Heaviside expansion

The next question is how to find the partial fractions in Eq. (2-25). One of the techniques is the so-called Heaviside expansion, a fairly straightforward algebraic method. We will illustrate three important cases with respect to the roots of the polynomial in the denominator (1) distinct real roots, (2) complex conjugate roots, and (3) multiple (or repeated) roots. In a given problem, we can have a combination of any of the above. Yes, we need to know how to do them all. 

The Heaviside expansion takes the following idea. Say if we multiply both sides by (s + 1), we obtain... 

We can generalize the Heaviside expansion into the fancy form for the coefficients... 

Double check a in the complex root Example 2.7 with the Heaviside expansion... 

In examples 8.3 and 8.4 Maple was used to invert from the Laplace domain to the time domain. Unfortunately, these two examples are very simple and, hence, we could invert to the time domain using Maple. For practical problems, inversion is not straightforward. The inversion to the time domain can be done in two different ways. In section 8.1.4, short time solutions will be obtained by converting the solution in Laplace domain to an infinite series. In section 8.1.5, a long time solution will be obtained by using the Heaviside expansion theorem. 

The short time solutions obtained in section 8.1.4 (examples 8.1.5 and 8.1.6) require only a few terms in the infinite series at short times to converge. However, at long times the series requires a large number of terms and cannot be used efficiently. The long time solution can be obtained using Heaviside expansion theorem.[l] If we denote the solution obtained in the Laplace domain as F(s) ... 

Differential Equations - Heaviside Expansion Theorem for Multiple Roots... 

Often inversion to time domain solution is not trivial and the time domain involves an infinite series. In section 8.1.4 short time solution for parabolic partial differential equations was obtained by converting the solution obtained in the Laplace domain to an infinite series, in which each term can easily inverted to time domain. This short time solution is very useful for predicting the behavior at short time and medium times. For long times, a long term solution was obtained in section 8.1.5 using Heaviside expansion theorem. This solution is analogous to the separation of variables solution obtained in chapter 7. In section 8.1.6, the Heaviside expansion theorem was used for parabolic partial differential equations in which the solution obtained has multiple roots. In section 8.1.7, the Laplace transform technique was extended to parabolic partial differential equations in cylindrical coordinates. In section 8.1.8, the convolution theorem was used to solve the linear parabolic partial differential equations with complicated time dependent boundary conditions. For time dependent boundary conditions the Laplace transform technique was shown to be advantageous compared to the separation of variables technique. A total of fifteen examples were presented in this chapter. 

Using partial-fractions expansion (also called Heaviside expansion) a sum of terms is obtained ... 

Method 3. The fastest and most popular method is called the Heaviside expansion. In this method multiply both sides of the equation by one of the denominator terms and then set s = -6/, which causes all... 

To find in (3-54), the Heaviside rule cannot be used for multiplication by s + 2), because 5 = -2 causes the second term on the right side to be unbounded, rather than 0 as desired. We therefore employ the Heaviside expansion method for the other two coefficients (tt2 and a3) that can be evaluated normally and then solve for by arbitrarily selecting some other value of 5. Multiplying (3-54) by (5 + if and letting 5 = -2 yields... 

However, the coefficients and 2 must be found by solving simultaneous equations, rather than by the Heaviside expansion, as shown as follows in Example 3.4. 

---