Mathematics

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Mathematics

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Subject: search.filters.subject.Convergence Analysis

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    Utilizing the Artificial Neural Network Approach for the Resolution of First-Order Ordinary Differential Equations
    (Penteract Technology, Malaysia, 2024-06-16) Khadeejah James Audu; Marshal Benjamin; Umaru Mohammed; Yusuph Amuda Yahaya
    Ordinary Differential Equations (ODEs) play a crucial role in various scientific and professional domains for modeling dynamic systems and their behaviors. While traditional numerical methods are widely used for approximating ODE solutions, they often face challenges with complex or nonlinear systems, leading to high computational costs. This study aims to address these challenges by proposing an artificial neural network (ANN)- based approach for solving first-order ODEs. Through the introduction of the ANN technique and exploration of its practical applications, we conduct numerical experiments on diverse first-order ODEs to evaluate the convergence rate and computational efficiency of the ANN. Our results from comprehensive numerical tests demonstrate the efficacy of the ANN-generated responses, confirming its reliability and potential for various applications in solving first-order ODEs with improved efficiency and accuracy.
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    Error and Convergence Analysis of a Hybrid Runge- Kutta Type Method
    (International Journal of Science and Technology Publications UK, 2015-04) Muhammad R; Y. A Yahaya,; A.S Abdulkareem
    Implicit Runge- Kutta methods are used for solving stiff problems which mostly arise in real life problems. Convergence analysis helps us to determine an effective Runge- Kutta Method (RKM) to use, but due to the loss of linearity in Runge –Kutta Methods and the fact that the general Runge –Kutta Method makes no mention of the differential equation makes it impossible to define the order of the method independently of the differential equation. In this paper, we derived a hybrid Runge -Kutta Type method (RKTM) for 𝑘=1, obtained the order and error constant and convergence analysis of the method.