Solving the Chemical Reaction Models with the Upadhyaya Transform

Introduction

Differential equations have frequently been solved using the integral transform. Due to their significance in mathematically expressing potential changes in scientific and engineering situations, differential equations have played a significant role in many scientific and engineering domains. Integral transforms are mathematical operations that convert a given function into a new representation, often in terms of a different variable or set of variables. These transforms are widely used in various branches of mathematics and physics to simplify the analysis and solution of complex problems. These problems involve finding the solution to a differential equation subject to certain boundary conditions. By applying an integral transform, the differential equation can be transformed into an equation involving the transform of the unknown function. This allows for the boundary conditions to be incorporated into the transformed equation, making it easier to find the solution. Lalit Mohan Upadhyaya¹ introduced the Upadhyaya transformation, a highly advanced transform within the Laplace class, which holds great promise for cutting-edge applications across various fields of research. Upadhyaya and his colleagues² further developed this theory, positioning it as one of the most robust and sophisticated transforms in contemporary mathematics literature. Its potential spans disciplines including mathematics, engineering, physics, chemistry, biology, pharmaceuticals, economics, sociology, and more. Despite its vast potential, many applications of this transformative tool remain untapped. This paper focuses on demonstrating the application of the Upadhyaya transform in solving chemical kinetics problems, showcasing its efficacy within the realm of chemical sciences. Murphy³ investigated consecutive chemical reactions of the first and second orders. Chrastil⁴used the final product to determine the rate constants of a first-order chemical process. Kalaiarasi et.al.⁵utilized the Kamal transform method to address Two Tank Mixing Problems, which involve solving first-order linear differential equations. To determine the concentration of chemical compounds, Higazy and Aggarwal⁶applied the Sawi transform to the mathematical model of the chemical reaction in series. Mousa⁷ utilized the Upadhyaya transform to determine the solution for the Volterra integral equation of the first kind. Patil and others ^{8, 9} applied the Soham and Kushare transforms in chemical sciences, while Peker et al.¹⁰utilized the Kashuri Fundo Transform to obtain solutions for various chemical reaction models. To obtain the solution for Volterra integral equations of the second kind, Dinesh and Prakash¹¹ employed the Upadhyaya transform. Kuffi and Mansour¹²solved certain cardiovascular models using the Emad-Falih integral transform.  Aggarwal et al.¹³utilized the Rishi Transform to ascertain the concentrations of the chemical compounds in a first-order successive chemical reaction. Anuj et al.¹⁴employed the Anuj transform to solve the problem of reactant concentrations in first-order successive chemical reactions analytically. To solve fractional integro-differential equations, Gunaseka and Prabakaran¹⁵ used the Mohand transform.Recently, Dinesh and Kuffi¹⁶ demonstrated the Upadhyaya transform’s ability to solve ordinary differential equations by applying it to cardiovascular models. The present study highlights the significance of the Upadhyaya transform in solving chemical reaction models.

Definition of Upadhyaya Transform

The Upadhyaya transform of the function f(t) is mathematically defined as¹:

The representation of the inverse Upadhyaya transform is as follows:

In the general formulation of the Upadhyaya transform, λ₁, λ₂ and λ₃ are complex parameters¹.

Some Useful Characteristics of Upadhyaya Transformation

Linearity property

If f₁(t) and f₂(t) be two functions with UT’s u₁(λ₁, λ₂, λ₃ )and u₂(λ₁, λ₂, λ₃ ) in terms of the parameters λ₁, λ₂, λ₃ and b₁, b₂ be any constants then¹

Convolution property

If the Upadhyaya transform of the functions f₁(t) and f₂(t) with respect to the parameters λ₁, λ₂, λ₃ are u₁(λ₁, λ₂, λ₃ ) and u₂(λ₁, λ₂, λ₃ ) , then convolution of Upadhyaya transform of the functions f₁(t)* f₂(t) is given by¹

Where f₁(t)* f₂(t) is given by

Upadhyaya transforms of some elementary functions¹: 

Upadhyaya transform of derivatives

If U[f(t)] then u(λ₁, λ₂, λ₃ ) from¹

Applications of the Upadhyaya Transform in Various Chemical Reaction Models

In this section, we apply the Upadhyaya transform to solve several chemical reaction models represented by ordinary differential equations, which play a crucial role in the realm of chemistry.

The zero-order chemical reaction model satisfies the initial value problem¹⁰

where

c(t) – concent ration of reacting substance at time t, k₀ – positive constant.

