1. Introduction
Fractional differential equations are an essential means of modeling complex processes in different fields of science [
1,
2,
3,
4,
5]. The Riemann–Liouville fractional integral of order
is defined as as follows:
When
n is a positive integer and
, the Caputo and Riemann–Liouville fractional derivatives of order
are defined as follows:
The Caputo and Riemann–Liouville fractional derivatives are related as
In this paper, we considered the case when the order of fractional differentiation is between zero and one. In the remainder of the paper, the parameter
. Without loss of generality, the lower limit of the integral in the definition of the fractional derivative is zero. The Caputo fractional derivative of order
is defined as follows:
Fractional derivatives and integrals are nonlocal and have a singularity at the endpoint. The properties of the fractional derivatives make them an important tool for describing memory processes. The exponential, sine, and cosine functions have Caputo derivatives:
where
is the Mittag–Leffler function:
Approximations of fractional derivatives involve the use of special functions. The Riemann zeta function is an analytic function that is defined for complex numbers with the real part greater than one as follows:
The polylogarithm function of order
is defined as follows:
and
has a series expansion:
Finite difference schemes for the numerical solution of fractional differential equations use discretizations of the fractional derivative. Two important discretizations of the Caputo fractional derivative are the Grünwald difference approximation and the L1 approximation [
6,
7,
8,
9,
10]. The Grünwald difference approximation has a first-order accuracy and a generating function
. Central difference approximations of integer-order derivatives have a second-order accuracy. Grünwald difference approximation is a second-order shifted approximation of the fractional derivative with a shift parameter
, and it is a generalization of finite difference approximations of integer-order derivatives [
11,
12].
Let
n be a positive integer,
, and
for
. The L1 approximation has an order
and is defined as follows:
where
and
The L1 approximation has a generating function
and a second-order expansion formula [
13]:
The construction of the L1 approximation uses central difference approximations of the first derivative on a uniform net. Over the past few decades, there has been an increasing interest in developing efficient numerical techniques for solving fractional differential equations. The growing interest in this field has resulted in the development of high-order approximations for fractional derivatives and innovative methods for investigating their properties and practical applications. Approximations of the fractional derivative of order
, which are called the
and
formulas, were constructed by Alikhanov [
14] and Gao et al. [
15]. In [
13,
16], we used the L1 approximation and approximations of the second derivative for the construction of second-order approximations of the fractional derivative. Finite difference schemes using L2 formula approximations of the fractional derivative of order
and their convergence were studied by Alikhanov [
17], Lv and Xu [
18], and Wong and Ren [
19]. High-order approximations of the fractional derivative and difference schemes for fractional differential equations were constructed in [
20,
21,
22,
23,
24,
25,
26,
27]. Navot [
28] used Taylor polynomials to deal with the singularity of the fractional integral
. He derived the asymptotic formula for Riemann sums on a uniform net, called the extended Euler–Maclaurin summation formula. The formula was extended by Navot [
29] to functions with a logarithmic singularity. In [
30], we derived the asymptotic formula of the Riemann sum of the fractional integral using the series expansion Formula (
2) of the generating function
. Many of the techniques, originally formulated for ordinary and partial differential equations, have been modified and extended to address fractional differential equations. Spectral methods employ orthogonally based functions such as the Jacobi and Chebyshev polynomials to approximate solutions of fractional differential equations [
31,
32]. Fractional multistep methods, which can be viewed as an extension of classical multistep methods, were studied in [
33,
34]. Methods using spline interpolation were presented in [
35,
36]. Numerical methods using fractional wavelets were explored in [
37,
38].
This paper is a continuation of the work in [
39]. In the paper, we study the properties of an approximation of order
of the Caputo derivative and the convergence of the corresponding difference schemes for the two-term ordinary fractional differential equation and the time fractional Black–Scholes equation for options pricing. A main approach for constructing approximations of fractional derivatives involves interpolating the function
f in the definitions of fractional derivatives over a suitable stencil on a uniform grid. Using this method, formulas for L1 and L2 approximations are derived, as well as high-order approximations of fractional derivatives. The approximations obtained in this way have generating functions that include the polylogarithm function, and their expansion formulas are derived using (
2). The direct application of this method leads to the construction of approximations of fractional derivatives with weights whose exponents are greater than
, as the fractional integral of a negative order is divergent. Naturally, the question arises of constructing and studying the properties of approximations whose weights contain terms with exponents smaller than
. A direct application of (
2) for the function
leads to the construction of an approximation of the fractional derivative and its expansion formula, which has weights of
. The construction of this approximation and the corresponding approximations of order
were presented in [
39]. The comparison performed with the L1 approximation shows that these approximations have the same order, similar properties of the weights, and comparable performance for the numerical solution of fractional differential equations. In this paper, we derive an estimate for the error of the approximation of order
of the fractional derivative with a generating function
and the corresponding difference schemes for the numerical solution of fractional differential equations. The motivation of the study this approximation stems from the simpler formulas of its weights, its mathematical properties and applications, and for constructing efficient methods for numerically solving fractional differential equations.
