Operational Calculus

The influence of Oliver Heaviside

One of the many difficulties engineers face is finding simple methods to verify our computer calculations. We often do this by checking for ballpark estimates or benchmarking against known solutions. One technique that I have found useful is to use concepts from Operational Calculus.

Although Structural Engineers are taught some these techniques in courses such as differential equations, we often don't apply them in practice and rather prefer the techniques that originated within our field such as Bending Moment Diagrams and Virtual Work applications. We are not wrong to default to tradition, but we shouldn’t forget that there are also other methods to solve these equations.

In contrast, Electrical Engineers are probably more familiar with the concepts of Operational Calculus, because the field did not originally emerge from pure mathematics. Rather, it was developed as a practical method to solve problems they encountered in the beginning of the last century as the electrification age arrived. The person who undoubtedly made the most significant contribution to this field was Oliver Heaviside.This is why any meaningful conversation about Operational Calculus must begin with a discussion of his practical methods.

What makes his achievements even more remarkable is that, like several other great scientists and mathematicians—such as Thomas Edison and Michael Faraday—Heaviside lacked formal higher education and was largely self-taught. His work, which continues to influence modern science and engineering, remains in my view, highly underappreciated.

Who was Oliver Heaviside?

Heaviside left formal schooling at the age of 16, already possessing a strong foundation in mathematics. Although he did not attend university, he pursued independent study, becoming well-versed in algebra, geometry, and physics. He began his career as a telegraph operator and engineer, gaining practical knowledge of electrical circuits and signals—knowledge that would later form the basis of his theoretical work. Heaviside became familiar with Maxwell's equations and, recognizing their significance, resolved to study them more deeply and reformulate them in a more practical, accessible form. Additionally he is credited as one of the original developers of vector calculus. His biographer referred to him as “the forgotten genius”.

In the late 19th century, electrical engineers faced significant challenges in analyzing complex circuits and systems. At the time, the available mathematical tools were insufficient for efficiently addressing these challenges at an industrial pace.

He introduced the idea of treating differential operators algebraically. For instance, he represented the derivative operator as D, where D=d/dt. He would then go on and pretend that it didn’t say anything special about advanced concepts like differentiation, until the last step where he would invert his functions.

This allowed engineers to manipulate differential equations in a way that was simpler and more intuitive than traditional calculus. By using this method, Heaviside could solve differential equations more efficiently and with greater ease.

An Example: Solving a Differential Equation

For example, consider the differential equation:

\(% Original Differential Equation \frac{d^2 y}{dt^2} + 5 \frac{dy}{dt} + 6 y = f(t) \)

Heaviside's approach would be to treat the operator D=dy/dt like a variable in an algebraic equation. The quadratic equation for the operator becomes:

\(\\ (D^2 + 5D + 6)y = f(t) \\\)

This equation has roots at D=2 and D=3. Thus, the general solution to the differential equation is with the roots projected to its exponents:

\(\quad y(t) = C_1 e^{-2t} + C_2 e^{-3t}+D\)

While the solution to such equations through characteristic equations was known, Heaviside’s innovation lay in treating the differential operator D as an algebraic entity, making the manipulation of such equations more intuitive for engineers.

A More Complicated Problem: The Heat Equation

Now consider a more complicated problem, such as the heat equation:

\(\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2} \)

The general technique that we are still taught today is to first separate the variables and then solve using Fourier transforms. To separate the variables, we assume a solution of the form.

\(u(x,t)=X(x)T(t)\)

Substituting this into the heat equation, and dividing both sides by T(t)X(x) we get.

\(\frac{1}{\alpha T(t)} \frac{dT(t)}{dt} = \frac{1}{X(x)} \frac{d^2 X(x)}{dx^2} = -\lambda \)

This leads to two ordinary differential equations, one for T(t) and one for X(x), with λ acting as the eigenvalue.

The solution to these equations is typically written as a Fourier series, with the coefficients An and Bn determined by the boundary conditions.

