This document discusses the one-dimensional heat equation and its general solution. It presents three cases for the constant in the heat equation: (1) = 0, (2) < 0, and (3) > 0. For each case, it derives the general solution and shows that the physical solution requires < 0. The document also provides an example of solving the heat equation with boundary and initial conditions and arrives at the solution as a half-range sine series.

1. G.H. Patel College of Engineering &
Technology
Subject : Advanced Engineering Mathematics (2130002)
Topic : Partial Differential Equations – One Dimensional Heat
Equation.
Prepared by :
Tejoy Vachhrajani (160110116057)
Khyati Valera (160110116059)
Bhavik Vashi (160110116061)
Submitted To : Prof. Bijal Khamar

2. HEAT EQUATION
The Heat Equation Is Mainly Used To Determine The Conduction Of
Heat Wires And Bars.
1-Dimension : =
Where = Thermal Diffusivity constant
2-Dimension Heat Equation In Steady State Condition :
+ = 0 (Laplace’s Equation)
1-Dimension Wave Equation :
=
= , T = Tangential force, = Mass of string per
unit length

3. General Solution Of Heat Equation In 1-Dimension
=
Derivation:-
Let u(x , t) = X(x)T(t) be the solution of given heat equation
It must satisfies the equation
Hence, X(x)T'(t) = X''(x)T(t)
= = ( = Constant)


4. Case 1 : = 0
Therefore, = 0
) = 0
Integrating with respect to t, We have
T(t) =
And = 0
Integrating with respect to x, We have
=
Integrating with respect to x, We have
= x +
General solution is
u(x , t) = (x + )


5. Case 2 : < 0 and = -
Therefore = -
) + X(x) = 0
It’s pure ordinary differential equation
So, (+)X = 0 D=
Now Auxiliary equation is + = 0
m = +IP, -IP
X(x) = +
And now other
= -
Integrating with respect to t, We have
= -t +
Hence T(t) =
So General solution is
u(x , t) = ( + )()

6. Case 3 : > 0 and =
=
) - X(x) = 0
It’s pure ordinary differential equation
So, (-)X = 0 D=
Now Auxiliary equation is - = 0
m = +P, -P
X(x) = +


7. And other
=
Integrating with respect to t, We have
= t +
T(t) =
So General solution is
u(x , t) = ( + )()
Since we have heat equation in that the time increase then
temperature decrease in the rod at looking at physical nature of the
problem
< 0 that is
u(x , t) = ( + )() is physible solution of given heat equation

8. QUESTION : Solve the heat equation
= subjected to boundaries condition
1) u(0,t) = 0 (i.e. zero temp. at the end x = 0)
2) u(L , t) = 0 (i.e. zero temp. at the other end x = L)
3) u(x,0) = ( - ) (0<x<L) ( = constant)
SOLUTION :
The given partially differential equation is one dimension heat
equation so General solution is
u(x , t) = ( + )() ….(1)


9. Now applying condition no. 1
u(0,t) = 0
substituting in equation 1, we have
() = 0
= 0
Substituting = 0 in equation 1, We have
u(x , t) = ()() …(2)
Now applying condition no. 2
u(L , t) = 0
substituting in equation 2, We have
()() = 0


10. = 0
PL = n n = …-2,-1,0,1,2,…
P=
substituting value of P in equation 2, We have
u(x , t) = ()()
= (= )
n = 1,2,3,…
= …(3)
Now applying condition no. 3
u(x,0) = ( - )


11. ( - ) = n = 1,2,3,… ….(4)
This form is half range sine series
Where = dx
= dx
= -
= -
=
=
=
0
L
0
L

12. Substituting value of in equation 3, we have
u(x , t)=
This is the solution of given function


13. THANK YOU