Introduction

In addition to its theoretical interest, it is well known that Dawson’s integral (or
Dawson’s function) arises naturally in several physical applications, whereby the
research related to this function is relevant. In this context, heat conduction
problems (^{Briozzo & Tarzia, 2010}; ^{Petrova, Domingo, Tarzia & Turner, 1994})
(see section 5), theory of electrical oscillations (^{McCabe, 1974}; ^{Weisstein, 2017}),
relativistic hydrodynamics (^{Scott, 2007}),
chemical physics in birefringence and dielectric relaxation phenomena in presence of
strong electric fields (^{Prigogine & Rice,
2001}), among many others, are mentioned.

The aim of this article is to propose a handy invertible and integrable approximation
for Dawson’s function. In the literature, there are several Dawson’s function
approximations reported; for instance, ^{Sykora
(2012)} proposed various approximation orders with good accuracy, although
they are not sufficiently simple nor are they hardly invertible and integrable.
Moreover, ^{Boyd (2008)} proposed an accurate
approximation for Dawson’s integral, by solving its differential equation using the
orthogonal rational Chebyshev functions of the second kind; nevertheless, its
rational approximation is hardly invertible and integrable. In the same way, ^{Cody, Kathleen & Thacher (1970)} proposed
rational approximations for Dawson’s integral, although the accuracy of the reported
results was verified only in finite intervals and not in the total domain of
Dawson’s function. ^{Lether (1997)} developed a
family of rational functions for computing Dawson's integral. Although this work got
approximations with a low relative error, its expressions are too large to be
invertible and integrable. Lether (1998) investigated the relation between some
rapidly convergent series of exponential functions for computing Dawson's integral.
Since the approximation is given in terms of an infinite sum of terms of the form ^{Franta, Necas, Giglia, Franta & Ohlídal
(2017)} proposed an extension of the universal dispersion model,
expressing the excitonic contributions in terms of linear combinations of Gaussian
and truncated Lorentzian terms. It appears that the real part of the dielectric
function is expressed by Dawson’s functions. Nevertheless, this work does not
present some analytical approximation for Dawson’s integral. In the same way, ^{Abrarov & Quine (2018)} proposed a rational
approximation for the Dawson’s integral, which can be implemented to calculate the
complex error function. Although this approach provides good accuracy, its rational
expression is too complicated to be invertible and integrable.

An example of the importance of inverting the Dawson’s integral is mentioned by ^{Scott (2007)}. In this work on relativistic
hydrodynamics, the time (t) measured by a reference observer is expressed in terms
of a scale factor (r), which determines the shape of an entropy profile through
Dawson’s function. His work mentions that in order to complement the solution of the
problem in this part, it is necessary to invert the mentioned expression in order to
obtain r as function of t, and, thus, obtain the velocity gradient. This work
targets this kind of applications.

In fact, the subject of inverting and approximating the integral for Dawson’s function is little addressed in the literature and this will be the main goal of this work.

The rest of this paper is organized as follows. In Section 2, the basic idea of the
power series extender method (PSEM) (^{Vazquez-Leal
& Sarmiento-Reyes, 2015}), which plays an important role in this work,
is introduced. Section 3 will provide a brief introduction to Dawson’s integral. In
Section 4, the deduction of a handy, accurate, invertible and integrable expression
for Dawson’s function is provided. Section 5 proposes an application of Dawson’s
integral to a non-classical Stefan problem in physics. Section 6 discusses the main
results obtained and includes a table for the benefit of the readers with the
relevant contributions found in this work. Section 7 provides a brief conclusion
and, finally, Section 8 resumes some results of cubic algebraic equations relevant
for this work.

Basic Concept of PSEM method

Here it is assumed the case of nonlinear differential equations, expressed in the form

with a boundary condition given by

*L* and *N* are linear and nonlinear operators,
respectively, *f*(*x*) is a known analytic
function, *B* is a boundary operator, Γ is the boundary of domain
Ω and ∂*u*/∂*η* denotes differentiation along the
normal drawn outwards from Ω (^{Cody et
al., 1970}).

In accordance with the PSEM methodology (^{Vazquez-Leal & Sarmiento-Reyes, 2015}), the solution of (1) is
expressed as a power series

Where *v*
_{
k
} (*k* = 0,1,2,…) denotes the coefficients of the power
series.

