Research Article
A Novel Analytic Method with Integral Transform for Solving Classes of Second and Third Order Ordinary Linear Differential Equations with Variable Coefficients
Mohamed Abdirahman Jama*
,
Kang'ethe Giterere,
Duncan Gathungu Kioi
Issue:
Volume 14, Issue 2, April 2025
Pages:
78-89
Received:
26 January 2025
Accepted:
18 February 2025
Published:
5 March 2025
DOI:
10.11648/j.acm.20251402.11
Downloads:
Views:
Abstract: Analytical solutions of second- and third-order non-homogeneous Ordinary Linear Differential Equations (OLDEs) with variable coefficients have been investigated using an established mathematical tool, the integral transform, together with a new analytic method developed in this study. This study aims to utilize the integral transform alongside the new analytical method. The new method was derived from the concept of exactness in higher-order ODEs. Specifically, second- and third-order ODEs with variable coefficients are exact if there exist first- and second-order linear ODEs whose derivatives correspond to the given equations, respectively. In this new analytic method, an integrating factor function formula for second-order ODEs has been carefully formulated and derived, making every second-order ODE with variable coefficients reducible to its lower-order form, specifically first-order ODEs. To ensure the accuracy of the new method, two well-known classes of second-order linear ODEs, namely the Whittaker second-order linear ODE and the Modified Bessel equation, were applied. The results demonstrated that the new analytic method effectively solves these equations, producing exact analytical solutions. To validate the effectiveness and efficiency of the new analytic method, a comparative analysis was conducted using illustrative examples, followed by graphical representations of the solution results.
Abstract: Analytical solutions of second- and third-order non-homogeneous Ordinary Linear Differential Equations (OLDEs) with variable coefficients have been investigated using an established mathematical tool, the integral transform, together with a new analytic method developed in this study. This study aims to utilize the integral transform alongside the ...
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Research Article
Gompertz Function Approach: Numerical Integration for Microbial Growth Problem
Issue:
Volume 14, Issue 2, April 2025
Pages:
90-100
Received:
28 February 2025
Accepted:
11 March 2025
Published:
28 March 2025
Abstract: In this work, we developed a numerical integrator using the Gompertz function model approach with the basic parameters as highlighted by Gompertz in finding and measure the growth in human cells as a basis function involving exponential, logarithmic, and polynomial, hence implemented the numerical integrator to solve problems arising in microbial growth staging. Microbial growth, synonymous to mildew or mold, which is a fungi family commonly found both indoors and outdoors. The indoors occur especially when there is humidity, moisture, oxygen, organic matters and low sunlight. Microbial growth which is the increase in the number of microbial cells which can also be in term of bacterial growth. It can be influenced by various factors to grow including temperature, Water, availability of oxygen, and other nutrient content. The growth staging can be in four phases such as lag, logarithmic, stationary and death phases. A culture of bacterial was taken, the approximate number of strand that was originally present and the growth were calculated using the numerical integration, the results obtained shows a significant, effective and robust improvement on the strand when compared the results with the exact solution. The properties of the integrator were analyzed, considering that Microbial Growth is an increase in the number of bacteria cells in a system when the proper nutrients and environment are provided. Therefore with the approach of Gompertz, the numerical integrator can be applied further to find the growth in each of the phases as they occurs.
Abstract: In this work, we developed a numerical integrator using the Gompertz function model approach with the basic parameters as highlighted by Gompertz in finding and measure the growth in human cells as a basis function involving exponential, logarithmic, and polynomial, hence implemented the numerical integrator to solve problems arising in microbial g...
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