This chapter is devoted to a fascinating method for solving initial value problems for linear differential equations with constant coefficients---called the Laplace transformation. It was invented and applied at the end of the nineteen century by the English self-taught electrical engineer, mathematician, and physicist Oliver Heaviside (1850--1925). The Laplace transformation method is widely used in circuit analysis and mechanical problems, control systems and feedback study, and many other areas.

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Oliver Heaviside (1850--1925).

Brief History of Laplace Transform

The Laplace transform is named after the French mathematician and astronomer Pierre-Simon Laplace (1749--1827). However, he did not actually invent what we now call the Laplace transform. Indeed, Laplace himself, a notoriously vain and selfish person in spite of his scientific genius, was careful to credit Leonhard Euler (1707--1783) with the basic formula. Well before the work of Laplace, however, mathematical genius Leonhard Euler had studied differential equations. One of his many noteworthy contributions in this field was the idea of transforming a function X(x) into a new function z via the equation

\[ z = \int e^{ax}\, X(x)\,{\text d} x , \]
which looks fairly similar to the modern Laplace transform, only with an indefinite rather than a definite integral. In a 1753 paper (entitled Methodus aequationes differentiales altiorum graduum integrandi ulterius promota -- it’s a good thing mathematicians don’t use Latin any more…), Euler used methods based on this transform to give a systematic method of solving second order linear differential equations. Later in his career, he further clarified the method and introduced the definite integral form
\[ y(u) = \int_a^b e^{K(u)\, Q(x)}\, P(x)\,{\text d} x . \]
In particular, this expression appeared in Euler’s 1768 Institutiones Calculi Integralis, vol. II, where he used it to solve the equation
\[ L\, \frac{{\text d}^2 y(u)}{{\text d} u^2} + M\, \frac{{\text d} y(u)}{{\text d}u} + N\,y(u) = U(u) , \]
\[ U(u) = R(a)\, e^{K(u)\, Q(a)} . \]

However, Euler did not pursue this topic very far. Joseph Louis Lagrange (1736--1813), born as Giuseppe Lodovico Lagrangia in Turin, Italy, who succeeded Euler (since Leonhard returned to Russia) as the director of mathematics at the Prussian Academy of Sciences in Berlin, began to study integrals in the form \( \int_0^{\infty} f(t)\,e^{-at}\,\mathrm{d}t \) in connection with his work on integrating probability density functions. Laplace was the next person to seriously work on this topic, and took a critical step forward by applying the idea of a "transformation" rather than just looking for a solution in the form of an integral. He looked for solutions with the following equation: \( \int x^{s}\,\phi (x)\,\mathrm{d}x = F(s) .\) In 1809, Laplace extended his transform to find solutions that diffused indefinitely -- giving us our popular Laplace transformation. Laplace appeared to have quickly understood the importance of his discovery, as he went on to use Laplace transforms numerous times in his later work and generalize his integrals to create Fourier and Mellin transforms as well. So while Laplace may not have invented his transforms, he certainly deserves credit for producing a systematic body of theory that went far beyond anything created by his predecessors.

Although the results had been published for at least 70 years, the transformation was not given a true physical and mathematical meaning until Oliver Heaviside (1850--1925) came up with completely new ideas on his own in the 1880s. His predecessors used (what we call now the Laplace) integral as an analytic tool and never even tried to establish the inverse Laplace transform---without this tool the theory is not complete at all. Oliver Heaviside did not actually use (and most likely was unfamiliar with) the Laplace integral in his derivations because it was not needed in his pioneering work in establishing the operator methods.