It should be mentioned that there is no single way of obtaining (3); thus, some
approximate methods from the literature could be employed for that purpose such
as Homotopy Peturbation Method (HPM), Homotopy analysis method (HAM),
Variational Iteration Method (VIM), Differential Transform Method (DTM), Adomian
Decomposition Method (ADM), Taylor Series Method (TSM), and Power Series Method
(PSM), among others (^{Vazquez-Leal,
Castañeda-Sheissa, Filobello-Niño, Sarmiento-Reyes & Sánchez-Orea,
2012}). Next, following the PSEM method, it is proposed that the
solution for (1) can be written as a finite sum of functions in the general
form

or

(^{Vazquez-Leal & Sarmiento-Reyes,
2015}), where *u*
_{
i
} are constants to be determined by PSEM, *f*
_{
i
} (*x,u*
_{
i
} ) are in principle arbitrary trial functions; and *n* and
2 *n* denote the orders of approximations (4) and (5),
respectively. It is agreed to denominate (4) and (5), from here on, as the trial
function (TF). Next, the Taylor series of (4) and (5) is calculated, resulting
in the power series

respectively, where Taylor coefficients *P*
_{
k
} are expressed in terms of parameters *u*
_{
i
} .

After equating/matching the coefficients of power series (6) or (7) with those
corresponding to (3), the values of *u*
_{
i
} are obtained. Finally, by substituting them into (4) or (5), it is
obtained the PSEM approximation. It is important to note that (4) or (5) can
separately be applied to obtain an approximate solution of (1). As a matter of
fact, the selection of TF depends on the nature of the problem under study. In
addition, it is important to remark that if the *f*
_{
i
} functions are chosen to be analytic, then (6) and (7) are convergent
series (^{Belser, 1999}; ^{Oberguggenberger & Ostermann, 2011};
^{Zill, 2012}).

Some rudiments of Dawson’s integral

Dawson’s integral is defined by ^{Khan (1990)}
as

As a matter of fact, it is not difficult to prove that
*F*(*x*) satisfies the following initial
condition problem:

On one hand, assuming a power-series expansion of the form *F*(*x*) near the origin through the following
series (^{Khan, 1990}):

On the other hand, it is possible to show that, after integrating by parts and
employing a breakpoint, *F*(*x*) is properly
expressed by the following asymptotic expansion for large values of
*x* (^{Khan, 1990}).

Nevertheless, it will be exposed a notable fact that it is possible to use (11),
keeping just the first two terms of the series to represent
*F*(*x*), even for relatively small values of
*x* with good accuracy, in order to get a handy
approximation, valid for *x ≥ 0* (see Section 4).

The function *F*(*x*) has just one extreme value, a
maximum that is *x*=0.923 where *F* adopts the
value *F*(0.923)=0.5410435224

Deduction of a handy accurate invertible and integrable expression for Dawson’s function

With the purpose of obtaining an approximated expression for Dawson’s function,
an algorithm will be followed by dividing the domain of
*F*(*x*) into two. In the first subinterval,
Dawson’s integral will be modeled by using the PSEM method, while in the second
one, it will be shown that employing the first two terms of (11) in order to
obtain a good approximation is sufficient.

To begin, equation (10) is used to obtain the following expression for Dawson’s
integral *F*(*x*), valid for values near the
origin:

In accordance with the PSEM algorithm (^{Vazquez-Leal & Sarmiento-Reyes, 2015}), it is proposed to model
the first part of *F*(*x*) near the origin by
means of the following rational function (see (5)):

Where *b*_{1}, *b*_{2},
ɑ_{1}, ɑ_{2}, and ɑ_{3} are
parameters to be adequately determined later on.

The following expression shows some terms of the Taylor series of (13):

Next, a system of algebraic equations will be deduced to calculate the values of the parameters mentioned above through the following criteria.

In order to ensure that *r*(*x*) correctly
represents the behavior of *F*(*x*) for values
near the origin, the Taylor series (12) and (14) will be matched by equating the
coefficients of powers *x* and *x*
^{2}. Since there are five parameters to be determined, then three
additional equations are necessary, chosen so that the proposed rational
function describes points of *F*(*x*) farther from
the origin.

