Laplace transformation and linear differential equations

[-> auf Deutsch <-]

The traditional theory of Laplace transformation (TLT), which in its currently prevalent form was put forward by
Gustav Doetsch, is in several fundamental respects deficient [70], [71], [77], [107]. TLT was asserted to constitute a
self-contained mathematical theory which provides a solid and rigorous foundation of operator calculus for the solution
of linear differential equations (DEs). This assertion turns out to be untenable.

As LT's transformation integral is "unilateral", i.e., extending from t=0 to t->+oo, the L-transform L{f(t)} of a real
function f(t) is independent of the behavior of f(t) for t<0. In TLT it is erroneously asserted that this kind of irrelevance
of the interval t<0 also holds when LT is used for the solution of linear DEs. More specifically: In TLT it is (explicitly or
implicitly) assumed that the solution of the linear inhomogeneous DE is independent of the excitation function's behavior
for t<0. This assumption is a severe mistake.

To understand what I am talking about, solve a first-order linear DE for the excitation function x(t)=const by the
conventional theory of the linear DE and compare the solution with the solution obtained by TLT. There is a pronounced
discrepancy. Essentially, the LT-based solution turns out to be not actually the solution for x(t)=const but for x(t)=const u(t),
where u(t) denotes the unit step function, i.e., u(t)=0 for t<0; u(t)=1 for t>0. Note that the discrepancy exists even in the
interval t>0.

This discrepancy originates from the fact that L-transforms, just as Fourier-transforms, represent t-domain functions for the
entire t-domain -oo<t, while the function actually represented by an L-transform invariably is causal. (In line with common
terminology, a function f(t) is termed causal if f(t)=0 for t<0.) When f(t) is L-transformed, the difference between a bilateral
function f(t) and its causal companion u(t)f(t) becomes neglected, as there holds L{f(t)}=L{u(t)f(t)}. The inverse L-transform
of L{f(t)} invariably is 0 for t<0. Thus, the inverse L-transform always equals u(t)f(t). It is in this sense that an L-transform
always represents the causal function u(t)f(t), i.e., irrespective of whether or not f(t) itself is causal. (If f(t) itself is causal then
there holds u(t)f(t)=f(t) [107]).

The correct solution of the linear inhomogeneous DE can by LT be obtained only for causal excitation functions

The LT-based solution of the inhomogeneous DE invariably is dependent on the function represented by L{x(t)}, i.e., on
the causal excitation function u(t)x(t). As the DE's correct solution for a bilateral excitation function x(t) is different from
that for u(t)x(t), the LT-based solution can be consistent only with u(t)x(t). Hence, if one cares about the solution to be correct,
one has to confine application of LT to DEs with causal excitation functions. This conclusion makes the distinction between
bilateral and causal functions crucial. It is in this sense that the behavior of x(t) for t<0 actually is highly significant.

In the realm of TLT, the fact that causal functions play a special role was not entirely missed. However, TLT is far from
appreciation of the aforementioned implications, let alone drawing consequences. TLT fails to provide the mathematical
infrastructure which is necessary to adequately account for the difference between bilateral and causal functions. This is why
the unilateral theory of unilateral LT constituted by TLT needs to be replaced with the bilateral theory of unilateral LT [107].
(The latter theory must not be confused with the theory of "two-sided" LT known from the literature.)

The evoked response is a causal function

The fact that the excitation function represented by L{x(t)}, i.e., the effective excitation function, is always causal, has the
important consequence that the contribution of x(t) to the DE's total solution - the evoked response function - is causal, as well.
The latter fact in turn has the consequence that conversion into the L-domain of the inhomogeneous DE for the evoked response
function is governed by the derivation theorem for causal functions. This derivation theorem differs from TLT's derivation
theorem as it does not include any constant. Therefore, when this theorem is used for conversion into the L-domain of the
original DE one obtains an algebraic linear equation for the L-transform of the evoked response function. This is how LT
provides for separate determination of the evoked response function, i.e., for that "particular particular" solution of the DE
that is solely dependent on the excitation function [107].

The method for solution of the linear DE

Once the evoked solution is determined as just outlined, the DE's total (general) solution is obtained by mere addition of
the general solution of the pertinent homogeneous DE. Thus, the new method described in [107] strictly adheres to the
familiar principle that the general solution of the inhomogeneous DE emerges from superposition of a particular solution
of the DE and the general solution of the pertinent homogeneous DE. (TLT inevitably violates this fundamental principle,)

Below, the new method is concisely outlined, i.e., for the inhomogeneous DE of the form

a0y(t) + a1y'(t )+ a2y''(t) + ... + aNy(N)(t) = b0x(t) + b1x'(t) + b2x''(t) + ... + bMx(M)(t).                (1)

x(t) denotes the excitation function; y(t) denotes the response function. The coefficients an, bm are presumed to be real
constants. N indicates the highest derivative of the response function (N = 1, 2, ...); M the highest derivative of the
excitation function (M = 0, 1, ...).

The evoked response

In [107] I have described that, and why, the evoked response function ye(t) pertinent to any DE of the form (1) is depicted
by the formula

ye(t} = L-1{ L{x(t)}B(s)/A(s) }        for    -oo < t,               (2)

where the operator L-1 indicates inverse L-transformation; and

A(s) =  a0 + a1s + a2s2 + ... + aNsN:               (3)

B(s) = b0 + b1s + b2s2 + ... + bMsM.               (4)

In the majority of applications it is only the evoked response which one actually is interested in. Then solution of (1) only
requires (a) determination of the L-transform of x(t); and (b) determination of the inverse L-transform of (2). Thus, the good
news is, that determination of the evoked solution of the DE is simple and requires only cursory knowledge about the theory of LT.
The bad news is, that inverse L-transformation of (2) may turn out to be just as difficult as is known from TLT. Anyway, the variety
of L-transforms that can be inverted by just using a table of LT-correspondences is considerable.

