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Chapter: Mathematics (maths) - Applications of Partial Differential Equations

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Solution of Laplace’s equation (Two dimensional heat equation)

Let u = X(x) . Y(y) be the solution of (1), where „X‟ is a function of „x‟ alone and „Y‟ is a function of „y‟ alone.

Solution of Laplace’s equation  (Two dimensional heat equation)

 

The Laplace equation is



Let u = X(x) . Y(y) be the solution of (1), where „X‟ is a function of „x‟ alone and „Y‟ is  a function of „y‟ alone.



Now the left side of (2) is a function of „x‟ alone and the right side is a function of „t‟ alone. Since „x‟ and „t‟ are independent variables, (2) can be true only if each side is equal to a constant.

 



Solving equations (3), we get

 

  (i) when „k‟ is positive and k = l2, say

 

X = c1 elx  + c2  e - lx

Y = c3 cosly + c4 sin ly

 


 

(iii)               when   „k‟   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 = ℓ and an end at right angles to them. The breadth of this edge y = 0 is ℓ 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 0°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 < ¥

(ii) u (8, y) = 0,     for 0 < y < ¥

(iii) u (x, ¥) = 0,   for 0 < x < 8

(iv) u (x, 0) = 100 Sin (¥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 0°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 < ¥

 

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

 

(iii) u (x, ¥) = 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 <<b

 

iii.           u(x,b) = 0 for 0 <<a

iv.          u(x,0) = sin3(px/ a) ,0 <<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 ££l   ii.   u(l,y) = 0, 0 ££

 

iii.         u(x,0) = 0, 0 ££l iv.  u(x,l) = f(x), 0 ££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 –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 maintained0‟atat 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 „l‟ 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 <<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 <<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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