With that purpose, the following points of Dawson’s function are proposed:
*A*(0.923,0.5410435224),
*B*(1.5,0.4282490711), and *C*(2.5,0.2230837222)
(it is noted that *A* corresponds to the point where
*F*(*x*) reaches its extreme value (see
section 3)).

The algebraic system of equations emanating from the above considerations is the following:

The numerical solution for the above system is

After substituting (16) into (13), it is obtained

As it can be seen afterwards, (17) describes
*F*(*x*) adequately for values of
*x*, from the origin to x≈2.6789.

For *x* greater than this value, it is proposed to model
*F*(*x*) with the first two terms of expansion
(11), that is,

It will be seen that, despite the character asymptotic of (18),
*A*(*x*) describes with good precision
Dawson’s function in the mentioned interval (Figure 1 and Figure 2). It is
emphasized that although a better approximation for
*r*(*x*) may be obtained, assuming a rational
function of greater order (see (13)) and increasing the accuracy of
*A*(*x*) keeping more terms of (11), the goal
is to obtain an accurate expression, as simple as possible, for Dawson’s
function, in such a way that it is invertible and integrable. As a matter of
fact, as an application, this last characteristic of this approximation for
*F*(*x*) will be employed in a case study
emanating from physics. On the other hand, in order to build a continuous
function from (17) and (18) it is necessary to find the point of junction of the
previous functions. The fact that the absolute values of relative errors of (17)
and (18) have to be the same in the point of intersection of
*r*(*x*) and
*A*(*x*), i.e., is proposed as criteria:

After applying condition (19), the value already mentioned is obtained:

In a sequence, by using the unit step function *δ* (^{Zill, 2012}), it is possible to express (21)
as

(Figure 1).

Next, it is noted that (21) is indeed invertible.

For the interval *x*>2.678915610,
*F*(*x*) is given by (18), in such a way that
after some algebraic steps, (18) is rewritten as

where, for the sake of simplicity, the dependence of *x* from
*A*(*x*) is omitted.

In Appendix A the basic aspects of cubic algebraic equations are summarized.

Next, *Q*, *R*, and *D* are
calculated (see equations (A1) and (A2)) so that

Since discriminant *D*>0 for the whole interval
*x*>2.678915610, it is inferred that (23) has just one
real root which, in accordance with the first equation of (A3), is given by

after using *F*(*x*) instead of
*A*(*x*).

On the other hand, for the interval 0≤*x*≤2.678915610,
*F*(*x*) is given by (17):

After some algebraic steps, (17) is rewritten in the form of the cubic equation

where *F*(*x*) has been employed instead of
*r*(*x*) and defined

The corresponding values for *Q* and *R* are given
by (see A [2])

On the other hand, in order to obtain *D*, it is just necessary to
substitute (28) and (29) into

nevertheless, a sort of cumbersome expression to *D* would be
obtained.

A better methodology is to graph the right-hand side of (30) (see Figure 3). From the mentioned Figure, it is
concluded that, *D*>0 in the interval of interest
0<*x*≤2.678915610 and from the theory for solving cubic
equations, (26) provides three real roots (see Appendix A).

Next, from the three above-mentioned roots, the last expression
*x*
_{3} of (A4) is provided as a solution for (26), because this supplies,
with good precision, the values of *x*, starting from the values
of Dawson’s function *F*:

for the interval 0<*x*≤2.678915610

Finally, it is noted that the approximation proposed to Dawson’s integral (21) is also integrable, in terms of elementary functions, that is to say,

(for real arguments *x*, *y*, the two-argument
function arctan(*y,x*) computes the principal value of the
argument of the complex number *x*+*iy*, so
(*π*<arctan(*y,x*)≤*π*.
Equation (32) was obtained with the assistance of Maple 17 built-in function
routine for integration.

In terms of step function, it is possible to rewrite (32) as

where the value 0.97010971726 expressed in the interval x>2.678915610
represents the value of

In order to prove the accuracy of (33), regarding the relevant contribution that
comes from 0≤*x*≤2.678915610, the area will be evaluated:

and later it is compared with the numerical value of (34) for Dawson’s function. As a matter of fact, the accuracy of the proposed approximate solution (33) is revealed.