Let us pause explanations of Heaviside's achievements and focus on his biography and historical circumstances that surrounded his science breakthrough. Oliver Heaviside was born on May 18, 1850 in Camden Town, London, England. He caught scarlet fever when he was a young child and this affected his hearing. This was to have a major effect on his life, making his childhood unhappy, with relations between himself and other children difficult. However, his school results were rather good, and in 1865 he was placed fifth from five hundred pupils. Academic subjects seemed to hold little attraction for Heaviside, however, and at age 16 he left school. Perhaps he was more disillusioned with school than with learning since he continued to study (Oliver was completely self-taught) after leaving school, in particular he learned Morse code, studied electricity, and studied further languages, in particular Danish and German. He was aiming at a career as a telegrapher; in this he was advised and helped by his uncle Charles Wheatstone (the piece of electrical apparatus known as the Wheatstone bridge is named after him).

In 1868, Heaviside went to Denmark and became a telegrapher. He progressed quickly in his profession and returned to England in 1871 to take up a post in Newcastle upon Tyne in the office of Great Northern Telegraph Company, which dealt with overseas traffic. Heaviside became increasingly deaf but he worked on his own research into electricity. While still working as chief operator in Newcastle he began to publish papers on electricity, the first in 1872, and then the second in 1873, which was of sufficient interest to James Clerk Maxwell (1831--1879) that he mentioned the results in the second edition of his Treatise on Electricity and Magnetism. Despite this hatred of rigor, Heaviside was able to greatly simplify Maxwell's twenty equations with twenty variables, replacing them by four equations with two vector variables (the electric field E and the magnetic field B). Today we call these 'Maxwell's equations forgetting that they are in fact 'Heaviside's equations.'

Heaviside went on to achieve further advances in knowledge, again receiving less than his just desserts. In a 1887 paper Heaviside gave, for the first time, the conditions necessary to transmit a signal without distortion. In Electromagnetic Theory (1893--1912), he postulated that an electric charge would increase in mass as its velocity increases, an anticipation of an aspect of Einstein’s special theory of relativity. Heaviside was elected a Fellow of the Royal Society in 1891, perhaps the greatest honor he ever received. In 1902 Heaviside predicted that there was a conducting layer in the atmosphere which allowed radio waves to follow the Earth's curvature. This layer in the atmosphere, the Heaviside layer, is named after him. Its existence was proved in 1923 when radio pulses were transmitted vertically upward and the returning pulses from the reflecting layer were received.

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On the eve of the twentieth century, new emerging physical ideas required new mathematical tools to formalize these theories. It is not an accident that many new mathematical notations and definitions were introduced by physicists, but not mathematicians. We cannot neglect to mention the famous Dirac's delta-function as well as the Einstein summation abbreviation. The latest example gave us the Internet, including the HTML protocol. Therefore, Heaviside's invention of operational calculus has naturally evolved to quantum mechanics and then as a byproduct, a useful technique for solving initial value problems for differential equations.

The Laplace transform is an integral transformation because it maps a function of a real variable t ∈ ℝ+ (as a rule, time) to a function of a complex variable λ (complex frequency):

