Date Published: March 8, 2019
Publisher: Public Library of Science
Author(s): Manuel Caravaca, Pilar Sanchez-Andrada, Antonio Soto-Meca, Esteban Tlelo-Cuautle.
In this article we introduce the software SimKinet, a free tool specifically designed to solve systems of differential equations without any programming skill. The underlying method is the so-called Network Simulation Method, which designs and solves an electrical network equivalent to the mathematical problem. SimKinet is versatile, fast, presenting a real user-friendly interface, and can be employed for both educational and researching purposes. It is particularly useful in the first courses of different scientific degrees, mainly Chemistry and Physics, especially when facing non-analytic or complex-dynamics problems. Moreover, SimKinet would help students to understand fundamental concepts, being an opportunity to improve instruction in Chemistry, Mathematics, Physics and other Sciences courses, with no need of advanced knowledge in differential equations. The potency of SimKinet is demonstrated via two applications in chemical kinetics: the photochemical destruction of stratospheric ozone and the chaotic dynamics of the peroxidase-oxidase reaction.
Differential equations allow a very convenient modelling of some essential natural phenomena that evolve over time in a continuous way. This is a prime reason why differential equations are ubiquitous in so many scientific and technological disciplines [1–3]. In this context, knowing how to pose and solve a particular set of differential equations is a crucial issue in order to describe a wide range of fundamental processes, as well as to recognize and control the variables governing them.
The electrical analogy of scientific problems is well recognized as a very useful and attractive educational subject, which constitutes a standard procedure in some undergraduate textbooks . This approach is no more than a formal equivalence between the governing equations of the problem and an electrical network. In this analogy, the terms of the original equations are usually identified with appropriate electrical devices. Within this framework, the mechanical-electrical analogies are well-known since James Clerk Maxwell’s era . A classical example, found in the subjects Physics and Mechanics belonging to first courses in Physics, Chemistry and Engineering degrees, is the equivalence between a non-forced damped oscillator and an electric circuit . The differential equation that describes the original problem, through Newton’s second law of motion, is:
where m is the mass, x is the position coordinate along the X axis, b is the damping constant and k is the restoring constant. On other hand, Kirchhoff’s second law for electric circuits can be easily applied to a series RLC circuit , thus obtaining:
where L is the inductance, q is the electrical charge, R is the resistance and C is the capacitance of the electric circuit, respectively. Eqs (1) and (2) present the same mathematical structure so, by establishing the particular equivalence m → L, b → R, k → 1/C and x → q, the analogy is completed. Then, the dynamics of the system can be followed through the resolution of Newton’s or Kirchhoff’s laws, equivalently. This particular example is easy to solve theoretically in both cases, and usually presents strictly academic interest. However, in some other problems, the use of the electrical analogy allows an easier handling of the theoretical solution, which can be particularly valuable in Lagrangian dynamics, when dealing with many-body systems [34,38]. Furthermore, the electrical analogy can be extended to other scientific areas, such as heat transfer, fluid flow, diffusion or chemical reactions [20,39,40].
SimKinet satisfies two basic requirements for simulating kinetic chemical differential equations. Firstly, it has a simple, editable and visual environment allowing an easy management. This enables the user to understand the sequential steps taking place in the simulation. Secondly, the program is fast and reliable, thus offering numerical advantages over traditional simulation algorithms. When employed for academic purposes, once students have solved a series of simple educational problems, they can naturally evolve into more complex systems.
In this section, we will show two practical cases of SimKinet involving researching and academic interest. The first one adresses a problem without analytical solution, guided by the Chapman mechanism for the formation and decomposition of atmospheric ozone. This example may be particularly useful in undergraduate Chemistry courses, where it can be employed to study in depth the order of chemical reactions. Furthermore, in the context of non-analytical solutions, SimKinet is a suitable tool for researchers who need to solve complex kinetic schemes [19,52]. The second practical example, the Olsen attractor, constitutes a very interesting illustration of chaotic dynamics. The study of oscillating chemical reactions becomes essential to understand some key aspects of the behaviour of living organisms. In this way, students can be introduced in nonlinear dynamics, a common feature for some crucial chemical reactions. Non-linear and chaotic dynamics are universal, and have special interest in other related subjects such as Physics or Mathematics at undergraduate courses . From the researching point of view, SimKinet simplifies the determination of chaotic patterns, as for example the insight of chaos in phase diagrams for Josephson junctions .
The software SimKinet, a powerful and versatile software, has been designed to solve kinetic chemical equations, and it can be successfully applied for educational and researching purposes. SimKinet is a free and user-friendly tool able to solve a wide range of problems involving differential equations. Its simple handling makes it also suitable for teaching at undergraduate levels without having to resort to theoretical approximations, and allowing the student to deepen into the mathematical nature of scientific models. Two interesting practical examples have been explained to illustrate the capabilities of the software: the Chapman model and the Olsen attractor. Among other interesting uses, it can be employed to determine the range of validity for theoretical approximations, to distinguish between chaotic and periodic behaviour in nonlinear dynamics, or to take control over the product composition in chemical reactions.