In [
39], we constructed an approximation of the Caputo fractional derivative and its asymptotic formula of order
:
Approximation (
5) has a generating function
. By substituting the derivative in the right-hand side of (1) using the first-order backward difference
, we find an approximation of the fractional derivative:
Approximations (
5) and (
6) have an order
when the function
f satisfies the condition
. In [
39], we extended the approximation (
6) to all functions in the class
by assigning the values of the last two weights.
where
. In this paper, we investigate the properties of the approximations (
6) and (
7) and applications of the approximations for the numerical solution of ordinary and partial fractional differential equations. The outline of the paper is as follows. In
Section 2, we study the properties of the weights of the approximation (
7) and we derive the following inequality:
In
Section 3, we derive estimates for the errors of the approximations (
6) and (
7). In
Section 4 and
Section 5, we construct finite difference schemes for numerically solving the two-term ordinary fractional differential equation and the time fractional Black–Scholes equation for option pricing. The convergence of the finite difference schemes is proven, and bounds for the errors of the methods are derived using Inequality (
11) and the estimate for the truncation error of the approximation.
The approximation (
5) is constructed using the series expansion Formula (
2) of the function
and can be extended to arbitrary orders. Equation (
2) holds for any value of
that is not an integer. This indicates that the approximations (
5)–(
7) also apply to fractional derivatives of order greater than one. One direction for future work related to the paper is to consider applications of the discussed approximations for numerically solving fractional differential equations of orders greater than one. The L1 approximation (4) and the approximation (
7) have an order
and similar performance and properties of the weights [
39]:
The properties of the L1 approximation and the approximation (
7) presented in
Section 2 enable an efficient analysis of the convergence and error of difference schemes for fractional differential equations. One approach for constructing approximations of integer-order and fractional derivatives was presented in [
16]. Another direction for future work is constructing high-order approximations for fractional derivatives that have the properties (
12) of the weights, investigating the properties of the constructed high-order approximations, and analyzing the convergence of the difference schemes for fractional differential equations. Additionally, it is important to examine the behavior of the numerical solution for values of the parameters where the denominator has a small value.
3. Estimate for the Error
In this section, we use the method from [
28] to derive estimates for the errors of Approximations (
6) and (
7). Let
and
The function
satisfies
. From Taylor’s Theorem,
where
for
. Let
Denote . The function satisfies , and its second derivative is not defined at the point x.
Lemma 2. Let . Then, Proof. The function
satisfies
In view of Equation (
20), the fractional derivative of the function
satisfies
which can be written in the form:
From (
21) and (
22), it follows that
□
The trapezoidal rule of a function
has a second-order accuracy when
. The error of the trapezoidal rule
satisfies [
42]
The second derivative of the function is undefined at the point x, which leads to a lower order of accuracy of the trapezoidal rule in the interval . Now, we estimate the error of the trapezoidal rule of the function . Let and be the error of the trapezoidal rule of in the interval .
Proof. Let
be the of trapezoidal rule of
in the interval
.
Let
be the error of the trapezoidal rule of
in the interval
.
The error
satisfies
From Equations (
23) and (
24), we obtain a bound for the error
of the function
in the interval
:
□
The trapezoidal rule of the function
satisfies
The function
has a value at zero:
Let
be the error of Approximation (
5).
Theorem 1. Let and . Then, Proof. From the definition (
20) of the function
,
From Equation (
19), we find
where
From Taylor’s Theorem,
where
for
. Therefore,
The terms on the right-hand side of (
25) satisfy the estimates:
Let
be the error of Approximation (
6).
Approximation (
6) is constructed from Approximation (
5) by substituting
with first-order backward difference approximation
.
Claim 5. Let and . Then, Proof. From Taylor’s Theorem,
where
. The error
of Approximation (
6) satisfies
□
Let
be the error of Approximation (
7).