When a table of LT-correspondences is used, it must be observed that inverse L-transforms invariably are causal functions. Most of
the t-domain functions included in the tables have to be multiplied by u(t). Exceptions from this rule are inherently causal functions
such as the unit step function u(t) and its (pseudo-)derivatives.

As an illustration of the simple procedure, consider the DE

ay(t) + y'(t) = u(t).               (5)

Notice that u(t) is a causal excitation function. According to (2-4) the evoked solution of (5) reads

ye(t) = L-1{ 1/[s(a+s)] } = u(t)[1-exp(-at)]/a          for       -oo < t.               (6)

(For causal functions such as ye(t) to be correct at t=0 it is indispensable that u(t) be defined at t=0; see [107].)

Notice that by (2) exactly the same solution (6) is obtained when in (5) the constant 1 is substituted for u(t). The reason is
that L{1}=L{u(t)}=1/s. However, for x(t)=1 this solution is incorrect. The correct evoked solution for x(t)=1 reads ye(t)=1/a.

The spontaneous response

When one subtracts from the original inhomogeneous DE (1) the pertinent DE for the evoked solution, there remains the
pertinent homogeneous DE. The general solution of the homogeneous DE is termed the spontaneous response, ys(t). The
spontaneous response is the pertinent system's "response to nothing". According to the familiar concept, this kind of
response is assigned to a preexcited initial state of the system which by definition is not accounted for by x(t).

A linear system's initial state has no effect on the evoked response. Therefore, determination of the spontaneous response
- i.e., in addition to that of the evoked response - is optional, coming only into play when the spontaneous response as such
is relevant to the application.

As determination of the spontaneous solution is independent of that of the evoked solution, for the spontaneous solution
still any mathematical method available can be used. In [107] I have described a new, LT-based method for determination of
the general solution of the linear homogeneous DE. This method yields for ys(t) the formula

u(t)ys(t) = L-1{ C(s)/A(s) }        for    -oo < t.               (7)

where A(s) is defined by (3), and

C(s) = c0 + c1s + c2s2 + ... + cN-1sN-1.               (8)

The coefficients cn of the polynomial C(s) are arbitrary real constants. Any particular set of constants defines a particular
initial state of the pertinent system.

The spontaneous response, i.e., the solution of the homogeneous DE of finite order N, invariably is an ordinary, derivable,
bilateral function. By contrast, the inverse L-transform is a causal function. This is the reason why by inverse
L-transformation of (7) one obtains at first u(t)ys(t) instead of ys(t) itself. However, as ys(t) is known to be an ordinary
derivable bilateral function, ys(t) can be retrieved for -oo<t by extrapolation. This is equivalent to merely "cancelling out"
the factor u(t) on both sides of (7), i.e., after inverse L-transformation.

For example, the spontaneous response of the DE (5) is determined by

u(t)ys(t) = L-1{ c0/(a+s) } = u(t)c0exp(-at),               (9)

such that the spontaneous response itself reads

ys(t) = c0exp(-at)         for      -oo < t.               (10)

The total response

As the total solution of (1) is composed of the evoked and spontaneous solutions, the N arbitrary constants contained in (7),
i.e., of the spontaneous response, are also arbitrary constants of the total solution. Thereby the sum

y(t)  = ye(t) + ys(t)                    (11)

becomes the DE's (1) general solution proper, and this solution holds in the entire t-domain, -oo<t.

The principle of superposition expressed by (11) holds for any kind of excitation function. However, when ye(t) is obtained
by LT, the solution holds only for causal excitation functions, as was emphasized above.

According to (6, 10, 11) the total solution of the first-order DE (5) reads

y(t) = u(t)[1-exp(-at)]/a + c0exp(-at)         for      -oo < t.               (12)

The constants

As was described in  [107], the coefficients of the polynomial (8) bear a well-defined, linear relationship to the
coefficients of the homogeneous DE and to the initial values of the spontaneous response function, as follows:

c0 = a1ys(0) + a2ys'(0) + ... + aNys(N-1)(0);

c1 = a2ys(0) + a3ys'(0) + ... + aNys(N-2)(0);

    . . .

cN-2 = aN-1ys(0) + aNys'(0);

cN-1 = aNys(0).               (13)

Concerning the general solution of (1), one has the choice either to treat the coefficients cn  per se as arbitrary constants; or
to define the initial values ys(0), ys'(0), etc. of the spontaneous response function, obtaining from them by (13) the coefficients
c0 ... cN-1.

For the present example, i.e., the solution (12) of the first-order DE (5), the constant c0 equals ys(0).

Comparison to the TLT-based solution

The solution of (5) obtained by the TLT-method reads

y(t) = [1-exp(-at)]/a + y(0)exp(-at)         for        t > 0.               (14)

This solution quite resembles (12), i.e., except for the constant and the definition interval. Thus, one may be tempted to conclude
that, by and large, the TLT-based solution is at least for t>0 correct, provided that the excitation function is causal. However,
this conclusion is wrong. The constant y(0) of (14) - the initial value of the total solution y(t) - is in general incompatible with
the constant c0=ys(0) of (12). The difference between the two kinds of constant is implied in (11). This discrepancy is the origin of
the so-called initial-value conflict which may occur in applications of TLT [107].

Author: Ernst Terhardt - Feb 2007

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