A novel approach to solving fractional fredholm-volterra integro-differential equations using the elzaki transform

1. Introduction

Fractional derivatives have an extensive background in mathematics, but they were not utilized in science for a very long period. The abundance of non-equivalent descriptions of a fractional derivative FD might be one factor contributing to its unpopularity (Bagley et al., 1991). Another issue is that because fractional derivatives are nonlocal, they lack a clear geometrical meaning (He, 1998). But in the past decade, mathematicians and engineers have become much more interested in fractional calculus. It was discovered that FDs may be used to cleanly replicate a variety of applications, especially interdisciplinary ones. For instance, FDs may be used to explain the nonlinear oscillation of earthquakes (Blackledge, 2010) and the flaw in the fluid-dynamic traffic model (He, 1999), brought on by the presumption of continuous traffic flow.

Differential equations for Fractional partial differential equations(FPDEs) and Asymptotic Expansions of Fractional Derivatives and Their Applications are given by Morita and Sato (2015) and Grigorenko, I and Grigorenko, E (2003), respectively, based on real data. Mainardi (1997) provides an overview of the various applications of fractional derivatives in statistical mechanics and continuous mechanics. Numerous authors have looked at the analytical findings about the presence and individuality of solutions to the fractional-order biological population model (Elzaki and Mohamed, 2024). Many techniques have been used in recent years to solve fractional Korteweg–De Vries equation-Burgers equations utilizing the Adomian decomposition methods (ADM) by (He, 1999) dynamic systems such as FDs (He, 1998), solving fractional partial differential equations using conformable Elzaki transform (Elzaki and Ahmed, 2025), FIDEs (Gejji and Jafari, 2007), and FPDEs (Grigorenko,I and Grigorenko, E, 2003), (Magzoub et al., 2024). These techniques include, among many others, ADM, Homotopy, and First Integral.

There aren’t many ways to solve FIDEs because they’re an emerging subject in mathematics. These methods include the ADM (Elzaki, 2014), fractional differential transform method (FDTM) (Ige et al., 2019), and collocation method (Ige et al., 2020).

This work focuses on the elzaki transform (ET) strategy (Elzaki and Ishag, 2019), a novel approach to solving Fredholm-Volterra integro-differential equations (FFVIDEs). The claim is not supported by numerous examples in this study.(Mahdy et al., 2023), investigated the following functional equations: Volterra–Fredholm Integral Equations, Difference Integro-Differential Equations with Variable Coefficients, (Mahdyet al . 2025) studying $(1 + 1) $ dimensional functional equations of mixed integro-differential type., (Mahdy and Mohamed, 2022) solve the Cauchy integral equations by using Lucas polynomials and (Mahdy et al. 2023) solve three-dimensional mixed volterra–fredholm integral equations, also (Al-Bugami et al., 2025) solve integral equations, and system of nonlinear partial differential equations and nonlinear partial differential equations by new laplace variational iteration method are studies by Elzaki and Biazar (2013) and Hilal and Elzaki (2014) respectively.

Kamran et al. (2025) and Kilbas et al. (2006) are studies Numerical investigation of a chlamydia epidemic model and Theory and applications of fractional differential equations. Dynamical investigation and numerical modeling of a fractional mixed nonlinear partial integro-differential problem in time and space it was taken by Mohamed et al. (2024). Momani and Aslam Noor (2006), Momani and Qaralleh (2006) and Momani and Shawagfeh (2006) also discussed the following topics, Numerical methods for fourth-order fractional integro-differential equations, solving systems of fractional integro-differential equations and Decomposition method for solving fractional Riccati differential equations. As for he studied (Nazari and Shahmorad, 2010) they studied the fractional differential transforms method to fractional-order integro-differential equations with nonlocal boundary conditions.

Additionally, Zeb et al. (2024), investigated models of fever and Chlamydia epidemics. While Rawashdeh (2006), Saha Ray et al. (2006) and Wang (2006) conducted a study fractional integro-differential equations by collocation method, nonlinear dynamic system containing fractional derivative by modified decomposition method and fractional KdV-Burgers equation by adomian decomposition method respectively.In physics, FFVIDEs are used to describe phenomena with memory and non-local effects, such as control systems, anomalous diffusion, particle dynamics, and viscoelastic materials. In fields such as fluid mechanics, electromagnetic, quantum physics, and continuum mechanics, where fractional calculus offers a superior framework to conventional integer-order differential equations, these equations are essential for characterizing complicated systems.