\begin{equation} \label{EqPart6.1} f^L (\lambda ) \equiv {\cal L} (f) = \int_0^{\infty} f(t)\,e^{-\lambda t} \,{\text d} t , \end{equation}
which mathematicians credited to Laplace. Although Pierre-Simon used this integral in his work, it was not considered by him as a transformation---a completely different idea because it requires the inverse transformation. Since O.Heaviside did not developed a systematic way to determine the inverse Laplace transform, it took about 20 years until a group of people came up with the answer. The first publication was made by the English mathematician Thomas John I'Anson Bromwich (1875--1929) who showed that the inverse Laplace transform can be expressed as the contour integral
\begin{equation} \label{EqPart6.2} {\cal L}^{-1} (F) = \mbox{P.V. }\frac{1}{2\pi {\bf j}} \, \int_{s-{\bf j}\infty}^{s+{\bf j}\infty} F(\lambda )\,e^{\lambda t} \,{\text d} \lambda = \frac{1}{2\pi {\bf j}} \, \lim_{\omega \to \infty} \int_{s-{\bf j}\omega}^{s+{\bf j}\omega} F(\lambda )\,e^{\lambda t} \,{\text d} \lambda \qquad ({\bf j}^2 = -1), \end{equation}
where the abbreviation P.V. stands for the Cauchy principal value, which indicates that the symmetrical regularization of the improper integration is done along the vertical line Reλ = s in the complex plane such that s is greater than the real part of all singularities of F(λ) and F(λ) is bounded on the line of integration. This integral is usually referred to as the Bromwich integral. The problem of the recovery of a real-valued function f(t) for t ≥ 0, given its Laplace transform
\[ \int_0^{\infty} f(t)\,e^{-s\, t} \,{\text d} t = g(s) \]
for real values of s ∈ ℝ is actually solving the integral equation of the first kind. It is an example of an ill-posed problem in the sense of J. Hadamard, who was the first to introduce the concept of a well-posed problem in 1902. Ill-posed problems may suffer from numerical instability when solved with finite precision, or with errors in the data. A typical example of an ill-posed problem is numerical differentiation. The Laplace transformation can be transformed into the convolution problem with the following substitution:
\[ s = a^x \qquad\mbox{and} \qquad t = a^{-y} , \quad\mbox{ where}\quad a > 1. \]
\[ g \left( a^x \right) = \int_{-\infty}^{\infty} \ln a \, e^{-a^{x-y}} f \left( a^{-y} \right) a^{-y} {\text d} y . \]
Multiplying both sides by 𝑎x, we obtain the convolution equation
\[ \int_{-\infty}^{\infty} K(x-y)\,F(y)\,{\text d}y = G(x) , \]
\begin{align*} G(x) &= a^x g \left( a^x \right) = s\, g(s) , \\ K(x) &= \ln a\, a^x e^{-a^s} = \ln a\,s\,e^{-s} , \\ F(y) &= f \left( a^{-y} \right) = f(t). \end{align*}
A convolution equation occurs widely in the applied sciences. Such equations are usually solved with the aid of filtered approximation, for example with Tikhonov regularization.

To this day, the Inverse Laplace Transform is the most difficult process to understand when solving differential equations with the operational technique. In this tutorial, we will not use the formal definition of the inverse Laplace transform; instead, we apply the residue method based on the novel approach proposed by Vladimir Dobrushkin.

The revolutionary idea of Heaviside's operational calculus consists in establishing the spectral decomposition of the (unbounded) derivative operator \( \texttt{D} = {\text d}/{\text d} t \) acting in the space of functions on the half line [0, ∞). This means that the operator was replaced by simple multiplication. We will meet this approach in the second part of this tutorial when spectral decomposition will be applied to square matrices and second order differential operators.

Due to his research, Heaviside dealt with multi-degree differential equations for electrical systems in the 1880s. To solve corresponding initial value problems, Heaviside substituted the derivative operator by a letter p (we use the contemporary notation \( \texttt{D} \) instead), which yields algebraic equation. He then quickly solved the algebraic equation and proceeded with a solution to the original differential equation. These kinds of equations (usually with 10 or more derivatives of a dependent variable) would usually take days or weeks for most people to solve. Heaviside was able to solve these kinds of equations within hours. Between 1880--1887, he invented operational calculus -- a new method for solving the differential equations.

Unfortunately, Oliver Heaviside failed to explain how he had derived the solutions to the initial value problems, and as often happens the work of geniuses is not often understood. It took many years for academia to accept his results because of a lack of a rigorous proof for his methods. He replied to this criticism with the famous statement "Mathematics is an experimental science, and definitions do not come first, but later on," claiming that "I do not refuse my dinner simply because I do not understand the process of digestion." As a result, mathematicians did not accept his technique and tried to explain what is going on behind his manipulations. After many years of work, it became clear that Heaviside used the integral transformation.

