

X = c1 elx + c2 e - lx

Y = c3 cosly + c4 sin ly

(iii) when â€žkâ€Ÿ is zero.

X = c9 x + c10

Y = c11 x + c12

Thus the various possible solutions of (1) are

Of these three solutions, we have to choose that solution which suits the physical nature of the problem and the given boundary conditions.

**Example 12**

An infinitely long uniform plate is bounded by two parallel edges x = 0 & x = â„“ and an end at right angles to them. The breadth of this edge y = 0 is â„“ and this edge is maintained at a temperature f (x). All the other 3 edges are at temperature zero. Find the steady state temperature at any interior point of the plate.

**Solution**

Let u (x,y) be the temperature at any point x,y of the plate.

Also u (x,y) satisfies the equation

Using (7) in (5), we get the required solution.

**Example 13**

A rectangular plate with an insulated surface is 8 cm. wide and so long compared to its width that it may be considered as an infinite plate. If the temperature along short edge y = 0 is u(x,0) = 100 sin (px/8), 0 __<__ x __<__ 8, while two long edges x = 0 & x = 8 as well as the other short edges are kept at 0Â°C. Find the steady state temperature at any point of the plate.

**Solution**

The two dimensional heat equation is given by

The boundary conditions are

(i) u (0, y) = 0, for 0 < y < Â¥

(ii) u (8, y) = 0, for 0 < y < Â¥

(iii) u (x, Â¥) = 0, for 0 < x < 8

(iv) u (x, 0) = 100 Sin (Â¥x/8,) for 0 < x < 8

Using conditions (i), & (ii), we get

Comparing like coefficients on both sides, we get

D1 = 100, D2 = D3 = . . . . = 0

Substituting in (4), we get

u (x,y) = 100 e(-py / 8) sin (px / 8)

**Example 14**

A rectangular plate with an insulated surface 10 c.m wide & so long compared to its width that it may considered as an infinite plate. If the temperature at the short edge y = 0 is given by

and all the other 3 edges are kept at temperature 0Â°C. Find the steady state temperature at any point of the plate.

**Solution**

The temperature function u (x,y) is given by the equation

The boundary conditions are

(i) u (0, y) = 0, for 0 __<__ y __<__ Â¥

(ii) u (10, y) = 0, for 0 __<__ y __<__ Â¥

(iii) u (x, Â¥) = 0, for 0 __<__ x __<__ 10

**Example 15**

A rectangular plate is bounded by the lines x = 0, x = a, y = 0 & y = b. The edge temperatures are u (0,y) = 0, u (x,b) = 0, u (a,y) = 0 & u (x,0) = 5 sin (5px / a) + 3 sin (3px / a). Find the steady state temperature distribution at any point of the plate.

The temperature function u (x,y) satisfies the equation

The boundary conditions are

(i) u (0,y) = 0, for 0 < y < b

(ii) u (a,y) = 0, for 0 < y < b

(iii) u (x, b) = 0, for 0 < x < a

(iv) u (x,0) = 5 sin (5px / a) + 3 sin (3px / a), for 0 < x < a.

**Exercises**

ii. u(a,y) = 0 for 0 <y <b

iii. u(x,b) = 0 for 0 <x <a

iv. u(x,0) = sin3(px*/* a) ,0 <x <a.

(2) Find the steady temperature distribution at points in a rectangular plate with insulated faces and the edges of the plate being the lines x = 0, x = a, y = 0 and y = b. When three of the edges are kept at temperature zero and the fourth at a fixed temperature ao C.

i. u(0,y) = 0, 0 Â£y Â£l ii. u(l,y) = 0, 0 Â£y Â£l

iii. u(x,0) = 0, 0 Â£x Â£l iv. u(x,l) = f(x), 0 Â£x Â£l

(6) A square plate is bounded by the lines x = 0, y = 0, x = 20 and y = 20. Its faces are insulated.

The temperature along the upper horizontal edge is given by u(x,0) = x (20 â€“x), when 0 < x <20,

while other three edges are kept at 0o C. Find the steady state temperature in the plate.

(7) An infinite long plate is bounded plate by two parallel edges and an end at right

angles to them.The breadth is p. This end is maintained0â€Ÿatat a c all points and the other edges are at zero temperature. Find the steady state temperature at

any point (x,y) of the plate.

(8) An infinitely long uniform plate is bounded by two parallel edges x = 0 and x = l, and

an end at right angles to them. The breadth of this edge y = 0 is â€žlâ€Ÿ and temperature f(x). All the other three edges are at temperature zero. Find the steady state

temperature at any interior point of the plate.

(9) A rectangular plate with insulated surface is 8 cm. wide and so long compared to its width that it may be considered infinite in length without introducing an appreciable error. If the temperature along one short edge y = 0 is given by u(x,0) = 100 sin(px*/* 8), 0 <x <8, while the two long edges x = 0 and x = 8 as well as the other short edge are kept at 0o C, show that the steady state temperature at any point of the plane is given by u(x,y) = 100 e-py*/* 8 sin px*/* 8 .

(10) A rectangular plate with insulated surface is 10 cm. wide and so long compared to its width that it may be considered infinite length. If the temperature along short edge y = 0 is given

u(x,0) = 8 sin(px*/* 10) when 0 <x <10, while the two long edges x = 0 and x = 10 as well as the other short edge are kept at 0o C, find the steady state temperature distribution u(x,y).

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