Some instances of the issues described are addressed in the general explanation of the suggested approach. Analytical results of FFVIDEs with initial conditions (ICs) provide a particularly challenging set of problems. The present work provides closed-form solutions by using a very basic and direct method to derive the analytical results of FFVIDEs.

By combining ideas of inherited characteristics with long-term dependencies that affect economic behavior over time, FFVIDEs are used in economics to simulate memory-dependent economic systems, such as financial markets and asset pricing. Compared to conventional differential equations, these equations provide a broader framework that makes it possible to analyze intricate phenomena like market swings, consumption-cutting habits, and coupled economic models with memory effects. To solve these fractional models, researchers create and use numerical approaches such as fixed-point theorems and other approximation methods.

The freshly developed method has two distinct features: it is simple to use and offers recommendations for choosing the initial estimate. This paper investigates the effectiveness of a special analytical technique in combination with the ET for FFVIDE resolution. This approach works well for solving functional equations that comes up when a variety of processes are analytically modeled.

The primary goal of this paper is to examine the structure of the solution that may be explicitly ascertained using a novel method with the ET. We also demonstrate the FFVIDE’s convergence.

Definition 1:

$f(q),$ $q>0$ is in the space $C_{\eta },\eta \in \Re$ if and only if $p>\eta ,$ such that: $f(q)=q^p{f}_1(q),$ where $f_1(q)\in C(0,\infty ),$ and it is said to be in the space $C_{\eta }^n$ if and only if $f^n\in C_{\eta },n\in N.$

Definition 2:

The Riemann-Liouville (R-L) fractional integral operator of order $\alpha >0,$ of $f\in C_{\eta },\eta \ge -1,$ is,

$\begin{array}{l}{J}^{ƛ}f(q)=\frac{1}{\Gamma (ƛ)}{\displaystyle \underset{0}{\overset{q}{\int }}{(q-v)}^{ƛ-1}f(v)dv,ƛ>0.}\\ J^{0}f(q)=f(q)\end{array}$

Definition 3:

Caputo states that the FD of $f(q),$

$D^{ƛ}f(q)=J^{m-ƛ}{D}^{m}f(q)$

For $m-1<ƛ\le m,$ $m\in N,$ $u>0,$ and $f\in C_{-1}^m.$

Caputo fractional derivative (CFD) computes a fractional integral after calculating an ordinary derivative to obtain the appropriate order of an FD. A linear operation, the R-L fractional integral operator is similar to the integer-order integration:

$J^{ƛ}\left({\displaystyle \sum _{i=1}^n{c}_i{f}_i(q)}\right)={\displaystyle \sum _{i=1}^n{c}_i{J}^{ƛ}{f}_i(q)}$

Where ${\left\{c_i\right\}}_{i=1}^n$ are constants.

[5,7] provides justification for the use of the CFD.

2. ET

For the function $\Upsilon (q)$, the ETis represented in Eq. (1),

Where, p is a complex value.

It is easy to apply the ET to FPDEs (Elzaki and Mohamed, 2024). ET may be useful in situations when DEs with variable coefficients cannot be solved using Sumudu and Laplace transforms (Elzaki and Ishag, 2019).

Also, the ET was used in solving thefractional-order biological population model (Elzaki and Ishag, 2019), and some Ψ-fractional derivative equations (Chamekh et al., 2024),

Theorem 1:

Let $T\left(p\right)$ is the ET of $\left[E\left(\Upsilon (q)\right)=T\left(p\right)\right].$ then:

$\begin{array}{l}\left(i\right){\rm E}\left[{\Upsilon }^{\prime }(q)\right]=\frac{T\left(p\right)}{p}-p\Upsilon (0)\\ \left(ii\right){\rm E}\left[{\Upsilon }^{″}(q)\right]=\frac{T\left(p\right)}{p^2}-\Upsilon (0)-p{\Upsilon }^{\prime }(0)\\ \left(iii\right){\rm E}\left[{\Upsilon }^{\left(n\right)}\left(q\right)\right]=\frac{T\left(p\right)}{p^n}-{\displaystyle \sum _{k=0}^{n-1}{p}^{2-n+k}{\Upsilon }^{\left(k\right)}\left(0\right)}\end{array}$

