Accurate analytic approximation for a
fractional differential equation with a
modified Bessel function term
arXiv:2502.14871v1 [math.GM] 31 Jan 2025
Byron Droguett1,a , Pablo Martin2,a Eduardo Rojas3,b Jorge Olivares 4,c
a
b
Department of Physics, Universidad de Antofagasta, 1240000 Antofagasta, Chile.
Department of Mechanical Engineering, University of Antofagasta, Antofagasta,
Chile
c
Department of Mathematics, University of Antofagasta, Antofagasta, Chile
1
3
byron.droguett@uantof.cl , 2 pablo.martin@uantof.cl ,
4
eduardo.rojas@uantof.cl ,
jorge.olivares@uantof.cl .
Abstract
A new approximation function is introduced to fit the solution of a fractional
differential equation of order one-half. The analyzed case includes a nonhomogeneous term defined by a modified Bessel function of the first kind.
The analytical solution of this equation corresponds to the product of two
modified Bessel functions. The new fitting function is developed using the
multipoint quasi-rational method, which uses both the series expansion of
the Bessel function and its asymptotic expansion. Additionally, a significant
modification is introduced to the structure of the fitting function to capture
two terms of the asymptotic expansion of the Bessel functions. We show an
example for fixed values, and the maximum relative error of the fitting function for the solution of the fractional differential equation of order one-half
is 0.18%, a remarkably low value considering that only six fitting parameters
are used.
1
Introduction
Fractional differential equations extend classical differential equations by allowing
derivatives of noninteger orders, providing a flexible framework to understand complex physical phenomena and their dynamics [1, 2]. These fractional derivatives are
particularly effective for modeling systems with memory effects and nonlocal interactions, making them an attractive tool to be applied in diverse fields in science and
engineering [3–5]. For example, the Caputo derivative has been extensively used
to model anomalous diffusion processes, capturing deviations from the classical diffusion behavior in systems such as porous media, biological tissues, and turbulent
flows [4, 6]. Similarly, in fluid and solid mechanics, fractional derivatives offer an
enhanced description of the material response. They contribute to accurately characterize complex processes like stress relaxation, creep, and delayed elastic responses
under deformation, as demonstrated in studies of polymers, biomaterials, and composite structures [7, 8]. Furthermore, fractional calculus has found applications in
control theory, signal processing, and electrochemical modeling, where it provides
a robust mathematical tool to capture real-world complexities beyond the reach of
classical approaches [9, 10].
On the other hand, the Bessel equation and its modified form play a fundamental
role in modeling a wide range of problems in science and engineering. These equations naturally emerge in various contexts, including: electrodynamics, where they
describe wave propagation in cylindrical geometries [11]; plasma physics for analyzing stability and wave behavior in magnetized plasmas [12]; heat transfer for solving
problems involving cylindrical or spherical symmetry [13]; and chemical engineering
in the study of diffusion-reaction systems [14]. The general solution of the modified
Bessel equation of the first kind, Iν (x), is commonly expressed as a power series.
However, for large or intermediate values of x, this series representation requires a
significant number of terms to achieve adequate accuracy, which makes computations time consuming in practical scenarios. To address these challenges, various
fitting functions have been developed to approximate the Bessel functions [15, 16].
However, the accuracy of these approximations is often limited to specific regions of
the domain, reducing their utility in broader applications. A notable improvement
to provide an extensive analytic approximation was introduced using the multipoint quasi-rational approximation (MPQA) technique [17], valid for all positive
values of x. This method achieves impressive generality and reduces the time of
mathematical operations for solving physical and engineering applications. The
study of this function with a fixed value of ν was studied in [18–20].
In this research, we address the fractional nonhomogeneous differential equation
with an arbitrary Caputo derivative, where the nonhomogeneous term is a modified
Bessel function of the first kind. When the Caputo derivative is set to 1/2, the
solution involves the product of two modified Bessel functions. By applying the
MPQA technique, we approximate the exact solution and introduce a highly precise
analytic approximation for Iν valid for all ν in the range (0, 1), using only six
2
fitting parameters. To obtain the new approximation additional information from
the asymptotic expansion of the Bessel function was incorporated. Specifically, it
combines the sum of two hyperbolic functions with rational functions to capture the
first two leading terms of the asymptotic expansion. This innovation significantly
improves accuracy, particularly for intermediate values of x, and it represents a
substantial advancement over existing methods, such as those proposed in [18–23].
