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Solving initial value problems of ordinary differential equations using Runge-Kutta methods in python-based scientific computing platforms
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DOI：10.1142/s1793962326500170.png)
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Abstract
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The solution of ordinary differential equations is an essential activity in most sciences because it represents the dynamics of different systems, both in the physics and biology spheres. Nonanalytical methods to solve ODEs, however, may be tricky, particularly the nonlinear equations. The proposed study suggests a system of solving initial value problems of ODEs with advanced Runge-Kutta algorithms, which are applied to scientific computing systems with Python. The study fills the knowledge gap on how to solve ODEs efficiently in terms of computational efficiency and accuracy. The main aim is to come up with a methodology that is balance in its precision and cost of computing and use both fourth-order Runge-Kutta and adaptive Runge-Kutta-Chebyshev (RKC) methods. The error control, stability analysis, and adaptive step-sizing techniques are combined to optimize performance in the methodology. The results demonstrate that the RK4 method achieves an error of 3.20 & times; 10(-4) at t = 0.32 under the prescribed error tolerance, whereas the adaptive RKC method satisfies the specified tolerance using 142 time steps and 568 function evaluations, indicating improved efficiency in handling stiff systems. The proposed framework shows more effectiveness in stiff problem solutions and is a sure solution to real-life applications. The present research offers a potent instrument to scientists in disciplines that need the numerical resolution of ODEs. Unlike conventional standalone implementations of Runge-Kutta schemes, the proposed framework integrates adaptive error control, dynamic stage regulation, and systematic performance benchmarking within a unified scientific computing architecture.
Keywords:
Runge-Kutta
ODEs
adaptive methods
computational efficiency
error control
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591