Solving the Economic Growth Acceleration Model with Memory Effects: An Application of Combined Theorem of Adomian Decomposition Methods and Kashuri–Fundo Transformation Methods

: The primary purpose of this study is to solve the economic growth acceleration model with memory effects for the quadratic cost function (Riccati fractional differential equation), using Combined Theorem of Adomian Polynomial Decomposition and Kashuri–Fundo Transformation methods. The economic growth model (EGM) with memory effects for the quadratic cost function is analysed by modifying the linear fractional differential equation. The study’s signiﬁcant contribution is to develop a linear cost function in the EGM for a quadratic non-linear cost function and determine the speciﬁc conditions of the Riccati fractional differential equation (RFDEs) in the EGM with memory effects. The study results showed that RFDEs in the EGM involving the memory effect have a solution and singularity. Additionally, this study presents a comparison of exact solutions using Lie symmetry, Combined Theorem of Adomian Polynomial Decomposition, and Kashuri–Fundo Transformation methods. The results showed that the three methods have the same solution. Furthermore, this study provides a numerical solution to the RFDEs on the EGM with memory effects. The numerical simulation results showed that the output value of Y(t) for the quadratic cost function in the economic growth model is signiﬁcantly affected by the memory effect.

Some studies related to the Adomian Decomposition Method (ADM) can be seen in Refs. [16][17][18][19][20][21]. Bhakelar and Listdar-Gejji [16] employed the ADM and homotopy perturbation technique (HPM) for solving the logistic fractional differential equation (FDE). Mahdy and Marai [17] obtained the approximate solution of RFDE using the combination of ADM and Sumudu integral transformation (SIT). Hu et al. [18] proposed the ADM for solving the linear fractional differential equation (FDE), and Daftardar-Gejji and Jafari [19] applied the ADM to obtain a solution to multi-order FDE. Additionally, Bildik and Bayramoglu [20] studied the ADM to solve the two-dimensionality of a non-linear differential equation. It was also considered for solving the Bagley Torvik equation by Ray and Bera [21].
Studies related to FDE have attracted the interest and attention of mathematicians. Ren et al. [22] proposed the iterative technique and established the convergence analysis

Adomian Decomposition Method
The DE of fractional order is as follows [36] D α t y(t) + Ny(t) + Ry(t) = g(t) with y(0) = c denoting the initial condition, where D α t is the CFD operator, the linear operator, given R, and the non-linear operator defined as D α t ≡ C D α t . The function y is to be determined, while the function g illustrates the non-homogeneity of DE and Equation (1) can be re-written D α t y(t) as the subject D α t y(t) = g(t) − Ny(t) − Ry(t) Meanwhile, D α t is a FD operator, and, thus, its inverse is a FI operator I α = D −α t , so, if we integrate both sides of (2) using I α , Theorem 3 can be written as follows: The ADM resumes decomposing y into an infinite series Such that y n can be obtained iteratively. The approach further suggests decomposing the non-linear operator N y into an infinite series of polynomial form where A n = A n (y 0 , y 1 , y 2 , . . . , y n ), it is defined as the Adomian polynomial.
A n (y 0 , y 1 , y 2 , . . . , where A n is a parameter. The polynomial Adomian A n can be described as follows: Substituting the initial conditions, Equations (4) and (5) into Equation (3), the formula can thus be rewritten as in Equation (6).
then obtained Then, the recursive relation obtained from the solving ODEs of the form (1) is as follows Thus, the approximate solution of (7)

Kashuri-Fundo Transformation
This section presents the fundamental theories and concepts related to the fractional calculus and the Kashuri-Fundo transformation. Definition 2. Given a set of functions [37] The Kashuri-Fundo transformation is defined as Furthermore, the inverse of the Kashuri-Fando transformation is Meanwhile, for α is a fractional number, then Theorem 3. The Kashuri-Fando transformation of the Caputo fractional derivative for a = 0 is defined as [37]

