Solutions of heat, wave, and Laplace's equations | Engineering Mathematics for Mechanical Engineering PDF Download

Wave Equation

Solution of Wave Equation

Wave equation is 

Let its solution be
y(x, t) = X(x) T(t)                                                               ….(2)
Substituting from Eq. (2) in Eq. (1), we get

Since, x and t are independent variables and left side of Eq. (3) is a function of t alone while right side is a function of x, Eq. (3) will be true only when each side is equal to a constant : 

When constant is negative (-k²)

∴ Solution is X =
Also,

∴ Solution is T = C cos kt + D sin kt
∴ Solution for y : y(x, t) = X(x) T(t)

When constant is zero

Similarly, T = (Ct + D)
Solution for y = y(x, t) = X(x) T(t) 

When constant is positive (k²)

Solution is
Solution for y : y(x, t) = X(x) T(t)

Heat Equation

Solution of the Heat Equation

The heat equation is

Let u(x, t) = X(x) T(t)                                                               ….(2)
where, X is a function of x only and T is a function of t only be a solution of Eq. (1). Then,

Substituting in Eq. (1),

Now, LHS of Eq. (3) is a function of x only and the RHS is a function of t only. Since, x and t are independent variable, this equation can hold only when each side is equal to a constant.

When constant is negative (-k²) 

Solution is X = A cos kx + B sin kx
Also,

Solution for u :
u(x, t) = X(x) T(t)

When constant is zero

When constant is positive (k²)

Solution for u :
u(x, t) = X(x) T(t)

Laplace Equation

Laplace equation in three dimensions is 

Solution of Laplace Equation in Two Dimensions

Laplace equation is

Let u(x, y) = X(x) Y(y)                                                      …(2)
where, X is a function of x only and y is a function of y only be a solution of Eq. (1). 

Substituting in Eq. (1) we have X"Y + XY" = 0

Now, LHS of Eq. (3) is a function of x only and the RHS is a function of y only. Since, x and y are independent variable, this equation can hold only when each side is equal to a constant.

When constant is negative (-k²)

Solution is X = A cos kx + B sin kx
Also,

Solution for u : u(x, y) = X(x) Y(y) 

When constant is zero

Similarly, Y = Cy + D
Solution for u : u(x, y) = X(x) Y(y)

When constant is positive (k²)

Solution for u :
u(x, y) = X(x) Y(y)