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The representation of the Dirac delta, obtained by differentiating the parametric equation of the unit step with a riser, is used to solve two examples referring to problems of a different physical nature, each with the product of two deltas as a forcing function. Each problem was solved by an entirely different procedure. In comparison with non-parametric solutions, the present solutions are both more accurate and truer representations of the physics involved.

One purpose of this paper is to emphasize the fact that the parametric delta is an exact representation, i.e., its value is zero everywhere except at one single point, and at that point its value is infinity. Another purpose is to illustrate the use and the effect of the parametric delta relating to two-dimensional domains, in space-time or in space-space; in these two cases, a product of deltas is involved, of course. Still another purpose is to present a problem example in which the operator action of the parametric delta facilitates the solution.

According to distribution theory, the Dirac delta is the result of differentiating the Heaviside unit step. The particular parametrization presented in [

It is well to keep in mind that the parametric equations of the delta confirm that its area has unit value, that they comply with the fundamental property and that they yield the correct Laplace [

In the solution of differential equations, the parametric equations are handled exclusively by calculus and algebra, both at an elementary level. The parametrized representation can be readily visualized geometrically. These two features should make these parametric equations particularly convenient as a useful research tool, and also, for the purpose of teaching the Dirac delta concept at an early stage in undergraduate school.

The parametric equations of the Dirac delta were developed by differentiating the unit step with a riser. The parametric representation of the unit step with a riser is given by [

These two functions would be continuous were it not for the fact that they are undetermined at the points

Where:

is the Cauchy limiting coefficient [

It is clear then that differentiating

Consequently:

This is the parametric Dirac delta, a more rigorous derivation of which was presented in [

Determine the deflection of a thin rectangular membrane clamped on all four edges and loaded by a force applied at point

“Solution:” The deflection is governed by the Poisson equation:

Subject to the boundary conditions:

Nomenclature:

x = position along the

y = position along the

a = location of the load in the x direction;

b = location of the load in the y direction;

u = x parameter;

v = y parameter;

P = load;

T = tension per unit length.

This problem will be solved by, what we will call, the Parametrized Eigenfunction Expansion Method.

Assuming that [

Substituting Equation (10) into Equation (8)

where

Equation (11) can be interpreted as the Fourier expansion of the product:

The Fourier coefficients are:

or equivalently:

introducing the parameters u and v:

simplifying Equation (15):

Substituting Equation (5) into Equation (16) results in

with

Or equivalently

Therefore the parametric solution is:

The non-parametric solution is Equation (20c), of course, notice that it is the same as the bilinear formula for Green’s function ([

Consider a one dimensional rod subject to an impulsive heat source concentrated at point

Subject to the boundary conditions:

and to the initial condition

Nomenclature:

T = temperature;

x = position along the rod;

t = time;

a = location of the heat source in the x direction;

u = position along the rod parameter;

w = time parameter;

Q = heat per unit area;

c = specific heat;

k = thermal conductivity;

ρ = mass density.

Solution: This problem will be solved by, what we will call, the Direct Parametric Method.

Separating the variables:

Recalling that

Introducing the parameter w into Equation (26), yields:

multiplying both sides of Equation (27) by

Specializing Equations (5) and (6) yields:

Substituting Equations (29) and (30) into Equation (28), yields the control equation:

or in accordance with Equation (25),

During the impulse instant,

Equation (31) becomes

Notice that, due to the parametric representation, the term referring to the energy conduction process has been eliminated by the operator action of the parametric delta,

According to the separation of variables method, Equation (33) implies

where

and

Integrating Equation (35) yields:

substituting Equations (36) and (37) into Equation (25), yields

At the “beginning” of the impulse instant,

But

At the “end” of the impulse instant, w = 1, Equation (40) reduces to:

At post impulse time,

The control Equation (32) becomes

Notice that, due to the parametric representation, the post impulse equation is homogeneous because the forcing function has been eliminated by the operator action of the parametric delta,

Which has the solution ([

:

Because of continuity requirements, the temperature at the “beginning” of the post-impulse time must be equal to the temperature at the end of impulse instant.

Thus the initial condition of post impulse time according to Equation (41) is

Substituting Equations (41) and (43) into Equation (44), yields

The right member of Equation (46) is recognized as the Fourier sine series of the left member; with coefficients

Substituting Equation (7) into Equation (47),

Substituting Equations (2), (5) and (6) into Equation (48),

or equivalently,

Complete parametric solution:

Collecting Equations (40), (43) and (51), the parametric solution for

The parametric Dirac delta representation was used to solve problems with forcing functions containing the product of two such deltas. The parametrized eigenfunction expansion method was used to solve problem (1) referring to the elastic deformation of a membrane subjected to a point load. The direct parametric method was used to solve problem (2) referring to the heat conduction in a metal rod subjected to the impulsive application of a concentrated heat source.

In the non-parametric eigenfunction expansion method, the integrals that constitute the values of the Fourier series coefficients contain the Dirac deltas. In the parametrized version, these deltas are substituted by the corresponding derivatives of the unit step and these, in turn, are expressed in terms of the parameters.

In the direct parametric method, in problems involving an impulsive forcing function represented by the time Dirac delta, the original differential equation is converted into two differential equations. The first of these equations refers to the impulse instant. Due to the operator action of the Dirac delta, the impulse instant equation may contain one term less than the original equation; furthermore, the Dirac delta is represented by a constant.

The second equation refers to the post-impulse time; and also due to the operator action of the Dirac delta, this equation becomes homogeneous. Thus, both the impulse and the post-impulse equations are easier to solve than the original equation.

In both problems, the accuracy was greater in the parametric solution than in the non-parametric solution. The magnitude reached by the error in problem (2) is striking and, contrary to the expected, increasing the number of terms in the series, increases the error.

The authors wish to express their gratitude for the continued support of the Dirección General de Apoyo al Personal Académico, UNAM.

Enrique J.Chicurel-Uziel,Francisco A.Godínez, (2015) Parametrization to Improve the Solution Accuracy of Problems Involving the Bi-Dimensional Dirac Delta in the Forcing Function. Journal of Applied Mathematics and Physics,03,1168-1177. doi: 10.4236/jamp.2015.39144