5.1. Numerical Methods
To approximate the solution of the HFDE, we will employ two methods—namely, Picard’s method and the general linear method (GLM).
Picard’s Method:
This is an iterative method for solving initial value problems of the form
It uses successive approximations until convergence, thereby, resulting in better approximations with more iterations. To approximate the solution of the above initial value problem, the initial value problem (
3) can be rewritten as an integral equation of the form
From this equation a sequence of approximations can be obtained as
Given , one can produce . This process is continued to some tolerance. For our purpose, the 1-differentiable and 2-differentiable problems will have the forms:
Using results from theory of differential equations, it can be proven that the sequence of approximations converges to the exact solution of IVP.
General Linear Method (GLM):
The name “general linear method ” applies to a large group of numerical methods for ordinary differential equations; more on these methods can be found in [
22,
23,
24]. We will consider
r-value
s-stage methods, where
for the Runge–Kutta methods and
for the linear multi-step methods. Each step of the computation takes, as input, a certain number
of items of data and generates, for output, the same number of items. The output items correspond to the input items but are advanced through one time step
Within a step, a certain number
of stages of computations are performed.
We now present a GLM based on linear k-step Adams–Bashforth-type schemes for solving fuzzy initial value problems. Assume that, for equally spaced points at , the exact solutions are indicated by and that are the approximate solutions. The k-step Adams–Bashforth methods can be written as:
The 1-differentiable system:
and the 2-differentiable system:
Now, the input and output approximations for the general linear methods are, respectively,
Under the 1- and 2-differentiability, the representations of the input vectors for the GLM are given by
and
, while the corresponding output vectors are indicated by
and
under 1 and 2-differentiability, respectively. Now, we consider the input approximation of the general linear methods in terms of 1-differentiability as:
and under 2-differentiability, we obtain the following input vectors:
Using the above input vectors, the fuzzy general linear methods can be formulated under 1-differentiability as:
and under 2-differentiability as:
where
and
are internal stages under 1- and 2-differentiability, respectively. Additionally,
We consider two examples of the Fuzzy GLMs form of k-step methods under strongly generalized differentiability for
. First, consider
. The input vectors for
under 1- and 2-differentiability are, respectively,
and
Similarly, for
, we obtain
and
To use the FGLMs
at step number
n, denote the input items by
,
and denote the stages computed in the step and the stage derivatives by
and
, respectively,
With this in mind, we present the methods in compact notation as
The stages are computed by the formula
and the output approximations by the formula
In each one of the above cases, the coefficients of the general linear formulation are presented in the
partitioned matrix
where
and
are the input and output approximations, respectively, and
For example, the matrices representing the Euler method and implicit Euler methods are, respectively,
5.2. Numerical Testing
For numerical testing, we consider HFDEs with selected types of fuzzy numbers as the initial conditions. Let us consider the following hybrid fuzzy IVP [
14,
17,
18],
where
We will impose initial fuzzy conditions on (
7), which are given as:
Triangular Fuzzy Number: Let
and
.
Trapezoidal Fuzzy Number: Let
and
Triangular Shaped Fuzzy Number: Let
5.2.1. Triangular Fuzzy Numbers
We solve the hybrid fuzzy differential equations (
7) using Picard’s method and the general linear method under 1- and 2-differentiable Hukuhara differentiation.
Now, consider the following examples:
Example 1. Let and Picard’s Method
Case 1: Under 1-differentiability:
We apply the Picard Method for hybrid fuzzy differential Equation (
5) with
when
and
, and then
and
We obtain
and
when
and
The exact and approximate solutions for this example under
differentiability are given in
Table 1,
Table 2 and
Table 3 and in
Figure 1. Note that the exact and approximate solutions agree up to
.
Case 2: Under 2-differentiability:
We again apply the Picard method for the hybrid fuzzy differential Equation (
6) with
when
and
, and then
and
. The results obtained are given in
Table 4,
Table 5 and
Table 6 and in
Figure 2.
Example 2. Let Case 1: Under 1-differentiability:
Thus, the Picard method with triangular fuzzy number as the initial conditions gave highly accurate results when used with high numbers of iterations.
General Linear Method (GLM):
We solved the same Examples 1 and 2 as in the previous subsection using the general linear method for hybrid fuzzy differential equations with
and
Similar results and similar accuracy are observed, and the results obtained for Example 1 for 1-differentiability and 2-differentiability are given in
Figure 4 and
Figure 5, respectively, and those for Example 2 for 1-differentiability are given in
Figure 6.
We also performed a comparison of the proposed GLM for Example 1 with the Runge–Kutta of order 5 method proposed by [
14,
15,
16]. The results for both models are presented in
Table 10 and
Table 11 for the 1-solution and in
Table 12 and
Table 13 for the 2-solution.
The results are almost identical except for slight differences in the 10th digits. This means that the GLM competes well with other methods in terms of accuracy, such as the Runge–Kutta, while in terms of the amount of work, it requires less computations. This is due to the fact that the GLM passes more than one piece of information between steps. Each step of the computation takes a certain number of data items as input and generates the same numbers of data items as output [
23,
24].
5.2.2. Triangular Shaped and Trapezoidal Fuzzy Numbers
We solve the hybrid fuzzy differential equation subject to triangular and trapezoidal fuzzy numbers.
1. Trapezoidal Fuzzy Numbers:
Let
= (0.5, 0.75, 1, 1.125) and
The solutions obtained using Picard’s method and the general linear method for
t = 1.0, 1.5 and 2.0 are given in
Figure 7 and
Figure 8, respectively.
When using the triangular fuzzy numbers of the form
The solutions obtained for Picard’s method and the general linear method are given in
Figure 9 and
Figure 10, respectively.
Again, accurate results are reported when using Picard’s method or the GLM with trapezoidal fuzzy numbers and triangular fuzzy numbers as the initial conditions. The general linear method gave accurate results with a lower number of steps compared with the Runge–Kutta method as reported by other authors.