4. Technique for Approximating Solutions
Using the Galerkin method [
36] and the generalized fractional-order B-poly basis set, we exploit a technique to seek practical solutions to fractional-order partial differential equations. We intend to apply Caputo’s fractional derivative to the fractional-order B-ploys. The examples of Caputo’s derivatives of the B-polys are provided in the last column of
Table 1. Using the recent technique, we transform the fractional-order linear partial differential equation into an operational matrix and the initial conditions and boundary conditions are applied to the operational matrix. The presumed approximate solution, Equation (3) is substituted into the fractional-order differential equation and by-products are separated in terms of integral products in both variables
x and
t. Finally, both sides of the fractional equation are multiplied with fractional B-polys basis elements,
, and the integrations are carried out using symbolic program, Mathematica, over the closed intervals [0, R] and [0, T], respectively. For example, the integration of the two fractional B-polys is given in the closed symbolic formula
Caputo’s derivative defined in Equation (2) is applied to the fractional B-ploy basis set, leading to the following closed results:
and the integrals of some arbitrary functions are given
With the help of these analytic formulas Equations (7)–(9), the operational matrix is constructed. The inverse of the operational matrix is required to find out the unknown coefficients of the linear combination in Equation (3). In the next section, we will describe our technique, as well as how to obtain a desirable result for the linear fractional-order partial differential equation. The technique will be employed in four examples to demonstrate that it works appropriately for approximating the accurate solutions. Plots of the approximate as well as exact solutions will be presented for comparison. An absolute error analysis of the fourth example will be introduced to show that, when the basis set of the fractional B-polys is enlarged, the accuracy of the solution is increased. Similarly, the error investigation can be carried out for other examples considered in this study. In the following section, for simplicity, we would like to drop off subscript and from the fractional B-polys, .
Example 1: Let us introduce a linear partial fractional-order differential equation of the form The ideal solution to Equation (10) is
. A numerical solution is sought out in the intervals
. The assumed solution,
, is substituted into the Equation (10) and the result is presented below:
Caputo’s derivative operator is applied to Equation (11). The product of fractional B-polys
from the basis set is multiplied on both sides of the Equation (11) and the integration on both variables is calculated over the intervals using a symbolic program. This operation provides the following equation
where
with
. The current technique leads to a system of
equations. The elements of matrix
are the unknown constants that are involved in those equations. After further simplification, the right-hand side column matrix
and the matrix elements of operational matrix
in terms of inner products of B-polys are given
The partial fractional-order differential Equation (10) is now transformed into a matrix equation
. By deleting the rows and corresponding columns of the equation (13), the initial conditions are imposed on the operational matrix equation
and the corresponding matrix W, so that the solution vanishes at
t = 0 and
x = 0. The operational matrix X was coded in the symbolic language Mathematica to determine its inverse. The inverse matrix was multiplied by the column matrix W to yield values of the unknown coefficients
. The emerging estimated result is composed of the linear combination of the B-poly basis set via Equation (3). The process provides a valid approximate solution
of the Equation (10) using B-polys of fractional-order
and fractional differential-order of
in Equation (10) is given below:
From the above result, it is noted that the approximate solution is very accurate. We have experimented with different values of fractional order
of the differential equation while keeping the same order
of the fractional polynomials basis set, the results remain the same with various values of
. To solve the fractional-order partial differential Equation (10), we choose
n = 1 and
order B-poly basis set
and
in variables
t and
x, respectively. The corresponding coefficient values we obtained are
. The Caputo’s derivative of the fractional B-poly basis set is
, as seen in the last column of
Table 1. A 3D plot of the estimated and the exact results of Equation (10) are presented in
Figure 1 for comparison, and an excellent agreement can be seen between both results at the level of machine accuracy. Note that when
t =
x is substituted into Equation (14), the absolute error can be observed in the order of
exhibiting the great aspect of constancy in one-dimension
x. In the example, the absolute error between the results, both exact and approximate, shows that both results have excellent reliability. The absolute error in the 3D graph is also presented on the right-hand side in
Figure 2. The 3D graph shows that the absolute error in the converged solution is of the order of
.
