Optimizing Budget Allocation Through First-Order Linear Differential Equations and Innovative Transform Techniques

Traditional cost calculation methods adopted by organizations typically assign expenses to each department based on their incurred costs within the year, often neglecting an analysis of departmental profitability [1, 2]. An alternative method, Activity-Based Costing (ABC), has been applied with some success, but it presents challenges in linking costs between activities and products, and lacks scalability [3, 4].

Kaplan and Anderson [3] proposed the Time-Driven Activity-Based Costing (TDABC) system, a derivative of the ABC method, in 1998. The TDABC system allocates indirect costs based on time but is limited to a specific period [5-7]. This study explores the use of ordinary differential equation systems for cost calculation, considering the relationship between costs and activities, and accommodating an indefinite time span. This approach provides accountants with a modern tool for developing practical and effective future budgets for project costs [8].

Integral transformations are accurate and efficient methods for solving mathematical systems, including ordinary differential equation systems. Various transformations exist, such as Laplace, Elzaki, Novel, and SEE transformations [9]. Introduced in 2016, the Novel transformation is an effective method for solving first-order linear mathematical systems and is derived from the Laplace transform [10].

Albukhuttar and Almasoudi [11] utilized systems of Euler's equations to model cost calculation problems in bank auditing. Meanwhile, Alshibley et al. [12] introduced first-order fuzzy linear systems to calculate costs when the initial project cost is fuzzy over a certain time period.

Several other integral transforms have been introduced and utilized to solve linear ordinary differential equations with constant coefficients, including the Aboodh transform by Aboodh [13] and the Kushare transform by Kushare et al. [14]. Jafari [15] proposed a new general integral transform that encompasses all classes of integral transforms in the Laplace transform class and explored its application in solving ordinary differential equations (ODEs) with constant and variable coefficients. Jafari's transform is particularly adept at handling fractional-order differential equations and integral equations.

The Anuj transforms, a recent development by Patil et al. [16], has been used to solve linear Volterra integral equations of the first kind, demonstrating its practicality and efficiency. Similarly, Patil and Khakale [17] introduced and applied the Soham transform to solve linear ordinary differential equations with constant coefficients.

In the realm of solving linear Volterra integral equations of the first kind, Aggarwal et al. [18] utilized the Kamal transform and provided applications to demonstrate its usefulness. They also applied the Mohand transform to solve convolution-type linear Volterra integral equations of the second kind, providing practical examples of its application [19].

Aiming to solve partial differential equations, Ahmed et al. [20] introduced the double Laplace - Sumudu transform (DLST), providing an alternative to other double transforms. The study establishes theorems on the modern properties of the DLST, discusses the convolution theorem, and demonstrates the solution of partial differential equations using these results. This investigation suggests that the DLST is superior in terms of effectiveness and utility when dealing with such equations.

For the purpose of solving differential equations in the time domain, Maitama and Zhao [21] presented the Shehu transform, a generalisation of the Laplace and Sumudu integral transforms. To demonstrate its ease of use, effectiveness, and high accuracy, the suggested integral transform was successfully deduced from the traditional Fourier integral transform. It was then applied to both ordinary and partial differential equations.

The scientific study of differential equations and their applications has garnered significant attention in recent years. The Sumudu transform, for instance, has been employed by Eltayeb and Kilicman [22] to solve linear ordinary differential equations, both without constant coefficients and with them. These equations, notably, included convolution terms. The authors optimally utilized the Sumudu transform for addressing various issues related to Spring-Mass systems, Population Growth, and finance.

In parallel, the exploration of the Shehu transform has also proven fruitful. Bokhari [23] introduced novel characteristics of the Shehu transform, applying this transformation to Atangana-Baleanu derivatives in the Caputo and Riemann-Liouville senses. Their work involved the resolution of various fractional differential equations.

In a pivotal study, Asiru [24] validated the convolution theorem for the Sumudu transform for functions expressible either as a polynomial or a convergent infinite series. Consequently, this theorem can be used to resolve integral equations of the convolution type. Arikoglu and Ozkol [25] resorted to the differential transform method for analysing integro-differential systems and integral equation systems. Their method, enriched with a formulation that addresses Fredholm integrals, has expanded the horizon for differential equation solutions. The method's robustness and competence have been exemplified through its application to various integral and integro-differential equations.

Atangana and Akgül [26] conducted a comparative analysis of various derivatives - the classical derivative, the Caputo derivative, the Caputo-Fabrizio derivative, and the Atangana-Baleanu derivative – to discern the differences between the Laplace and Sumudu transforms. Their findings illustrated that the Sumudu transform-derived Bode diagrams essentially function as a low-pass filter, while those derived from the Laplace transform mainly function as a high-pass filter.

Emphasizing the effectiveness of integral transforms, Patil et al. [27] successfully applied the Kushare integral transform to solve the Faltung Type Volterra Integro-Differential Equation of the First Kind. The results achieved provided evidence of the integral transform's efficacy in solving first-class integro-differential equations.