Proof:

• (i)

  ${\rm E}\left[{\Upsilon }^{\prime }(q)\right]=p{\displaystyle \underset{0}{\overset{\infty }{\int }}{\Upsilon }^{\prime }(q)}{e}^{\frac{-q}{p}}dq$, Integrating by parts to find:

  ${\rm E}\left[{\Upsilon }^{\prime }(q)\right]=\frac{T\left(p\right)}{p}-p\Upsilon (0)$

• (ii)

  Let, $g\left(q\right)={\Upsilon }^{\prime }(q),$ then:

  ${\rm E}\left[g^{\prime }\left(q\right)\right]=\frac{1}{p}{\rm E}\left[g\left(q\right)\right]-pg\left(0\right),$

using (i) to get:

${\rm E}\left[{\Upsilon }^{″}(q)\right]=\frac{T\left(p\right)}{p^2}-\Upsilon (0)-p{\Upsilon }^{\prime }(0)$

• (iii)

  May be demonstrates by mathematical induction (Table 1).

We shall now show how a few lemmas may be used to deduce the function from its ET.

Lemma 1:

R-L fractional operator for $ƛ>0$ can be written as:

$E[J^{ƛ}\Upsilon (q)]=p^{ƛ}T(p)$

Proof:

Taking ET to obtain:

$E[J^{ƛ}\Upsilon (q)]=E\left[\frac{1}{\Gamma (ƛ)}{\displaystyle \underset{0}{\overset{q}{\int }}{(q-\alpha )}^{ƛ-1}}\Upsilon (v)dv\right]=\frac{1}{\Gamma (ƛ)}\frac{1}{p}T(p)G(p)=p^{ƛ}T(p),$

Where $G(p)=E[q^{ƛ-1}]=p^{ƛ+1}\Gamma (ƛ)$

Lemma 2:

Let, $m-1<ƛ\le m,$ $m\in IN,$ then:

$E[D_{}^{ƛ}\Upsilon (q)]=p^{m-ƛ}\left[\frac{T(p)}{p^m}-\frac{\Upsilon (0)}{p^{m-2}}-\frac{\frac{\partial \Upsilon (0)}{\partial p}}{p^{m-3}}-\cdots -p\frac{{\partial }^{m-1}\Upsilon (0)}{\partial p^{m-1}}\right]$

Or, $E[D_{}^{ƛ}\Upsilon (q)]=\frac{1}{p^{ƛ}}E[\Upsilon (q)]-{\displaystyle \sum _{k=0}^{m-1}\frac{{\partial }^k\Upsilon (0)}{\partial p^k}}{p}^{2-ƛ+k},$ $m-1<ƛ\le m$

Lemma 3:

If, $ƛ,\beta >0,$ $a\in C$ and $\frac{1}{p^{ƛ}}>\left|a\right|,$ then:

$E^{-1}\left[\frac{p^{\beta +1}}{1+a{p}^{ƛ}}\right]=q^{\beta -1}{\in }_{ƛ,\beta }(-a{q}^{ƛ})$

${\in }_{ƛ,\beta }(q)$, are the Mittag-Leffler functions,they are described as:

${\in }_{ƛ,\beta }(q)={\displaystyle \sum _{n=0}^{\infty }\frac{q^n}{\Gamma (nƛ+\beta )}},{\in }_{ƛ,1}(q)={\in }_{ƛ}(q)$

Proof:

$\frac{p^{\beta +1}}{1+b{p}^{ƛ}}=p^{\beta +1}\frac{1}{1+b{p}^{ƛ}}=p^{\beta +1}{\displaystyle \sum _{j=0}^{\infty }{(-b)}^j{(p^{ƛ})}^j}={\displaystyle \sum _{j=0}^{\infty }{(-b)}^j}{p}^{jƛ+\beta +1},$

then

$\begin{array}{c}{E}^{-1}\left[\frac{p^{\beta +1}}{1+b{p}^{\alpha }}\right]=E^{-1}\left[{\displaystyle \sum _{j=0}^{\infty }{(-b)}^j{p}^{j\alpha +\beta +1}}\right]={\displaystyle \sum _{j=0}^{\infty }\frac{{(-b)}^j{q}^{j\alpha +\beta -1}}{\Gamma (j\alpha +\beta )}}\\ =q^{\beta -1}{\displaystyle \sum _{j=0}^{\infty }\frac{{(-b{q}^{ƛ})}^j}{\Gamma (jƛ+\beta )}=q^{\beta -1}{\in }_{ƛ,\beta }(-b{q}^{ƛ})}\end{array}$

3. The proposed method

The steps for solving FFVIDEs are as follows:

Apply the ET to the equation, use the ICs, applying the inverse ET, formulate the recurrence relationship and finally, find the solutions of FFVIDE.