The Effect of Memory on the Economic Growth Mode
In the natural economic growth model, Y(t) represents a function for the output value at time t. It is assumed the unsaturated market forecast shows that all products produced have been sold out, thereby not affecting the goods price (P), i.e., P > 0 constant. The function expressing the net investment is denoted by I(t). This investment is considered for the development of production.
The mathematical model of economic growth involving the memory effect can be seen in Equation (9 Furthermore, if the linear production cost C(t) = aY(t) + b, Equation (9) reduces to Thus, if the cost function C(t) is linear, then Equation (10) is called a linear DE of fractional order α > 0. Furthermore, Equation (11) becomes where k 1 = m v (P − a) dan k 2 = − mb v , with initial condition is y(0) = c. In Equation (9), the linear cost function is assumed to be C(t) = aY(t) + b. Then, we propose a new nonlinear cost function with C(t) = aY 2 (t) + kY(t) + b with a, k representing the margin costs and b denoting the independent costs, then Equation (9) becomes If the cost function C(t) is quadratic, then Equation (13) is a non-linear RFDEs of order α > 0 with memory effect. Furthermore, Equation (13) reduces to where The parameters of a system (9) can be defined as follows: Y(t) = the value of the number of products produced during the production process d. m = net investment figure (0 < m < 1), i.e., the sharing of profit process for net investment e. a = marginal cost (additional cost if production increases/depends on the value of output) f. k = marginal cost g. b = independent costs (costs that do not depend on the number of products produced h. v = positive constant which is referred to ratio of investment describing the acceleration rate (v > 0) The evaluation results showed that Riccati fractional differential equation in EGM with memory effect (11) has a solution and singularity for each value of k, k 1 , k 2 and k 3 R. The value of k < 0 means that there is a marginal cost reduction in the output value of Y(t). Meanwhile, if k = 0, then there is no additional cost or marginal cost reduction, and if k > 0, then there is an additional marginal cost at the output value of Y(t).

Main Theorems
This section presents the Combined Theorem of Adomian Polynomial Decomposition and Kashuri-Fundo Transformation methods for solving Riccati fractional differential equations.

Theorem 4. (Combined Theorem).
Given the Riccati fractional differential equation as follows: with initial conditions w(0) = c and D α x are Caputo's fractional derivative operator, where 0 < α ≤ 1, then the solution of Equation (14) is Proof. Based on Equation (14), the Riccati fractional differential equation is as follows (14) with the Kasturi-Fundo transformation, such that using Theorem 4, we obtain Then, using the inverse of the Kashuri-Fundo transformation and Equation (8) in Equation (15), we obtain The Adomian decomposition method (4) assumes that the function w can be decomposed into an infinite polynomial series as follows where w n can be specified recursively. This method also assumes that the non-linear operator w 2 can be decomposed into an infinite polynomial series so that based on Equation (5), we obtain: where A n is an Adomian polynomial, defined as where λ is a parameter. Adomian polynomial A n can be described as follows Substituting Equations (17) and (18) into Equation (16), we obtain So, that the combined theorem is obtained from the recursive relation of the solution of the fractional ordinary differential equation using the Adomian Decomposition Method and the Kashuri-Fundo Transform (8) as follows (20) and the proof is completed.

The Comparison of the Exact Solution of the Symmetry Lie with Odetunde and Taiwo
In this subsection, we present the exact solution of the Riccati differential equation using the concept of Lie symmetry as in [38] and compare the solution with the exact solution from Odetunde and Taiwo [39]. Example 1. Given the Riccati fractional differential equation as follows [39] We will determine the exact solution for y(0) = 0 f or 0 ≤ x ≤ 1 and its graph using the concept of Lie Symmetry, then compare it with the exact solution and graph from Odetunde and Taiwo [39].

a.
Tangent Vector With the help of MAPLE, the vector tangent at the point (x, y) is obtained Then, the vector tangent at the point (x, y) is (ξ(x, y), η(x, y)) = (1, 0). So, the tangent vector is the vector tangent of Reduced Characteristics The reduced characteristic equation is presented as follows Because Q(x, y) = 0, then Lie symmetry with vector tangent (ξ(x, y), η(x, y)) = (1, 0) is non-trivial Lie symmetry.