Concept Overview

The Laplace transformation, denoted by ℒ, is the mapping of a set X of integrable functions on the half line [0, ∞) that grow no faster than exponentials into another set Y of holomorphic complex-valued functions defined on the half plane Rezh, so ℒ: XY. However, when the Laplace transform is used in differential equations, the set X is naturally identified with the set of functions on the real axis ℝ that vanish for negative t. The Laplace transform is usually applied to a subset of X that contains piecewise smooth functions having derivatives almost everywhere. Let \( \texttt{D} = {\text d}/{\text d}t \) be the derivative operator. Then the Laplace transform provides the spectral decomposition to the unbounded derivative operator, that is,
\[ \left( {\cal L}\texttt{D} f\right) (\lambda ) = \lambda\,{\cal L}f (\lambda ) . \]
This means that the Laplace transform converts differentiation into a multiplication operation. In general, when \( Q[\texttt{D}] = a_0 \texttt{D}^0 + a_1 \texttt{D} + \cdots + a_n \texttt{D}^n \) is a linear differential operator with constant coefficients, then the Laplace transform converts it into multiplication by a function:
\[ \left( {\cal L}\, Q\left[ \texttt{D}\right] f\right) (\lambda ) = Q\left[ \lambda\right] {\cal L}f (\lambda ) . \]

Note that this relation is valid only in a vector subspace of functions that vanish at the origin. Essentially, we will be doing almost the same thing that Heaviside had done-- we will take a linear differential equation, subject to the initial conditions

\begin{equation} \label{EqPart6.3} a_n \frac{{\text d}^n y}{{\text d} t^n} + a_{n-1} \frac{{\text d}^{n-1} y}{{\text d} t^{n-1}} + \cdots + a_0 y(t) = f(t) , \qquad y(0) = y_0 , \ \dot{y} (0) = y_1 , \ldots ,\left. \frac{{\text d}^{n-1} y}{{\text d} t^{n-1}} \right\vert_{t=0} = y_{n-1} , \end{equation}
and associate the differential operator \( Q[\texttt{D}] = a_n \texttt{D}^n + a_{n-1} \texttt{D}^{n-1} + \cdots + a_1 \texttt{D} + a_0 \texttt{D}^0 \) with the differential equation using the operator notation \( \texttt{D} = {\text d}/{\text d} t . \) Then application of the Laplace transform to the IVP \eqref{EqPart6.3} creates an algebraic equation
\begin{equation} \label{EqPart6.4} \left( a_n \lambda^n + a_{n-1} \lambda^{n-1} + \cdots + a_0 \right) y^L = f^L + a_n \left( y(0) + \lambda \dot{y} (0) + \cdots + y_{n-1} \right) + \cdots , \end{equation}
with respect to \( y^L (\lambda ) = {\cal L}\left[ \,y \,\right] , \) the Laplace transform of unknown solution y(t) to the IVP \eqref{EqPart6.3}. Solving this algebraic equation \eqref{EqPart6.4}, we represent yL, the Laplace transform of unknown solution y(t) as a ratio of two polynomials in λ:
\begin{equation} \label{EqPart6.5} y^L (\lambda ) = f^L \, \frac{1}{{\it Q} (\lambda )} + \frac{P(\lambda )}{{\it Q} (\lambda )} , \end{equation}
where \( {\it Q}(\lambda ) = a_n \lambda^n + a_{n-1} \lambda^{n-1} + \cdots + a_0 \) is the characteristic polynomial for the differential operator \( {\it Q}(\texttt{D}) \) of order n and P(λ) is the polynomial of degree less than n in λ depending on the initial conditions. Finally, we apply the inverse Laplace transform to obtain the required solution. Since the initial value problems (IVP) with specified initial conditions have unique solutions, any method is legimitimate to obtain these solutions. If the initial conditions are not specified, the Laplace transfrom provides the general solution containing arbitrary constants. Note that the solutions to ODEs obtained by the Laplace transformation are identically zero for negative t. Therefore, all expressions for these solutions should be multiplied by the Heaviside function.

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