Example 2: Consider another example of fractional-order linear partial differential equation with different initial condition The ideal solution of the Equation (15) is
. The function
, is called the Mittag–Leffler function [
39] and is described as
. In the summation of Mittag–Leffler function, we only kept
k = 15 in the summation of terms. Therefore, the accuracy of the numerical solution will likely depend on the number of terms that we would keep in the summation of the Mittag–Leffler function. According to Equation (3), an estimated solution of Equation (15) using the initial condition may be assumed as
After substituting this expression into the Equation (15). The Galerkin method, [
29] and [
32], is also applied to the presumed solution to obtain
Caputo’s fractional derivative is applied to Equation (16), and the product of fractional B-polys
from the basis set is multiplied on both sides of the Equation (16). The resulting integration of both variables (
t and
x) is calculated over the intervals
and
, respectively. After further simplification of the Equation (16), we obtain
where the fractional-order derivative of the Mittag-Leffler function
, with
is used. The current technique leads to a system of
equations. This system of equations may be summarized in the matrix equation
, where the elements of matrix
are the unknown constants. The right-hand side column matrix elements of
and the matrix elements of operational matrix
are given as
By deleting the rows and corresponding columns of Equation (18), the initial condition was imposed on the operational matrix, to make sure the solution vanishes at
x = 0 and
t = 0. The operational matrix
X is inverted using Mathematica symbolic program and multiplied with the column matrix
W to solve equation
and yield values of the unknown coefficients
. The emerging estimated result is composed of the B-poly basis set and the coefficients via Equation (3). The process provides an approximate solution
to Equation (15) using B-polys of fractional-order
and a fractional differential-order of
. The final approximate solution is provided below:
To solve the fractional order partial differential Equation (15), we chose
n = 15 and
order B-poly basis set in both variables
t and
x. The corresponding fractional-order B-poly basis set is given in terms of variable
x,
To obtain the B-polys basis set in variable
t, we replace
x =
t. We have verified that as we enlarge the number of fractional-order B-polys set, the accuracy of the numerical solution increases. A 3D plot of the absolute error, between the estimated and exact solution of Equation (15), is presented for comparison in
Figure 3, which presents the reliability between both results at the level of
. Note that when
is substituted in the solution Equation (19), the absolute error can be observed of the order of
in one-dimension. The absolute error between the solutions, both approximate and exact, is found to be in good agreement.
Example 3: Consider another partial fractional differential equation with an initial condition depending on the generalized fractional-order sine function A numerical solution is pursued using the initial condition
in intervals
. The generalized definitions of sine and cosine function [
3] are given
In this example, the assumed approximate solution contains an initial condition that has a generalized sine function as defined above. The efficiency of the numerical result is based on the number of terms that are kept in the summation of the generalized function. For this example, we kept k = 15 terms in the summation of the generalized sine function.
The exact solution of Equation (20) is
. The approximated solution
is substituted into the Equation (20) and the result is given below:
The Caputo’s derivative is applied to the above expression and the product of fractional B-polys
from the basis, sets are multiplied on both sides of the Equation (21). The integration on both variables (
x and
t) is carried out over the intervals, respectively, and after further simplification, Equation (21) may be written in the form
The fractional-order derivative of the sine function is
with
. The technique leads to a system of
operational matrix. This system of equations may be summarized in the matrix equation of the form
, where the elements of matrix
are the unknown constants. After further simplification, the right-hand side column matrix
and the matrix elements of operational matrix
in terms of inner products of B-polys are given as
To construct an appropriate solution to Equation (20), the partial fractional order differential Equation (20) is transformed into a matrix equation
. By deleting rows and the corresponding columns of the Equation (23), the initial conditions are imposed on the operational matrix equation,
so that the solution vanishes at
t = 0 and
x = 0. The operational matrix
X is programmed in the symbolic language Mathematica to determine its inverse. The inverse of the matrix
X is multiplied by the column matrix W and, by solving the matrix equation
, the values of the unknown coefficients
are determined. The resulting approximate solution is composed of the product of the B-poly basis set and the expansion coefficients, as in Equation (3). The technique provides the approximate solution
to Equation (20) using 16 B-polys with
n = 15, fractional-order
and fractional differential-order of
. The approximate solution is provided,
From the above result, it is noted that the desired approximate solution is converged and reached the desired accuracy. To find the solution to fractional-order partial differential Equation (20), we used the same fractional-order B-poly basis set as in Example 2. With the higher number (
n) of fractional B-polys, a higher order of accuracy is achievable at the expense of computer CPU time. A 3D plot of the estimated and exact results of Equation (20) is presented in
Figure 4 for the purpose of comparison. The plot shows an excellent agreement between both solutions at the level of
. Note that when
t =
x is substituted in the Equation (24), the absolute error can be observed at the same level as
exhibiting the great aspect of constancy in one-dimension
x.