c.
Canonical coordinates (r, s) The canonical coordinates (r, s) can be found with So, we obtain r = y.

d. Solving the Riccati Equation
The following is the solution to the Riccati Equation using ds dr .
Furthermore, converting coordinates (r, s) to coordinates (x, y), with r = y and s = x, we obtain If y(0) = 0, we obtained So, the exact solution of Riccati differential equation using the concept of Lie Symmetry is Meanwhile, the exact solution of Odetunde and Taiwo [39] is The Comparison of Exact Solution between Lie Symmetry and Odetunde and Taiwo [39] is presented in Figure 1. Based on Figure 1, the graph of the exact solution (green dot) using the concept of Lie Symmetry coincides with the graph of the exact solution (red line) from Odetunde and Taiwo [39]. It shows that RFDEs exact solution using the Lie Symmetry concept has the same solution as the exact solution from Odetunde and Taiwo [39]. So, the exact solution of Riccati differential equation using the concept of Lie Symmetry is Meanwhile, the exact solution of Odetunde and Taiwo [39] is [39] is presented in Figure 1. Based on Figure 1, the graph of the exact solution (green dot) using the concept of Lie Symmetry coincides with the graph of the exact solution (red line) from Odetunde and Taiwo [39]. It shows that RFDEs exact solution using the Lie Symmetry concept has the same solution as the exact solution from Odetunde and Taiwo [39].

Example 2. Given the Riccati fractional differential equation as follows
with the initial condition is (0) = 0, and the exact solution of Odetunde and Taiwo [39], i.e., We will draw the graph of the solution to the Riccati Fractional differential equation ( ) using the Combined Theorem for = 0.7; 0.8; 0.9 1 0 ≤ ≤ 1 to determine whether the solution graph of the ( ) using the combined method for = 1 coincides with the exact solution of Odetunde and Taiwo [39].
Based on the Combined Theorem, the approximate solution of the Riccati fractional differential Equation (21)

Example 2.
Given the Riccati fractional differential equation as follows with the initial condition is w(0) = 0, and the exact solution of Odetunde and Taiwo [39], i.e., We will draw the graph of the solution to the Riccati Fractional differential equation w(x) using the Combined Theorem for α = 0.7; 0.8; 0.9 and 1 f or 0 ≤ x ≤ 1 to determine whether the solution graph of the w(x) using the combined method for α = 1 coincides with the exact solution of Odetunde and Taiwo [39]. Based on the Combined Theorem, the approximate solution of the Riccati fractional differential Equation (21) is obtained: where A n is the Adomian polynomial of the non-linear operator Nw = w 2 , which can be described as follows: The following is a description of the approach solution (23): Figure 2 shows the approximate solution to the tenth iteration of the Riccati fractional differential equation using the Combined Theorem (23) for different values assisted by MAPLE software. Additionally, Figure 2 shows the exact solution (22) in black, coinciding with the approximate solution, using the Combined Theorem for α = 1 and the Maple application help as follows:

Numerical Simulation of Economic Growth Model using Combined Theorem
Example 3. Given the Riccati fractional differential equation in the economic g follows Figure 3. The exact solution (green line) coincides with the approximate solution using the Combined Theorem for α = 1 (black dot), for 0 ≤ x ≤ 1.