This paper is organized as follows. In Sect. 2, we present the fractional differential equation. In Sect. 3, we apply the multi-point quasi-rational approximation
technique. In Sect. 4, the optimization procedure is developed, and in Sect. 5, we
present our conclusions.
2
Fractional differential equation and analytic solution
The nonhomogeneous fractional differential equation under analysis is expressed as
C
Dxα y(x) = Iν (x) ,
(2.1)
where the nonhomogeneous term Iν (x) represents the modified Bessel function of
first kind with arbitrary order ν, while α denotes the order of the Caputo derivative.
This equation is particularly significant in the study of fractional calculus because
of the compatibility of the Caputo derivative with the physical boundary and initial
conditions. The solution to this fractional differential equation can be derived using
the Laplace transform. The Laplace transform of the Caputo fractional derivative
and modified Bessel function are given by respectively:
⌈α⌉−1
L{
C
Dxα y(x)}
α
= s ỹ(s) −
√
L {Iν (x)} =
X
sα−k−1 y (k) (0) ,
(2.2)
k=0
s2 − 1 + s
√
s2 − 1
−ν
,
(2.3)
where ỹ(s) is the Laplace transform of y(x) and y (k) (0) represents the initial conditions of the solution. This expression highlights the utility of the Caputo derivative
in retaining classical interpretations of initial conditions. The general solution is
obtained by the inverse Laplace transformation
√
[α]−1
X
π
x2
xk
α
ν
y(x) = α+2ν Γ(1 + ν)x (ix) 2 F̃3 (a1 , a2 ; b1 , b2 , b3 ; ) +
y (k) (0) .
2
2
Γ(1
+
k)
k=0
(2.4)
The function 2 F̃3 is the regularized generalized hypergeometric function defined by
x2
) =
F̃
(a
,
a
;
b
,
b
,
b
;
2 3 1 2 1 2 3
2
3
x2
2 F3 (a1 , a2 ; b1 , b2 , b3 ; 2 )
Γ(b1 )Γ(b2 )Γ(b3 )
,
(2.5)
where
1+ν
1
b3
α
1
= a2 − = ,
b1 = a1 + = b2 − .
(2.6)
2
2
2
2
2
When the Caputo derivative takes the value α = 1/2 the Eq. (2.5) simplifies to the
product of two modified Bessel functions of the first kind. Therefore, the solution
to the Caputo equation (2.1) has the form
a1 =
y(x) = i
ν
r
x
x
πx
I 1 (2ν−1)
I 1 (2ν+1)
.
2 4
2 4
2
(2.7)
By summing the indexes terms of the multiplied Bessel functions, the result is ν.
Therefore, this value of the nonhomogeneous term is conserved.
3
Approximation procedure
The modified Bessel function of the first kind is a solution to the modified Bessel
differential equation. Its series and asymptotic behavior are given by [24]
∞
x ν X
x 2k
1
,
Iν (x) =
2 k=0 k!Γ(k + ν + 1) 2
ex
4ν 2 − 1 (4ν 2 − 1)(4ν 2 − 9)
Iν (x) ∼ √
1−
+
− ... .
8x
2!(8x)2
2πx
(3.1)
(3.2)
To simplify the computation process of the Bessel series, an accurate fit is proposed based on an extended multi-point quasi-rational approximation method. This
method captures the behavior of the modified Bessel function for both small and
large argument values. A combination of rational expressions, elementary hyperbolic functions, as well as, fractional powers of the x variable are used for writing
the following general approximation function:
x ν (p + p x2 ) cosh(x) + (p + p x2 ) sinh(x)
0
2
1
3
x
e
,
Iν (x) =
2
2
ν/2+1/4
2
Γ(ν + 1)(1 + λ x )
(1 + qx2 )
(3.3)
where pi , q and λ are fitting parameters. The parameters pi and q are set through
an analytic procedure, while λ constitutes a free parameter for optimization. Thus,
this parameter is free, allowing the identification of the optimal fitting function that
minimizes the error. The new fitting function considers the sum of two hyperbolic
functions weighted by polynomials. This modification allows us to capture two
terms of the asymptotic expansion for large values of the argument, instead of just
one, as in previous work [19].
