Abstract
Solving ordinary differential equations (ODEs) is vital in diverse fields. However, it is difficult to obtain the exact analytical solutions of ODEs due to their changeable mathematical forms. Traditional numerical methods can find approximate solutions for specific ODEs. Unfortunately, they often suffer from ODEs’ forms and characteristics. To approximate different types of ODEs, this paper proposes a generic method based on adaptive differential evolution. Besides, in order to further reduce the error of the obtained approximate solutions, an improved Fourier periodic expansion function is developed, which is then combined with the least square weight method to formulate the ODEs as an optimization problem. Since the proposed method is not limited to ODEs’ forms and constraint conditions, it can be used to approximate any ODEs, including linear ODEs and nonlinear ODEs. The proposed method is evaluated on twenty popular test cases. The results indicate that the proposed method is able to accurately approximate different ODEs with better performance compared with other methods.
Highlights
Differential equations are widely applied in various fields, including chemistry, engineering, economics, demography and other disciplines
ODEs can be subdivided into linear ODEs (LODEs) and nonlinear ODEs (NLODEs) based on the relationship between independent variables and dependent variables, where several LODEs or NLODEs can constitute a system of ordinary differential equations (SODEs)
TEST PROBLEMS AND PERFORMANCE CRITERIA We evaluate the proposed approach for 20 different types of ODEs with initial value problem (IVP) and boundary value problem (BVP) conditions
Summary
Differential equations are widely applied in various fields, including chemistry, engineering, economics, demography and other disciplines. A number of analytical methods have been employed to determine approximate solutions of ODEs, including the Homotopy-perturbation methods [11], Taylor expansion approach [12], variable separation method [13] and so on These approaches are often inefficient in solving complicated multi-order ODEs. In the recent few decades, the use of meta-heuristic algorithms for solving ODEs has attracted increasing attention. The transformed optimization problem can be optimized by meta-heuristic algorithms [14], [15] From this viewpoint, these algorithms are mesh-less and more generic to deal with different types of ODEs. In [16], the genetic algorithm was used to solve the first-order ODEs. Tsoulos et al [17] proposed a novel method based on the grammatical evolution to solve ODEs with the aid of genetic programming (GP).
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