Numerical Simulation of Economic Growth Model using Combined Theorem
Example 3. Given the Riccati fractional differential equation in the economic growth model as follows with m = 1 2 , P = 42, a = −2, b = 20, v = 10, k = 2 and the initial condition Y(0) = 1, and the exact solution, i.e., We draw the graph of the solution to the Riccati Fractional differential equation Y(t) using the Combined Theorem for α = 0.7; 0.8; 0.9 and 1 f or 0 ≤ x ≤ 1 as follows.
We will show that the solution Y(t) using the combined method for α = 1 coincides with the exact solution. Note that, Thus, the Riccati fractional differential equation in Equation (24) is as follows Based on the Combined Theorem, for the approximate solution of the Riccati fractional differential Equation (26) in the economic growth model, the output value is obtained as follows where A n is the Adomian polynomial of the non-linear operator NY = Y 2 , which can be described as follows The following is a description of the approximate solution (27): where A n is the Adomian polynomial of the non-linear operator NY = Y 2 . Figure 4 shows the approximate solution graph (up to the twentieth iteration) of the Riccati fractional differential equation using the Combined Theorem for α = 0.7; 0.8; 0.9 and 1. Additionally, Figure 5 displays the exact solution (green line) coinciding with the approximate solution using the Combined Theorem for α = 1 (black dot) using MAPLE software.
50Γ ( + 1)Γ(3 + 1)Γ (5 + 1) where is the Adomian polynomial of the non-linear operator = . Figure 4 shows the approximate solution graph (up to the twentieth iteration) of the Riccati fractional differential equation using the Combined Theorem for = 0.7; 0.8; 0.9 and 1. Additionally, Figure 5 displays the exact solution (green line) coinciding with the approximate solution using the Combined Theorem for = 1 (black dot) using MAPLE software.  Figure 4 shows a graph of the approximate solution to Riccati's fractional differential equation (until the twentieth iteration) using the Combined Theorem with a value close to 1. The graph is close to the exact solution (the blue graph is = 1), which shows that the output value Y(t) for the quadratic cost function in the economic growth model is significantly affected by the memory effect.  Figure 5 captures the exact solution (green line) graph coinciding with the approximate solution, using the Combined Theorem (black dot line graph) for =1. It shows that using the Combined Theorem of the Adomian Decomposition Method and the Kashuri-Fundo Integral Transformation are very accurate, useful, and easy to solve Riccati's fractional differential equation on the economic growth model.
The numerical simulation results presented in Table 1 for t are close to 1. The output value Y(t) increases as it reaches closer to 1, then Y(t) becomes smaller. This shows that the output value of Y(t) for the quadratic cost function in the economic growth model is significantly affected by the memory effect.   Figure 4 shows a graph of the approximate solution to Riccati's fractional differential equation (until the twentieth iteration) using the Combined Theorem with a value close to 1. The graph is close to the exact solution (the blue graph is α = 1), which shows that the output value Y(t) for the quadratic cost function in the economic growth model is significantly affected by the memory effect. Figure 5 captures the exact solution (green line) graph coinciding with the approximate solution, using the Combined Theorem (black dot line graph) for =1. It shows that using the Combined Theorem of the Adomian Decomposition Method and the Kashuri-Fundo Integral Transformation are very accurate, useful, and easy to solve Riccati's fractional differential equation on the economic growth model.
The numerical simulation results presented in Table 1 for t are close to 1. The output value Y(t) increases as it reaches closer to 1, then Y(t) becomes smaller. This shows that the output value of Y(t) for the quadratic cost function in the economic growth model is significantly affected by the memory effect.

Conclusions
This paper has successfully developed the Riccati fractional differential equation in the new economic growth acceleration model with a memory effect for the quadratic cost function. We have identified that the RFDEs in the economic growth model with memory effect for the quadratic cost function in Equation (11) has a solution and singularity. The value of k < 0 means a marginal cost reduction in the output value of Y(t). Meanwhile, if k = 0, then there is no additional cost or marginal cost reduction, and if k > 0, then there is an additional marginal cost at the output value of Y(t). Additionally, we have presented a comparison of exact solutions using Lie Symmetry, Combined Theorem of Adomian Polynomial Decomposition, and Kashuri-Fundo Transformation methods. The exact solution showed a similar result. In addition, we have tested the developed Riccati fractional differential equation numerically on an EGM involving memory effects. The numerical simulation results showed that the output value of Y(t) for the quadratic cost function in the economic growth model is significantly affected by the memory effect.