To better understand the main idea of the suggested method, we examine the following explanation of FFVIDE using the CFD in this section Eqs. (2) and (3)

With ICs,

$F\left(q\right),P\left(q\right),$ are known, $K_1\left(h,q\right),K_2\left(h,q\right),$ are the kernels of the integral equation, ${}^{C}D{}^{ƛ}$ is the $ƛ$ th order derivative of $\Upsilon (q)$, in the sense of CFD, ${\Psi }_1,{\Psi }_2,$ are continuous functions and $\Upsilon (q)$ is the continuous function to be determined,

Using Eq. (3) and taking ET to Eq. (2), we obtain

$\begin{array}{l}E[{}^{C}D{}^{ƛ}\Upsilon (q)]=E\left[F\left(q\right)\right]+E\left[P(q)\Upsilon (q)\right]+E\left[{\displaystyle \underset{0}{\overset{q}{\int }}{K}_1\left(h,q\right)}{\Psi }_1\left(\Upsilon \left(h\right)\right)dh\right]\\ +E\left[{\displaystyle \underset{0}{\overset{1}{\int }}{K}_2\left(h,q\right)}{\Psi }_2\left(\Upsilon \left(h\right)\right)dh\right],\\ \frac{1}{p^{ƛ}}E\left[\Upsilon (q)\right]-p^{2-ƛ}\Upsilon (0)-p^{3-ƛ}{\Upsilon }^{\prime }(0)-p^{4-ƛ}{\Upsilon }^{″}(0)\\ =E\left[F\left(q\right)\right]+E\left[P(q)\Upsilon (q)\right]\\ +E\left[{\displaystyle \underset{0}{\overset{q}{\int }}{K}_1\left(h,q\right)}{\Psi }_1\left(\Upsilon \left(h\right)\right)dh\right]+E\left[{\displaystyle \underset{0}{\overset{1}{\int }}{K}_2\left(h,q\right)}{\Psi }_2\left(\Upsilon \left(h\right)\right)dh\right],\\ E\left[\Upsilon (q)\right]=p^2{\delta }_1+p^3{\delta }_2+p^4{\delta }_3+p^{ƛ}E\left[F\left(q\right)\right]+p^{ƛ}E\left[P(q)\Upsilon (q)\right]\\ +p^{ƛ}\left\{E\left[{\displaystyle \underset{0}{\overset{q}{\int }}{K}_1\left(h,q\right)}{\Psi }_1\left(\Upsilon \left(h\right)\right)dh\right]+E\left[{\displaystyle \underset{0}{\overset{1}{\int }}{K}_2\left(h,q\right)}{\Psi }_2\left(\Upsilon \left(h\right)\right)dh\right]\right\},\end{array}$

Take the inverse ET to get:

$\Upsilon (q)=G(q)+E^{-1}\left\{p^{ƛ}E\left[\left[P(q)\Upsilon (q)\right]+\left[\begin{array}{l}{\displaystyle \underset{0}{\overset{q}{\int }}{K}_1\left(h,q\right)}{\Psi }_1\left(\Upsilon \left(h\right)\right)dh\\ +{\displaystyle \underset{0}{\overset{1}{\int }}{K}_2\left(h,q\right)}{\Psi }_2\left(\Upsilon \left(h\right)\right)dh\end{array}\right]\right]\right\},$

$G(q)$ is the term that originates from all or a portion of $F(q)$ and the ICs.

Use the relationships listed below to solve this problem iteratively Eq. (4)

The solution of Eq. (2) is:

${\Upsilon }_0\left(q \right),{\Upsilon }_1\left(q\right),{\Upsilon }_2\left(q\right),\mathrm{...},$ are deduced from Eq. (4).

The solution may then be found using Eq. (5).