Equating the two first leading terms of Eqs. (3.2) and (3.3) for asymptotic
behavior to respect to x with ν fixed we obtain
4ν 2 − 1
ex
1
ex
√
1−
≈ √
(p2 + p3 /x) ,
(3.4)
8x
2πx
2πx f (λ, ν)
4
where
f (λ, ν) =
21+ν λν+1/2 Γ(1 + ν)
√
.
2π
(3.5)
From Eq. (3.4) the following expressions are obtained:
p2 = qf (λ, ν) ,
p3 =
1 − 4ν 2
f (λ, ν)q .
8
(3.6)
To find the other parameters an expansion around x = 0 is performed. The series
of the hyperbolic function and the binomial series are given by
sinh(x) =
x2n+1
,
(2n
+
1)!
n=0
X
cosh(x) =
∞ X
β
(1 + (λx) ) =
(λx)2n ,
n
n=0
2 β
X x2n
(2n)!
n=0
,
|x| < 1 ,
where β = 12 (ν + 1/2) and the generalized binomial coefficient are given by
1
β
= β(β − 1) · · · (β − (n − 1)) .
n
n!
(3.7)
(3.8)
(3.9)
Let us compare the following terms in Eq. (3.1)
x ν x 4
1
1
1 x 2
Iν (x) =
+
··· ,
1+
Γ(1 + ν) 2
1+ν 2
2(1 + ν)(2 + ν) 2
(3.10)
with the approximation function
∞ X
(p0 + p2 x2 )
(p1 + p3 x2 ) 2n
+
x = (1 + qx2 )
(2n)!
(2n
+
1)!
n=0
X
∞ x 4
1 x 2
1
β 2n 2n
× 1+
+
···
λ x . (3.11)
1+ν 2
2(1 + ν)(2 + ν) 2
n
n=0
We compare the coefficients of the polynomials and derive the following system of
equations for the fitting parameters:
5
p0 + p1 = 1 ,
(3.12)
p0 p1
1
+
+ p2 + p3 = q + βλ2 +
,
(3.13)
2
6
4(1 + ν)
p0
p1
p2 p3
1
1
4
2
+
+
+
=
(β − 1)βλ + q βλ +
24 120
2
6
2
4(v + 1)
2
1
βλ
+
,
(3.14)
+
4(v + 1) 32(v + 1)(v + 2)
p2
p3
1
βλ2
1
4
+
= q
(β − 1)βλ +
+
24 120
2
4(v + 1) 32(v + 1)(v + 2)
βλ2
(β − 1)βλ4
+
.
(3.15)
+
8(v + 1)
32(v + 1)(v + 2)
In the system, we have truncated terms of degree six or higher in Eq. (3.11). Our
aim is to avoid zeros in the denominator of the fitting function, the so called ”defects” in Padé approximation technique [15,16]. The parameter q must be restricted
to positive values, along with λ. By solving the system of equations, we obtain the
q function as a function of the parameter λ for all ν. Therefore, the q function is
given by:
pπ
(−4ν 2 − 240(β − 1)βλ4 (ν + 1)(ν + 2) + 24βλ2 (ν + 2)(2ν − 3) + 1)
2
q(λ, ν) =
√
1
4 2ν (4ν 2 − 49) λν+ 2 Γ(ν + 3) + 3 2π(ν + 2) (20βλ2 (ν + 1) − 2ν + 3)
(3.16)
In Fig. (1), the parameter q is shown as a function of the pair (λ, ν). The condition
is that q must be positive. It can be observed that there is a region between the
asymptotes where this function is positive for all values of the parameter ν ∈ (0, 1).
The objective is to avoid zeros in the denominator of the fitting function.
6
Figure 1: Parameter q as a function of the parameters λ and ν.
4
Optimization
The criterion for determining λ is by comparing the maximum error of the fitting
function for different values of λ and ν. Two types of error definition are established,
a punctual error εp (x, λ, ν) and a global error ε(λ, ν) evaluated on an interval [a, b]
of the independent variable x. Theses are respectively
|Iν (x) − Ieν (x, λ)|
,
Iν (x)
ε(λ, ν) = max [εp (x, λ, ν)]x∈[a,b] .
