Numerical Approximations of ODEs Initial Value Problem; A Case Study of Gluconic Acid Fermentation by Pseudomonas ovalis
Journal of Advances in Mathematics and Computer Science,
Across different sections of life, physical and chemical sciences, differential equations which could be ordinary differential equations (ODEs) or partial differential equations (PDEs) are used to model the various systems as observed. Some types of ODEs, and a few PDEs are solvable by analytical methods with much difficulties. However, the great majority of ODEs, especially the non-linear ones and those that involve large sets of simultaneous differential equations, do not have analytical solutions but require the application of numerical techniques. This work focused on exemplifying numerical approximations (Adams-Bashforth-Moulton, Bogacki-Shampine, Euler) of ODEs Initial value Problem in its simplest approach using a case study of gluconic acid frementation by Psuedonomas Ovalis. The performance of the methods was checked by comparing their accuracy. The accuracy was detrermined by the size of the discretization error estimated from the difference between analytical solution and numerical approximations. The results obtained are in good agreement with the exact solution. This work affirms that numerical methods give approximate solutions with less rigorous work and time as there is room for flexibility in terms of using different step sizes with the Euler solver as most accurate.
- numerical simulation
How to Cite
Constatinides A, Mostoufi N. Numerical methods for chemical engineers with MatLab applications. Prentice Hall PTR, upper Saddle River, New Jersey 07458; 2005.
Boyce WE, DiPrima RC. Elementary differential equation and boundary value problems, John Wiley and Son, Hoboken; 2001.
Gilat A. Matlab: An Introduction with application. John Wiley and Sons; 2004.
Kockler N. Numerical methods and scientific computing. Clarendon Press. Oxford London; 1994.
Stephen MP. To compute numerically, concepts and strategy. Little Brown and Company, Canada; 1983.
Abdel-Halim Hassan IH. Application to differential transformation method for solving systems of differential equations. Applied Mathematical Modelling. 2003;32:2552–2559.
Ihoeghian N.A, Osagiede CA, Ikogwe KI. Numerical simulation of the non-isothermal plug flow reactor design for the dehydrogenation of ethylbenzene to styrene. Journal of the Nigeria Association of Mathematical Physics. 2018;44(1):311-318.
Meburgerac S, Niethammer M, Bothe D, Schäfer M. Journal of Non-Newtonian Fluid Mechanics. 2021; 287(1):104451.
Atcovi. Adams-Bashforth and Adams-Moulton methods; 2017.
Available:https://en.wikiversity.org/w/index.php?title=Adams-Bashforh_and_Adams-Moulton_methods&oldid=1761775, (Accessed; 17 October 2018)
Autar K, Garapati SH. Physical Problem for partial differential equations for general engineering. University of South Florida; 2011.
Bogacki P, Shampine LF. A3(2) pair of Runge-Kutta formulas, Applied mathematics Letters. 1989;2(4):321-325.
Lambert JD. Numerical method for ordinary systems of Initial value problems, John Wiley and Sons, New York; 1991.
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