In a recent advancement, Saadeh et al. [28] introduced a new integral operator transform, the ARA transform. Their research validated the existence of this transform, as well as its inverse. They also demonstrated its applicability in solving ordinary and partial differential equations, which are frequently encountered in various research fields, including physics and engineering.

In the realm of cost dynamics, the current study proposes a novel method for modelling various organizational activities using first-order linear differential equations. This method is expected to enable the accurate calculation of activity-based costs over a three-year period. In the context of current research, the present study aims to apply this proposed method to answer questions concerning cost calculation and budget preparation.

$\left\{\begin{array}{c}\mathcal{N}_{\mathrm{i}}\left(\mathcal{H}_1\right)=\frac{1}{\Delta}\left|\begin{array}{cccc}\frac{\mathcal{H}_1(0)}{\mathcal{P}}+\mathcal{N}_{\mathrm{i}}\left(\mathrm{w}_1\right) & -\mathrm{c}_{12} & \cdots & -\mathrm{c}_{1 \mathrm{n}} \\ \frac{\mathcal{H}_2(0)}{\mathcal{P}}+\mathcal{N}_{\mathrm{i}}\left(\mathrm{w}_2\right) & \left(\mathcal{P}-\mathrm{c}_{22}\right) & \cdots& -\mathrm{c}_{2 \mathrm{n}} \\ \vdots & \vdots & \cdots & \vdots \\ \frac{\mathcal{H}_{\mathrm{n}}(0)}{\mathcal{P}}+\mathcal{N}_{\mathrm{i}}\left(\mathrm{w}_{\mathrm{n}}\right) & -\mathrm{c}_{\mathrm{mn}} & \cdots & \left(\mathcal{P}-\mathrm{c}_{\mathrm{mn}}\right)\end{array}\right| \\ \mathcal{N}_{\mathrm{i}}\left(\mathcal{H}_{\mathrm{n}}\right)=\frac{1}{\Delta}\left|\begin{array}{cccc}\left(\mathcal{P}-\mathrm{c}_{11}\right) & -\mathrm{c}_{12} & \cdots & \frac{\mathcal{H}_1(0)}{\mathcal{P}}+\mathcal{N}_{\mathrm{i}}\left(\mathrm{w}_1\right) \\ -\mathrm{c}_{21} & \left(\mathcal{P}-\mathrm{c}_{22}\right) & \ldots & \frac{\mathcal{H}_2(0)}{\mathcal{P}}+\mathcal{N}_{\mathrm{i}}\left(\mathrm{w}_2\right) \\ \vdots & \vdots & \cdots & \vdots \\ -\mathrm{c}_{\mathrm{m} 1} & -\mathrm{c}_{\mathrm{m} 2} & \ldots & \frac{\mathcal{H}_{\mathrm{n}}(0)}{\mathcal{P}}+\mathcal{N}_{\mathrm{i}}\left(\mathrm{w}_{\mathrm{n}}\right)\end{array}\right|\end{array}\right.$                        (13)

Then, taking the inverse of the Novel transform for $\mathcal{N}_{\mathrm{i}}\left(\mathcal{H}_{\mathrm{i}}\right), \,\,\mathrm{i}=1,2,3 \ldots, \mathrm{n}$, yields the set solution of problem (3.1).

3.2 A homogeneous system which has the formula $\mathcal{H}^{\prime}=\mathbf{C} \mathcal{H}$

From the previous formula (3.1), the vector $\vec{W}=0$, so the solution of (3.1) has the from:

$\left\{\begin{array}{c} \Delta=\left|\begin{array}{ccccc}\left(\mathcal{P}-c_{11}\right) & -c_{12} & \cdots & -c_{1 \mathrm{n}} \\ -c_{21} & \left(\mathcal{P}-c_{22}\right) & \cdots & -c_{2 \mathrm{n}} \\ \vdots & \vdots & \cdots & \vdots \\ -c_{\mathrm{m} 1} & -c_{\mathrm{m} 2} & \cdots & \left(\mathcal{P}-c_{\mathrm{mn}}\right)\end{array}\right|\\\mathcal{N}_{\mathrm{i}}\left(\mathcal{H}_{\mathrm{n}}\right)=\frac{1}{\Delta}\left|\begin{array}{cccc}\left(\mathcal{P}-\mathrm{c}_{11}\right) & -\mathrm{c}_{12} & \ldots & \frac{\mathcal{H}_1(\mathrm{U})}{\mathcal{P}} \\ -\mathrm{c}_{21} & \left(\mathcal{P}-\mathrm{c}_{22}\right) & & \frac{\mathcal{H}_2(0)}{\mathcal{P}} \\ \vdots & \vdots & \ldots & \vdots \\ -c_{\mathrm{m} 1} & -\mathrm{c}_{\mathrm{m} 2} & \ldots & \frac{\mathcal{H}_{\mathrm{n}}(0)}{\mathcal{P}}\end{array}\right|\end{array}\right.$                        (14)

Accordingly, in similar way by taking the inverse of the Novel transform for $\mathcal{N}_{\mathrm{i}}\left(\mathcal{H}_{\mathrm{i}}\right), \,\,\mathrm{i}=1,2 \ldots, \mathrm{n}$ yields the set solution of problem (3.2).