The weights of Approximation (7) are defined with (8)–(10). Denote
The function
satisfies
, and the second and third derivatives of
f and
are equal. Therefore,
Lemma 4. Let . Then, 5. Time Fractional Black–Scholes Equation
The time fractional Black–Scholes equation is a fractional partial differential equation, which is used for modeling the prices of the options [
49,
50,
51,
52,
53,
54,
55].
where
C is the options price,
T is the expiry time,
r is the risk-free rate,
is the volatility, and
S is the underlying stock price. Fractional Black–Scholes Equation (
39) has terminal and boundary conditions:
and the fractional derivative is
The terminal condition is transformed to an initial condition with the substitution [
55]:
By applying the substitution to (
40), we find [
55]
In view of the definitions of the Riemann–Liouville and Caputo fractional derivatives and (
1)
The function
satisfies the partial fractional differential equation:
The substitution
transforms (
41) into a linear partial fractional differential equation [
55]:
Finite difference schemes for the time fractional Black–Scholes equation were constructed in [
55,
56,
57,
58]. Now, we construct an implicit finite difference scheme, which uses the approximation (
7) of the fractional derivative and central difference approximations of the partial derivatives for the following time fractional Black–Scholes equation:
where
. Equation (
43) has initial and boundary conditions:
Let
M and
N be positive integers and
be a rectangular grid on
, which has a step size
in space and
in time:
Denote
. The central difference approximations of the partial derivatives of Equation (
43) have second-order accuracy:
where
and
. The numbers
and
are the bounds for the third- and fourth-order partial derivatives:
The numerical solution
of Equation (
43) on layer
m of the grid
satisfies
for
and has has initial conditions:
Denote
and
The numerical solution of the time fractional Black–Scholes Equation (
43) is a solution of the system of linear equations:
Denote by
the error of the finite difference scheme (
44) at the point
and by
and
the the truncation errors of the approximations of the left-hand and right-hand sides of (
43). The errors
on row
of the grid
satisfy
The errors are equal to zero,
at the boundary points of
. Let
be the
-dimensional tridiagonal matrix with entries
on the main diagonal and
and
below and above the main diagonal. The error vector
of row
m is a solution of the matrix equation:
where
The numerical solution on the first row of
is computed with Approximation (
28):
Denote
. Then,
where
and
The errors of the numerical solution of the fractional Black–Scholes Equation (
43) on the first row of
are the solutions of the system of linear equations:
Let
be the
-dimensional tridiagonal matrix with elements
, and
. The error vector for the first row satisfies
where
5.1. Convergence and Error Estimate
Let for all and . The norm of a matrix is the maximum of the absolute row sums.
Theorem 4. Let and . Then,for all . Proof. When
, the gamma function satisfies [
59]
When
, we have that
The matrices
and
are
M-matrices. From the Ahlberg–Nilson–Varah bound [
60,
61],
The
norm of the error vector of the first row of
satisfies the bound:
Suppose that Inequality (
46) holds for all
. From (
45),
Corollary 2. Let . Then,for all . 5.2. Numerical Examples
Numerical results for the error and order of the numerical solution of time fractional Black–Scholes Equation (
43) are given below.
Example 3. Consider the following time fractional Black–Scholes equation: Equation (
47) has a solution
. The experimental results of the numerical solution of Equation (
47) with parameters
are given in
Table 5 and
Table 6. The orders of convergence of the finite difference scheme are
in time and
in space.
The order of the error of the finite difference scheme for the fractional Black-Scholes equation (
47) in space is greater than the order in time. When the step size
h is small, the errors in the temporal direction dominate the errors in space. The errors of the numerical solution to (
47) for different values of
N and
M are plotted in
Figure 1.
Example 4. Consider the time fractional Black–Scholes Equation (
41)
for pricing European call options, with a source term and initial and boundary conditions: The European call option premium curves for different values of
are given in
Figure 2, for values of the parameters
,
,
,
,
(year), and strike price
. Regarding near-the-money options, lower values of
price them lower and evaluate higher out-of-the-money and in-the-money options, compared to the classical Black–Scholes dynamics.
The graphs of the call options and put options premiums for
and all
t are given in
Figure 3 and
Figure 4. To conclude the section with computational simulations, we compare the numerical solutions to (
41), using the scheme explained in the paper (7)–(10) and the L1 scheme, as defined, for example, in [
55]. We plot the difference as the solution for
, obtained from the L1 scheme, subtracted from the solution, obtained by the
scheme (
7). The results for
and for all
t are given on
Figure 5.