\(u(x, t) = \sum_{n=1}^{\infty} \left( A_n \cos\left( \frac{n \pi x}{L} \right) + B_n \sin\left( \frac{n \pi x}{L} \right) \right) e^{-\left( \frac{n \pi}{L} \right)^2 \alpha t} \)

Heaviside approach to the heat equation

Heaviside’s approach to solving the heat equation would treat the differential operators algebraically.

For example, he might have written the heat equation as:

\(\frac{\partial }{\partial t}u(x,t) = \alpha \frac{\partial^2 }{\partial x^2}u(x,t) \)

Then, the differential operators Dt and Dx ​would be treated as follows.

\( {D_t} = \frac{\partial }{\partial t} , {D_x^2} = \frac{\partial^2 }{\partial x^2} \)

Through substituting Dt and Dx, the heat equation therefore becomes:

\({(D_t-\alpha D_x^2)}{u(t,x)=0} \)

By following the algebraic approach, we would arrive at.

\({D_t=\alpha D_x^2}\)

If we assume that there is a point at which both must be constant. This is the same as determining the eigenvalue as before.

\(D_x = -\lambda, D_t = \alpha\lambda^2.\)

With a bit of guesswork, the solution that satisfies U(t,x) is a product of exponential terms with factors related to Dx​ and Dt. This step is similar to the method of separation of variables. The only difference is that the notation is simpler.

\(u(t, x) = e^{\lambda x}e^{-\alpha \lambda^2 t} \)

If you were to expand the first term using Fourier transform:

\(e^{\lambda x} = \sum_{n=1}^{\infty} \left( A_n \cos\left( \frac{n \pi x}{L} \right) + B_n \sin\left( \frac{n \pi x}{L} \right) \right) \)

Then, it would result in the same equation as the general solution to the original heat equation.

\(u(x, t) = \sum_{n=1}^{\infty} \left( A_n \cos\left( \frac{n \pi x}{L} \right) + B_n \sin\left( \frac{n \pi x}{L} \right) \right) e^{-\left( \frac{n \pi}{L} \right)^2 \alpha t} \)

The key advantage of Heaviside’s approach lies in its notation and simplicity. This approach bridged the gap between abstract mathematical theory and practical engineering problems.

The Skepticism Heaviside Faced

What I found most remarkable when reading about Heaviside’s contributions, that we now take for granted, is the extent of skepticism he faced from mathematicians of his time. Many of them claimed that he lacked mathematical rigor. Heaviside himself admitted that his methods were not always rigorously formal but argued that they were effective in practice, and he viewed mathematics more as an experimental discipline than a strictly formal one. Interestingly, nearly 50 years before Heaviside, mathematicians like Fourier and Laplace had already developed the theoretical foundations upon which Heaviside built his work. However, their ideas had remained confined to theoretical mathematics and were not yet applied to practical engineering problems until Heaviside's intervention.

It’s a bit of a stretch to suggest that they were forgotten, but rather they simply hadn’t yet diffused. Heaviside popularized the techniques as a solution to the practical problems of his day. It wasn’t until after Heaviside’s death that Thomas Bromwich helped to formalize the inverse Laplace transform and demonstrate that it produced the same results as manipulating operators algebraically.

Over time, Fourier and Laplace transforms, which provide a more rigorous framework for solving differential equations, became a core subject in virtually every engineering curriculum.

Why simplicity is key

I have often observed how many engineers struggle with complex differential equations and feel intimidated by them. Equations such as the heat equation, wave equation, and moment-curvature equations can be difficult to grasp intuitively, and the differential operators involved can seem abstract and challenging. This is because differential equations often require a higher level of abstraction and it takes time to think about it.

In my experience, the default solution today is to walk away from the methods that we were taught at University and often to turn to computer software that employ known techniques such as finite element analysis. The challenge with this approach is that computers are black boxes that are known to produce numerical errors.

Engineers are constantly in search for simplify methods to check themselves and verification becomes increasingly difficult when working with large structures in complex domains. More often it is the user and not the computer that creates the error.

Operator calculus on the other hand provide relatively simplistic methods to solve complex equations. I recommend the story of Heaviside to anyone doing numerical simulations, who is searching for a matchbox calculation to verify himself. If we treat complex operators like algebraic tools, then the equations become relatively simple to solve, resulting in more optimal designs.