Note: Selecting the first iteration ${\Upsilon }_0$ is crucial since it can initiate the method’s convergence and enable the calculation of the exact solution in a limited number of steps. This strategy depends on how we choose the first iteration ${\Upsilon }_0$ that yields exact solutions in a few stages. If the ICs are exactly zero, we can choose ${\Upsilon }_0$ all or a subset of the $F(q)$.

4. Providing examples

Here, we used the methods outlined in this article to produce an exact solution to the FFVIDE.

Example 1:

Consider the FIDE,

With ICs,

Taking ET of Eq. (6), and using the ICs in Eq. (7), to get:

$\begin{array}{c}E\left[D^{2.3}\Upsilon (q)\right]=E\left[\frac{\Gamma \left(4.5\right)}{\Gamma \left(2.2\right)}{q}^{1.2}\right]-E\left[\frac{q^2}{99}\Upsilon (q)+\frac{8}{1155}\sqrt{q}{\Upsilon }^{‴}(q)\right.\\ \left.-\frac{1}{4}{\displaystyle \underset{0}{\overset{q}{\int }}(q-h)\Upsilon \left(h\right)dh}-\frac{1}{2}{\displaystyle \underset{0}{\overset{1}{\int }}qh\Upsilon \left(h\right)dh}\right],\\ \frac{1}{p^{2.3}}E[\Upsilon (q)]=\frac{\Gamma \left(4.5\right)}{\Gamma \left(2.2\right)}\Gamma \left(2.2\right)p^{3.2}-E\left[\frac{q^2}{99}\Upsilon (q)+\frac{8}{1155}\sqrt{q}{\Upsilon }^{‴}(q)\right.\\ \left.-\frac{1}{4}{\displaystyle \underset{0}{\overset{q}{\int }}(q-h)\Upsilon \left(h\right)dh}-\frac{1}{2}{\displaystyle \underset{0}{\overset{1}{\int }}qh\Upsilon \left(h\right)dh}\right];\\ E[\Upsilon (q)]=\Gamma \left(4.5\right)p^{5.5}-p^{2.3}E\left[\frac{q^2}{99}\Upsilon (q)+\frac{8}{1155}\sqrt{q}{\Upsilon }^{‴}(q)\right.\\ \left.-\frac{1}{4}{\displaystyle \underset{0}{\overset{q}{\int }}(q-h)\Upsilon \left(h\right)dh}-\frac{1}{2}{\displaystyle \underset{0}{\overset{1}{\int }}qh\Upsilon \left(h\right)dh}\right];\end{array}$

Inverse ET implies that:

$\Upsilon [(q)]=q^{\frac{7}{2}}+E^{-1}\left\{p^{2.3}E\left[\begin{array}{l}\frac{1}{4}{\displaystyle \underset{0}{\overset{q}{\int }}(q-h)\Upsilon \left(h\right)dh}+\frac{1}{2}{\displaystyle \underset{0}{\overset{1}{\int }}qh\Upsilon \left(h\right)dh}\\ -\frac{q^2}{99}\Upsilon (q)-\frac{8}{1155}\sqrt{q}{\Upsilon }^{‴}(q)\end{array}\right]\right\};$

Using the last equation and Eq. (4), to state the iteration formula Eq. (8)

Eq. (8) gives us:

$\begin{array}{l}{\Upsilon }_1(q)=E^{-1}\left\{p^{2.3}E\left[\begin{array}{l}\frac{1}{4}{\displaystyle \underset{0}{\overset{q}{\int }}(q-h){\Upsilon }_0\left(h\right)dh}+\frac{1}{2}{\displaystyle \underset{0}{\overset{1}{\int }}qh{\Upsilon }_0\left(h\right)dh}\\ -\frac{q^2}{99}{\Upsilon }_0(q)-\frac{8}{1155}\sqrt{q}{{\Upsilon }^{‴}}_0(q)\end{array}\right]\right\}=0\\ \text{and}\text{then}:{\Upsilon }_2\left(q\right)=0,{\Upsilon }_3\left(q\right)=0,{\Upsilon }_4\left(q\right)=0,\mathrm{...},\end{array}$

Eq. (5) gives:

$\Upsilon (q)={\displaystyle \sum _{n=0}^{\infty }{\Upsilon }_n(q),}\Rightarrow \Upsilon (q)=q^{\frac{7}{2}}.$