εp (x, λ, ν) =
(4.1)
(4.2)
In Fig. (2), the global error is shown as a function of the parameters (λ, ν). Several
global and relative minima are indicated in the white region of the figure. These
values can be modeled by the linear equation ν = 24.5(0.265 − λ), represented by
the red dashed line. Therefore, the optimal values for each parameter of the fitting
function can be determined.
7
1.0
= 24.5(0.265
)
0.004592
0.004082
0.8
0.003571
0.003061
0.6
0.002551
0.002041
0.4
0.001531
0.001020
0.2
0.000510
0.0
0.000000
0.200 0.225 0.250 0.275 0.300 0.325 0.350 0.375 0.400
Figure 2: Relative global error ϵ as a function of the optimization parameter λ and
ν for the fitting function
In Fig. (3) (a), the global minimum error is shown as a function of the ν
parameter. We can observe that the error for all values of ν is of the order of 10−3 .
In Fig. (3) (b), the q parameter is shown as a function of ν and is always positive.
Therefore, there are no issues with poles in the denominator.
(a)
0.00200
(b) 10
0.00175
8
0.00150
0.00125
q
0.00100
6
4
0.00075
0.00050
2
0.00025
0.00000
0.0
0.2
0.4
0.6
0.8
0
0.0
1.0
0.2
0.4
0.6
0.8
1.0
Figure 3: (a) shows the global minimum error as a function of ν, where for each
fixed ν, the optimal λ parameter is modeled by ν = 24.5(0.265 − λ). In Fig. (b),
we can observe that using this linear approximation, the values of the q parameter
remain positive for all ν.
We present a specific example for the fixed value ν = 7/10, where Eq. (2.7)
8
takes the form:
y(x) = i
7/10
r
x
x
πx
I1/10
I3/5
.
2
2
2
(4.3)
To apply the approximation method to the solution for this fixed value of ν, the
minimum error εmin is located at λmin = 0.236. In Fig. 4, the relative punctual
error εp is shown for the different analytical fitting functions developed in this study.
The maximum errors for Ie1/10 (x) and Ie3/5 (x) are 0.1% and 0.06%, respectively.
Finally, the maximum relative error for the product of these two modified Bessel
functions, Ie1/10 (x)Ie3/5 (x), which corresponds to the error of the solution of the onehalf nonhomogeneous fractional differential equation addressed in this work, reaches
0.18%.
0.00200
0.00175
0.00150
I1/10(x/2)
I3/5(x/2)
I1/10(x/2) I3/5(x/2)
0.00125
p
0.00100
0.00075
0.00050
0.00025
0.00000
10 1
100
101
x
102
103
Figure 4: The relative punctual error εp as a function of the independent variable
x for the general solution.
5
Conclusion
In this work, we obtain the exact solution in a general form for all ν to a nonhomogeneous fractional differential equation, which was solved using the Laplace
transform. Subsequently, we evaluate the equation considering the Caputo derivative of order one-half. The solution is expressed as the product of two modified
Bessel functions, where the sum of their respective orders equals the order ν of the
nonhomogeneous term, which itself is a modified Bessel function of order ν.
We present a new approximation for the modified Bessel function of the first
kind. To ensure the accuracy of the solution for all values of x and ν, both asymptotically and near zero, we introduce the multipoint quasi-rational method (MPQA).
9
This method is valid for all values of ν ∈ (0, 1). The proposed method improves
upon previous approaches by incorporating the sum of hyperbolic functions into
the fitting function. This inclusion allows capturing two terms of the asymptotic
series instead of just one, which enhances the precision of the approximation. To
determine the optimal parameters of the fitting function, we apply an optimization
process that minimizes the global error. This process provides a nearly linear approximation of the minimal global error as a function of certain parameters, thus
improving the accuracy of the solution. To demonstrate the effectiveness of the
MPQA method, we present a specific example with ν = 7/10. Using this value,
we approximate the modified Bessel functions I1/10 and I3/5 , which form part of
the analytical solution to the fractional differential equation. The MPQA method
achieves maximum relative errors of 0.1% and 0.06% for I1/10 and I3/5 , respectively.
For the total solution, the maximum relative error is just 0.18%. These results are
notable given that only six parameters are used to fit each Bessel function, and the
new approximation remains valid across the entire domain.
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