The numerical value of (34) turns out to be 0.9635924825, while the value
obtained after employing the proposed analytical approximation (33) is
0.9701971726. What is more, Figure 4 shows
the plot of the area function *x*≤2.678915610 of (33), and Figure 5 shows that the biggest absolute
error committed is about 7×10^{-3}, proving the accuracy of this work’s
approximation.

Application to a nonclassical Stefan problem in physics

A one-phase Stefan problem for a semi-infinite material is a free boundary problem for the heat equation, which aims to find the temperature distribution for the case of melting and solid phases, as well as to determine the evolution of the free boundary.

Following ^{Briozzo & Tarzia, 2010}, it is
briefly considered the nonclassical heat conduction problem for a semi-infinite
material given by the conditions

where *u*=*u*(*x,t*) is the
temperature, *x*=*s*(*t*) is a free
boundary, *k*, *ρ*, *C, l,* and
*γ* are certain positive thermal coefficients, the boundary
temperature is denoted by *f>0*, and *λ*
_{0} is a constant. With the purpose of getting an explicit solution of
a similarity type, the following substitution is proposed (^{Briozzo & Tarzia, 2010}):

where ɑ^{2}=*k/ρc* is the diffusion coefficient of
the phase change material.

After using (36), it can be noted that (35) adopts the form

where the dimensionless parameter

has been defined, and *s*(*t*) must be of the
form

The value of the parameter *η*
_{0} is determined later.

From ^{Briozzo & Tarzia (2010)}, the
solution of (37) that satisfies the conditions (38) and (39) is given by

where the following is defined:

In (44), *erf*(*x*) denotes the error function

and *F*(*x*) the Dawson’s integral (see (8)).

By substituting (43) and (44) into (40), it is obtained the following equation
for the unknown parameter *η*
_{0} (^{Briozzo & Tarzia,
2010}):

where the Stefan’s number given by
*Ste*=*fc/l*>0. has been introduced.

It is noted that one of the main contributions of this work is to provide an
analytical approximation for the integral of Dawson's function (see (33)) with
good precision and, for the same reason, this result may be employed into (44).
On the other hand, ^{Vazquez-Leal,
Castañeda-Sheissa, Filobello-Niño, Sarmiento-Reyes & Sanchez-Orea
(2012)} provided the following handy accurate analytical approximation
to the error function *erf*(*x*) (see
discussion):

Thus, unlike ^{Briozzo & Tarzia (2010)},
who just provided a symbolic solution to problems (37)-(40), the authors in this
study are in a position to propose an analytical solution for the same problem
with good precision by substituting (33) and (47) into (43).

Finally, approximations (22), (33) and (47) may be substituted into (46) in order
to obtain a numerical approximate solution for *η*
_{0} and, with this result, the problem is concluded.

For sake of simplicity, the following values of Stefan’s number
*Ste*=1 and *λ*=1 are proposed as a case
study. After performing the numerical solution of the above-mentioned equation,
the value *η*
_{0}=1.479186840. is obtained.

Discussion

This article successfully accomplished the purpose originally proposed: to get a
handy analytical approximate solution for the Dawson’s integral, which, as seen
before, describes for example the solution of heat conduction problems (^{Briozzo & Tarzia, 2010}; ^{Khan, 1990}). Although the proposed approximation
has an acceptable precision, it introduces two advantageous characteristics, which
are not presented in other approximations of Dawson’s function from the literature:
being invertible and integrable in terms of elementary functions. To achieve this
goal, a piece wise-like approximation was proposed, for which the interval of
definition [0,∞) is divided into two subintervals: [0,2.678915610] and
(2.678915610,∞).

