New Refinements for the Error Function with Applications in Diffusion Theory

In this paper we provide approximations for the error function using the Padé approximation method and the Fourier series method. These approximations have simple forms and acceptable bounds for the absolute error. Then we use them in diffusion theory.


Introduction and Motivation
The error function er f is a special function. It is widely used in many areas of research, such as: statistical computations, mathematical models in biology, mathematical physics, and diffusion theory.
In this work we intend to draw attention to the Padé approximant method and to the Fourier series method by some inequalities and approximations which involve the even function exp(−x 2 ) and the error function.
The error function is related to the function expression for a Gaussian distribution and has the form The complementary error function is The integral in the error function cannot be evaluated analytically, so values for er f and er f c are computed via approximations and often available in tables.
There are two main aspects in the study of the error function. The first aspect refers to the establishment of some bounds for the error function. The second aspect is to approximate the error function with different elementary functions such that the absolute error function is as small as possible.
Pólya [8] proved that the inequality holds for all x > 0. Neumann [15] proved that the double inequality holds for all x > 0.
Recently, Yang and Chu [13] improved the above inequalities by the following inequalities for all x > 0.
Furthermore, in sigmoid functions theory, the function √ π 2 er f (x) is compared with the following algebraic functions The aim of our work is to refine these inequalities. First, we will use the Padé approximant method in order to obtain a new approximation of the error function. The proposed approximation has a simple rational form and it is easy to programme.
We notice that the expressions A (x), B (x), C (x) are positive for all x > 0.
After several attempts we obtain that the best approximation for the function The second idea is that the function exp −x 2 is even, so it can be expanded in Fourier trigonometric series, e.g., In the following we will present our algorithm for the first expansion. We define the function The power series expansion of exp −x 2 − F 1 (x) near 0 is In order to increase the speed of the function F 1 (x) approximating exp −x 2 , we vanish the first coefficients as follows: and we obtain a = − 4 3 and b = 5 6 . Therefore we have

Main Results
In the first part of this section we will prove our main results obtained using Padé approximant method for Theorem 1. The following inequalities hold: In order to find all positive critical points of g, first we compute g (x).
Solving the equation g (x) = 0 yields x ≈ 3.77096. Now we evaluate g (x) at the critical points and the endpoints of the domain, taking limit at the ∞: The function g has only one root x ≈ 4.418.
Since er f (4.418) = 0.999999999584246, the error function can be considered er f (x) = 1 for x ∈ [4.418, ∞), hence our rational approximation offer good bounds near the origin for the error function.
The proof of Theorem 1 is complete.
In the following we will prove that our rational inequalities are more precise than Neuman's type inequalities. holds.
(ii) For every x ≥ 0, the inequality holds.
Proof. (i) The inequality from (i) takes the equivalent form: for all x > 0. It is sufficient to prove that for all x > 0, or, equivalently, for all x > 0. The above inequality is obviously true.
(ii) We consider the function To find all positive critical points, first we compute h (x): Solving the equation h (x) = 0 yields the positive roots: Moreover, the function h (x) exists everywhere. Evaluate h (x) at the critical points and take the limit: Summarizing the results, we deduce that h (x) ≤ 0 for all x ≥ 0.
(iii) We introduce the function In order to find all positive critical points, first we compute p (x): The equation p (x) = 0 has the positive roots: Evaluate p (x) at the critical points and take the limit: Our upper bound for the error function er f (x) also offers a better approximation near the origin than Pólya's inequality, due to the following proposition.
hold for all x > 0 or, equivalent: Using the Padé approximation for er f (x), we obtain an improved version of Komatsu's inequalities as follows.
Proposition 3. The following inequalities hold: Proof. (i) The inequality from (i) takes the equivalent form: The function from the left-hand side has only one positive root x ≈ 1.28089 and it is positive for x ∈ (0, 1.28089).
(ii) The inequality from (ii) can be rewritten as follows: The function from the left-hand side has two positive roots: x ≈ 1.33740 and x ≈ 4.37287 and it is positive for x ∈ (0, 1.33740) ∪ (4.37287, ∞). Therefore, we improve Komatsu's inequalities near the origin providing sharp bounds of rational types for the error function.
In the second part of this section we will establish our approximations of the error function using Fourier trigonometric series expansions.

Theorem 3. The following inequalities
hold for all x > 0.
In order to prove the left-hand side inequality, first we notive that the function has the positive roots x = 0, x ≈ 2.57617. Since f 3 π 2 > 0, it follows that f 3 > 0 only on (0, 2.57617), hence we have to show the left-hand side inequality only for x ∈ (0, 2.57617).
In the following we derive approximation to three decimals of precision for the function er f (x) on a large neighbourhood of the origin.
We note by and In Theorem 3 we proved that for all x > 0. Now we consider the convex combination or, equivalently, The derivative of θ (x) is θ (x) = 1.12838 e −x 2 − 0.848334 cos x + 0.0399333 cos 2x − 0.145444 cos 3x − 0.045555 .

Application Case
As an example to apply the error function, one case is considered for the diffusion and reactions in sediments.
We assume that oxygenated water comes in contact with sediments at the bottom of a lake at the time of "fall turnover". The water contains 300 µmoles of O 2 /liter. We suppose that no reactions consume the oxygen. We want to find what is the concentration of oxygen in sediments after one day, one week and one month after the turnover event.
Mathematically, the diffusion problem for a constant source region diffusing into a space of unlimited extent ("semi-infinite half-space") is to solve given the boundary conditions that the concentration at the sediment-water interface C 0 is always fixed and no reactions consume the oxygen, and initial condition that the concentration below the boundary is initially everywhere 0. This linear PDE, having a solution that requires one initial condition and two boundary conditions is called Fick's second law for isotropic one-dimensional diffusion with D independent of concentration.
The solution to this problem involves the error function: where C (x, t) is the concentration at a given depth and time, C 0 is the concentration at the interface, D is the diffusion coefficient, x is depth below the interface, and t is time. An appropriate value of the diffusion constant D is 2 × 10 −5 cm 2 s −1 . We choose a depth range x of 0-20 cm. Since the units of time in the diffusion constant are seconds, we calculate our valued for concentration using seconds for t to obtain a good result.
In order to solve this equation, we will use our approximation for the error function: where η (x) is defined by (2) or C (x, t) = C 0 1 − ϕ , where ϕ (x) is defined by (4).

Conclusions
In our work we use Padé approximation method and Fourier series method in order to obtain approximations for the error function. Using Padé approximation method, we obtain the following bounds: er f (x) ≥ 2 √ π A (x), for all x ≥ 0 and er f (x) ≤ 2 √ π B (x), for all x ∈ [0, 4.418) and the following approximation to four decimals of precision for the function er f (x) on a large neighbourhood of the origin: with the absolute error |ε 1 (x)| < 9.28402 × 10 −4 , where C (x) = 302,379x 5 +1,027,320x 3 +10,242,540x 52,595x 6 +758,625x 4 +4,441,500x 2 +10,242,540 . Furthermore, using Fourier series method, we obtain the following bounds: Funding: The APC was funded by "Dunȃrea de Jos" University of Galaţi, Romania.

Conflicts of Interest:
The author declares no conflict of interest.