It is further noted that, from the traditional trigonometric identity, we know that
However, in Ref. [
39] the authors state that, in fractional calculus, this kind of trigonometry identity does not hold. In this example, we have computationally proven that the above identity is no longer valid in fractional calculus, i.e.,
For further verification, we have plotted both sides of the identity Equation (26) and
Figure 5 and they seem to disagree. For example, for
, we show the graphs of both sides of the identity at
x =
t,
(blue curve) and
(yellow curve). From the graphs of both sides of the identity, we see that the blue and the yellow curves do not match. However, we know that when
takes integral values, both curves overlap. We tried different
and
n values of B-polys; these curves still did not overlap. It is concluded that, in fractional calculus, certain traditional trigonometry identities may not be valid.
Example 4: We consider a final example of the partial fractional-order differential equation with an initial condition as the generalized fractional-order cosine function, A numerical solution is sought using initial condition
[
3], in the intervals
. The assumed approximate solution contains an initial condition that has a generalized cosine function. The exact solution of Equation (27) is
. The assumed solution,
, is substituted into Equation (27) and the result is presented below:
After further simplification and applying the Caputo’s derivative, we obtain
where
with
. The current technique leads to a system of
equations. This system of equations may be summarized in the matrix equation of the form
, where the elements of matrix
are the unknown constants. The matrix elements of the column matrix
, and operational matrix
are given as
The partial fractional-order differential Equation (27) is now converted into an operational matrix equation
. By deleting the rows and corresponding columns of Equation (30), the initial condition is imposed on the operational matrix equation
, so that the result has the correct behavior at
t = 0 and
x = 0. The inverse of matrix
X is multiplied by the column matrix
W to solve the matrix equation
to yield the values of the unknown coefficients
. The resulting approximate solution is composed of the product of the expansion coefficients and the B-poly basis set, as given in Equation (3). The technique provides the approximate solution
of Equation (27) using
n = 15 B-polys of fractional-order
and fractional differential-order
.
From the above result of Equation (31), it is noted that the approximate solution is converged and accurate. We have experimented with various values of fractional-order γ of the differential equation while keeping the same fractional-order polynomials basis set; the result remained the same at the desired level of accuracy. It is noted that when
n = 6 set of B-polys is used, the absolute error is 10
−3, and when
n = 15 set of B-polys is used, the absolute error reduces to 10
−7. It is concluded that with the increasing number of
n sets of B-polys, a higher order of accuracy is attainable. In
Table 2, we compare our calculated
values with the exact values
at various points for
x and
t. The absolute differences between the solutions are provided in the last column of
Table 2, which shows excellent agreement. A 3D plot of the estimated and exact results of Equation (31) is presented in
Figure 6 for the purpose of comparison. Note that when
t =
x is substituted in Equation (31), the absolute error in one dimension also goes to
. The absolute error in the 3D graph is also presented in
Figure 6 showing that the error in the converged solution is of the order of
From the traditional trigonometric rule, we know that the following is a valid identity
However, in Ref. [
39], the authors state that, in fractional calculus, this trigonometry identity may not be true. In this example, we have computationally proven that the above identity is no longer valid in fractional calculus, i.e.,
For further verification, we have plotted both sides of the identity Equation (33) and they seem to disagree as shown in
Figure 7. For example, for
, we show the graphs of both sides of the identity at
x = t, (blue curve) and
(yellow curve). The graphs of both sides of the identity show that the blue and the yellow curves do not agree. However, we know that when
takes integral values, both curves overlap. We tried different
n values for B-polys and fractional values of
, these curves still did not overlap. It is concluded that, in fractional calculus, this trigonometry identity may not be valid.