For the case of [0,2.678915610] it was required the application of the PSEM method
(^{Vazquez-Leal & Sarmiento-Reyes,
2015}) in order to adequately model Dawson’s function in this interval. Such
as it was explained in Section 4, it was proposed to model Dawson’s function near to
the origin by means of the rational function (13), provided with five parameters to
be determined. Next, a system of algebraic equations was deduced to calculate the
above mentioned parameters, first, by equating the coefficients of powers
*x* and *x*
^{2} from Taylor series of the proposed solution (14) and the expansion for
Dawson’s integral (12), valid for *x* values near the origin. In
order to obtain three additional equations, the points denominated as *A,
B,* and *C* into (13) were substituted. After solving the
resulting system of equations (15), it was obtained (17). It is emphasized that the
procedure mentioned ensures that (44) describes adequately and with good precision
Dawson's function for *x*values from the origin to 2.678915610. With
the purpose to model Dawson’s function for the interval
*x*>2.678915610, the first two terms of its asymptotic expansion
(11) were kept (see (18)). It is remarkable the accuracy with which the above
mentioned truncated asymptotic expression of only two terms (18) describes Dawson’s
integral, starting from relatively small values of x, in absolute value. Although a
better approximation for *r*(*x*) may be obtained,
regarding a rational function of a greater order than five and increasing the
accuracy for values *x*>2.678915610, considering even more terms
for the truncated series (18), the goal of this research was to obtain an
approximation, as simple as can be, to get an invertible and integrable Dawson’s
analytical approximation.

Thus, from the above discussion it was natural the proposal of a piece-wise
approximation of the form (22), where the step function in order to get a compact
expression for *F*(*x*) was employed. Although from
Figure 1 it is clear the accuracy of the
proposed approximation (22), Figure 2 shows
that the biggest relative error committed employing (22) is about 2.5%, from which
it can be inferred that the proposal in this research provides a good precision. As
a matter of fact, the possibility of providing an invertible and integrable Dawson’s
analytical approximation is indeed complicated to obtain, and it was noted that (22)
indeed turns out to have both characteristics.

For the interval *x*≥2.678915610,
*F*(*x*) is given by (18). After carrying out some
algebraic steps, it was possible to express (18) in terms of cubic equation (23),
and from Appendix A, it was concluded that (23) just owns one real root, because
*D*>0 and it is given by (25). On the other hand, for
0≤*x*≤2.678915610, (17) is rewritten in the form (26). After
taking into account the results of Appendix A, it was concluded that (26) provides
three real roots (because *D* is negative for all values of
*x* in the interval), and the root that better describes the
values of *x*, in terms of values of the Dawson’s function, is given
by (31). It was noted that instead of calculating the value of *D*
directly, the novel procedure of plotting the right hand side of (30) was chosen
(Figure 3) with the result mentioned above.
Once again, it is emphasized that handiness of (17) and (18) allowed to get an
invertible expression for Dawson’s integral.

Next, it is noted that our approximation to Dawson’s integral (21) is also
integrable, in terms of elementary functions through the remarkable results (32) or
(33). In order to prove the accuracy of the relevant contribution of (33) that comes
from the interval 0≤*x*≤2.678915610, it was considered to evaluate
the area integral (34) employing (33), and the resulting value was compared with the
numerical value of (34). The numerical value of (34) turned out to be 0.9635924825,
while the value obtained after employing the proposed analytical approximation (33)
was 0.9701971726. In a sequence, Figure 5 shows
that the biggest absolute error committed is about 7×10^{-3}, from which it
is deduced the accuracy of approximation proposed here.

Finally, in order to show the usefulness of the results presented in this work, the
case of a one-phase Stefan problem for a semi-infinite material was introduced.
These problems involve the heat equation, which aims to find the temperature
distribution for the case of melting and solid phases, as well as to determine the
evolution of the free boundary (^{Briozzo & Tarzia,
2010}).

The procedure proposed by ^{Briozzo & Tarzia
(2010)} express the original non-classical heat conduction problem for a
semi-infinite material (35), in terms of the ordinary differential equation (37),
which obeys the conditions (38)-(40), through substitutions (36). Following Briozzo
& Tarzia (2010), the solution of (37) that satisfies the conditions (38) and
(39) is given by (43). It is remarkable that (43) is expressed in terms of integrals
of Dawson's function and error function. It is noted that the integral of the
Dawson's function can be expressed in terms of the generalized hypergeometric
function ^{Murley & Saad, 2008} ), but
as opposed to the proposed solution (32) (or (33)), which is expressed just in terms
of elementary functions (and for the same reason it turned out to be useful for
practical applications), the generalized hypergeometric function is not an
elementary concept. In fact, it requires a special mathematical background to be
employed, because _{
q
}
*F*
_{
p
} is expressed in terms of a special powers series in which the ratio of
successive terms is a rational function of the summation index (Murley & Saad,
2008). Instead, the accuracy of (33) was proved, regarding the relevant contribution
which comes from 0≤*x*≤2.678915610, by evaluating the area (34) after
employing (32), and comparing this value with the corresponding (34), obtained by
numerical methods.

