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Applications of the Laplace transform in solving integral equations. Laws for convolution. Abel’s integral equation. The tautochrone problem. Integro-differential equations. Conversion of linear differential equations into integral equations.

Theorem 1. Let

where a
x
b and f is assumed to be integrable on [a, b]. Then the function* *F(x) is continuous
and F'(x) = f(x) at each point where f(x) is continuous.

Theorem 2. Let L[F(t)] = f(s). Then

Proof. Let

Then by Theorem 1 we have

and

3) G"(t) = F(t).

Since G(0) = G'(0) = 0, we get

4) L[G"(t)] = s^{2} L[G(t)] - sG(0) - G'(0) = s^{2} L[G(t)]

Since from 3), G"(t) = F(t), we have

5) L[G"(t)] = f(s)

From 4) and 5) we obtain

6) s^{2} L[G(t)] = f(s)

Consequently,

or

QED

This result can also be written as

and can be generalized in the following theorem

Theorem 3. Let L[F(t)] = f(s). Then

Theorem 4. Let L[F(t)] = f(s). Then

Proof. By the convolution theorem

Thus

By Theorem 2 above

Thus

QED

This result can be written as

and can be generalized in the following theorem

Theorem 5. Let L[F(t)] = f(s). Then

Laws for convolution

Theorem 6. Given functions F(t), G(t) and H(t), convolution obeys the following laws:

F*G = G*H Commutative law

F*(G*H) = (F*G)*H Associative law

F*(G + H) = F*G + F*H Left distributive law

(F + G)*H = F*H + G*H Right distributive law

Def. Integral equation. An integral equation is an equation in which an unknown function occurs under an integral sign. It has the general form

where F(t) and K(u, t) are known functions, a and b are either given constants or functions of t, and Y(t) is an unknown function to be determined. A function Y(t) may or may not exist that satisfies the equation.

The function K(u, t) is called the kernel or nucleus of the equation. If a and b are constants, the equation is called a Fredholm integral equation. If a is a constant and b = t, it is called a Volterra integral equation.

Def. Integral equation of convolution type. An integral equation of type

is said to be of convolution type. It can be written as

Y(t) = F(t) + K(t)*Y(t)

If we take the Laplace transform of both sides we find, assuming L[F(t)] = f(s) and L[K(t)] = k(s) both exist, that

y(s) = f(s) + k(s) y(s)

or

The required solution can then be found by inversion.

Example. Solve the integral equation

Solution. The equation can be written

Y(t) = t^{2} + Y(t)*sin t

Taking the Laplace transform and using the convolution theorem, letting y = L[Y], we get

Solving for y we get

Inverting

Abel’s integral equation. The tautochrone problem. An important integral equation of convolution type is Abel’s integral equation

where Y(u) is the unknown function to be determined, G(t) is given, and α is a constant such that 0 < α < 1. The equation is associated with a problem known as the tautochrome problem in which it is desired to find the shape of a frictionless wire lying in a vertical plane such that a bead placed on the wire slides to the lowest point in the same time regardless of where the bead is placed initially. Solution of the problem reveals the shape to be that of a cycloid. See Murray R. Spiegel. Laplace Transforms. (Schaum) for details.

Def. Integro-differential equation. An integro-differential equation is an integral equation in which various derivatives of the unknown function Y(t) can also be present. For example,

is an integro-differential equation. The solution of such equations subject to given initial conditions can often be obtained by Laplace transform methods.

Example. Solve the equation

where Y(0) = 2.

Solution. The equation can be written

Y'(t) + 5 cos 2t * Y(t) = 10

Taking the Laplace transform and letting y = L[Y], we get

which can be reduced by the method of partial fractions to

Inverting we get

Conversion of linear differential equations into integral equations. It is possible to convert a linear differential equation into an integral equation. See the following example.

Example. Convert the differential equation

Y"(t) - 3Y'(t) + 2Y(t) = 4 sin t

where Y(0) = 1, Y'(0) = -2, into an integral equation.

Solution. We give two methods.

Method 1. Let Y"(t) = V(t). Since by the definition of an integral

we obtain, after evaluating the constant c,

Similarly,

Now using Theorem 4 above we obtain

Thus the differential equation becomes

or

Method 2. We first integrate both sides of the differential equation

Y"(t) - 3Y'(t) + 2Y(t) = 4 sin t

where Y(0) = 1, Y'(0) = -2.

We thus obtain

Substituting in Y'(0) = -2 and Y(0) = 1, we get

Integrating again from 0 to t as before, and using Theorem 4, we obtain

or

References

Murray R. Spiegel. Laplace Transforms. (Schaum)

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