Performing the Upadhyaya transform bilaterally in equation (3) yields

Let U[c(t)] = u(λ₁, λ₂, λ₃ ). Using the derivative property of Upadhyaya transform and initial condition (4), we obtain equation (6)

After applying bilaterally the inverse Upadhyaya transform to equation (6), we will find the solution to the given initial value problem as

Figure 1: The solution behaviour of application Click here to View Figure

Figure 1 illustrates the concentration of a reacting substance over time in a zero-order chemical reaction. Starting from an initial concentration , the concentration decreases linearly with time, indicating a constant reaction rate. The slope of the graph represents the rate constant , with a negative value denoting a decrease in concentration over time. This linear decrease in concentration showcases the characteristic behavior of zero-order reactions, where the rate remains unaffected by changes in concentration.

The First-order chemical reaction model satisfies the initial value problem¹⁰

We apply the Upadhyaya transform to solve the given first-order chemical reaction model.

Applying the Upadhyaya transform bilaterally to equation (8) yields

Let U[c(t)] = u(λ₁, λ₂, λ₃ ) . By rearranging equation (10) in accordance with the initial condition (9) and Upadhyaya transform’s derivative property, we obtain

After applying bilaterally the inverse Upadhyaya transform to equation (11), we will find the solution to the given initial value problem as

Figure 2: The solution behaviour of application. Click here to View Figure

Figure 2 depicts the concentration of a reacting substance over time in a first-order chemical reaction. As time progresses, the concentration exponentially decreases, following a decay curve characterized by the rate constant (k₁), The initial concentration (c₀), determines the starting point of the curve, with higher initial concentrations leading to slower decay rates.

Now we determine the solutions for the systems of differential equations that govern the successive chemical reactions of the first order given by¹⁰

With the initial conditions

In equation (2), c₁(t) is the concentration of substance A1 at time t , which breaks down to generate the new substance A2 with concentration c₂(t), and c₃(t) is the concentration of a new element derived from .

In the prior application, we determined that the differential equation of the function c₁(t) has a solution as c₀e^-k₁^t . When we plug c₀e^-k₁^t solution into the differential equation of the c₁(t) function, we get

Applying the Upadhyaya transform bilaterally to equation (15) yields

Let U [c₂(t)] = u(λ₁, λ₂, λ₃ ) . By rearranging equation (16) in accordance with the initial condition (14) and Upadhyaya transform’s derivative property, we obtain

After applying bilaterally the inverse Upadhyaya transform to equation (17), we get

We now have the solutions to the differential equation of the function c₂(t) . Substituting c₂(t)=K₁c₀/k₂ – k₁[e^-k1t – e ^–k2t] into the differential equation for the c₃(t) function produces the following result:

Applying the bilateral Upadhyaya transform to equation (19), we obtain

Let U [c₃(t)] = u(λ₁, λ₂, λ₃ ) and using the derivative property of Upadhyaya transform, we obtain

Equation (21) can be expressed as follows by using the initial condition (14).

Rearranging the equation (22), we get

After applying bilaterally the inverse Upadhyaya transform to equation (23), we get

Considering the previous solutions, the outcome can alternatively be stated as follows:    

Figure 3: The solution behaviour of application. Click here to View Figure

Figure 3 depicts the evolution of concentrations in a system undergoing successive first-order chemical reactions. Initially, the concentration of A1(c₁) is high and gradually decreases exponentially over time as it transforms into A2(c₂) . A2’s concentration rises initially, reaching a plateau as it accumulates, then slowly declines. Meanwhile, B’s concentration  steadily increases over time as it forms from (c₃) , eventually leveling off. This dynamic interplay among the substances highlights the intricate kinetics of first-order reactions and the transformation of reactants into products.

Conclusion

This research showcases how employing the Upadhyaya transform for chemical reaction models has proven to be effective and elegant, streamlining the solution process of ordinary differential equations commonly encountered in chemical kinetics and various other branches of chemistry. By providing a streamlined approach that requires minimal computational burden, this transform offers a valuable tool for researchers and practitioners in understanding reaction kinetics. The inclusion of graphical representations has further enhanced the interpretability of results, facilitating deeper insights into the temporal evolution of reactant concentrations. Moving forward, there is potential for extending the utility of the Upadhyaya transform to more complex chemical systems and exploring its integration with advanced computational techniques for real-time simulation and optimization. This underscores its significance as a versatile method with promising implications for advancing our understanding and application of chemical kinetics.

Conflict of Interest

The authors declare no conflict of interest.

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