As it was already mentioned, the value obtained after employing the proposed
analytical approximation (33) was 0.9701971726, and the corresponding absolute error
committed by using (32) was only 7×10^{-3} (Figure 5). Thus, it is concluded that our proposed approximation has a
good precision and it is expressed in terms of elementary functions.

On the other hand, ^{Vazquez-Leal et
al. (2012}) provided a handy accurate analytical approximate
solution for the error function *erf*(*x*) (47). The
mentioned article compares different approximations for the error function presented
in the literature. From them, (47) turned out to have a relative error lower than
2×10^{-4}, and for the cumulative error function, the maximum committed
error for region *x*>0 is lower than 9×10^{-5}. Therefore,
(47) has a high level of accuracy, comparable to other approximations found in the
literature; nevertheless, the proposed approximation has such mathematical
simplicity that allows to be used on practical engineering applications and sciences
with good precision. Thus, unlike ^{Briozzo &
Tarzia (2010}) who just provides a symbolic solution to problem (37)-(40),
an analytical approximate solution was provided for the same problem and, from the
above mentioned, with good precision by using (33) and (47) approximations into
(43).

Finally, with the purpose of completing the solution to the proposed problem,
approximations (22), (33), and (47) were substituted into (46) in order to obtain a
numerical approximate solution for *η*
_{0}. As a case study, it was proposed the following values of Stefan’s
number: *Ste*=1 and *λ*=1. The numerical solution of
the above-mentioned equation provided the value *η*
_{0}=1.479186840.

Next, a table is provided with the relevant contributions found in this work.

Proposed approximation to Dawson’s Function:

Proposed approximation to Inverse Dawson’s Function:

Proposed Integral for Dawson’s Function in Terms of Elementary Functions:

Conclusions

This work proposed a novel analytical approximation for Dawson’s function in the form
of a piecewise type function (22). Given the different nature of
*F*(*x*) for values of *x* close to
the origin and far from it, it is natural to propose a solution by sections.
Nevertheless, it is emphasized that although a better approximation for Dawson’s
function may be obtained, assuming a rational function of greater order (see (13))
and increasing the accuracy of *A*(*x*) (see (18)),
keeping more terms of series (11), the goal of this study was to obtain an accurate
expression as simple as possible, in order to get an invertible and integrable
*F*(*x*) function with good precision. To achieve
this goal, the domain of this study was divided into two intervals: [0,2.678915610]
and (2.678915610,∞).

With the purpose of modeling *F*(*x*) in
[0,2.678915610], the PSEM method was used (^{Vazquez-Leal & Sarmiento-Reyes, 2015}), whose methodology required to
employ three known points of Dawson’s integral in order to propose a rational
approximation provided with five parameters, which were successfully determined.
Unlike the above mentioned, for *x*>2.678915610 it was found that
it was possible to model Dawson’s function by keeping the first two terms of the
asymptotic expansion (11). In fact, the approximation in this work has just a
maximum relative error of 2.5% from which it was deduced that the proposal here is
adequate for the purpose of this work. Given the mathematical simplicity of the
expressions mentioned, it was obtained the invertible and integrable Dawson’s
analytical approximations (21) or (22). Finally, it was shown the usefulness of
integral approximation (32) (or (33)) in the interesting case study of a
nonclassical heat conduction problem for a semi-infinite material. Unlike the
symbolic solution presented by ^{Briozzo & Tarzia
(2010)}, an analytical approximate solution with good precision was
presented. It was employed in this work the analytical approximations (22) and (33)
for Dawson’s function and (47) for error function
*erf*(*x*), in order to provide an analytical
approximation with good accuracy for the important physics problem mentioned above.
This suggests that future research should aim to find accurate approximations to
other special functions of mathematical physics.