Example 2

Consider the nonlinear FFVIDE Eqs. (9) and (10),

With the ICs,

By the same methods as in example 1, we have:

$\begin{array}{c}E[\Upsilon (q)]=\frac{\Gamma \left(4\right)}{\Gamma \left(1.8\right)}\Gamma \left(1.8\right)p^5+p^{2.2}E\left[\frac{1}{3}{\displaystyle \underset{0}{\overset{q}{\int }}(q+h){\Upsilon }^2\left(h\right)dh}\right.\\ \left.+\frac{1}{4}{\displaystyle \underset{0}{\overset{1}{\int }}(q-h){\Upsilon }^3\left(h\right)dh}-\frac{5}{56}{q}^5\Upsilon (u)-\frac{1}{240}{\Upsilon }^{″}(q)+\frac{1}{284}{\Upsilon }^{‴}(q)\right];\end{array}$

And,

$\Upsilon (q)=q^3+E^{-1}\left\{p^{2.2}E\left[\begin{array}{l}\frac{1}{3}{\displaystyle \underset{0}{\overset{q}{\int }}(q+h){\Upsilon }^2\left(h\right)dh}+\frac{1}{4}{\displaystyle \underset{0}{\overset{1}{\int }}(q-h){\Upsilon }^3\left(h\right)dh}\\ -\frac{5}{56}{q}^5\Upsilon (u)-\frac{1}{240}{\Upsilon }^{″}(q)+\frac{1}{284}{\Upsilon }^{‴}(q)\end{array}\right]\right\};$

The iteration formula acts as follows:

From Eq. (11), we obtain,

$\begin{array}{l}{\Upsilon }_1(q)=E^{-1}\left\{p^{2.2}E\left[\begin{array}{l}\frac{1}{3}{\displaystyle \underset{0}{\overset{q}{\int }}(q+h){\Upsilon }_0^2\left(h\right)dh}+\frac{1}{4}{\displaystyle \underset{0}{\overset{1}{\int }}(q-h){\Upsilon }_0^3\left(h\right)dh}\\ -\frac{5}{56}{q}^5{\Upsilon }_0(q)-\frac{1}{240}{{\Upsilon }^{″}}_0(q)+\frac{1}{284}{{\Upsilon }^{‴}}_0(q)\end{array}\right]\right\}=0;\\ \text{and}\text{so},{\Upsilon }_2(q)=0,{\Upsilon }_3(q)=0,\mathrm{....};\end{array}$

So, $\Upsilon (q)=q^3.$

Example 3

Consider the FFVIDE Eqs. (12) and (13)

With the IC,

By following the same steps as in the previously described examples, you may get:

$E[\Upsilon (q)]=6p^5+p^{0.75}E\left[\begin{array}{l}{\displaystyle \underset{0}{\overset{q}{\int }}h{e}^q\Upsilon \left(h\right)dh}+{\displaystyle \underset{0}{\overset{1}{\int }}(4-h^{-3})\Upsilon \left(h\right)dh}\\ -\frac{q^2}{5}{e}^q\Upsilon (q)\end{array}\right];$

After calculating the inverse of ET, we can use the following expression to represent the recurrent connection:

Based on Eq. (14), we have: ${\Upsilon }_1=0,{\Upsilon }_2=0,{\Upsilon }_3=0,\mathrm{.....},$ then: $\Upsilon (q)=q^3.$

We see that in Examples 1-3, the FFVIDEs may be solved using this approach in a single step.

5. Conclusions

We have offered an analytical method for FFVIDEs utilizing an integral transform to handle particular problems fast and simply. The converted integral is typically used in the restricted circumstance when dealing with linear and nonlinear equations that only have an initial condition. The focus of this study has been on developing a simple procedure to obtain an explicit analytical solution about the FFVIDEs utilizing a fundamental computational method by only one step. The method was applied to solve problems identified in the literature.

In addition to being more accurate, the evaluated results also exhibit the best agreement with the existing ones. The accuracy of a series solution is evaluated at a comparatively faster rate of convergence as additional terms are included.

Our results show that the current method is efficient for FFVIDEs using an integral transform, which allows it to be applied to additional linear and nonlinear FFVIDEs in the future. Furthermore, expanding the use of this method or combining it with other numerical methods or symbolic computing tools to create a more complete application that may be utilized is one of our main goals moving ahead.