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All Journal CAUCHY: Jurnal Matematika Murni dan Aplikasi Science and Technology Indonesia Indonesian Journal of Mathematics and Applications
Ardiana, Dita
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Biochemistry, Genetics & Molecular Biology Chemical Engineering, Chemistry & Bioengineering Computer Science & IT Education Environmental Science Materials Science & Nanotechnology Mathematics Physics Other

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Comparison Study between Shooting and Finite Difference Methods for Solving Linear Boundary Value Problem with Dirichlet, Neumann, and Robin Boundary Conditions Ardiana, Dita; Rachman, Alifira Meliana; Nurkarimah, Dwi; Habibah, Ummu
Indonesian Journal of Mathematics and Applications Vol. 3 No. 1 (2025): Indonesian Journal of Mathematics and Applications (IJMA)
Publisher : Universitas Brawijaya

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.21776/ub.ijma.2025.003.01.2

Abstract

This study conducts a comparative analysis of the Shooting and Finite Difference methods for solving boundary value problems (BVPs) in ordinary differential equations (ODEs). The findings indicate that the Shooting method offers superior accuracy, particularly for smaller step sizes, whereas the Finite Difference method is more straightforward to implement and exhibits greater computational efficiency. The results further demonstrate that the Shooting method is particularly highly appropriate for problems with Dirichlet boundary conditions, as it achieves the lowest mean absolute error (MAE) across various step sizes. Conversely, the Finite Difference method attains higher computational efficiency for the same problem type but performs less advantageously in cases involving other boundary conditions. In contrast, the Shooting method demonstrates greater efficiency in solving problems with Neumann and Robin boundary conditions. The selection of an appropriate numerical method depends on the specific characteristics of the problem, necessitating a balance between accuracy and computational cost. This study provides a comprehensive evaluation of these numerical approaches to support the selection of the most suitable method for efficiently and accurately solving BVPs.
An Improved Fifth-Order Runge-Kutta Method with Higher Accuracy and Efficiency for Solving Initial Value Problems Habibah, Ummu; Medrano, Fermin Franco; Permana, Adith Chandra; Ardiana, Dita; Trisilowati
Science and Technology Indonesia Vol. 10 No. 3 (2025): July
Publisher : Research Center of Inorganic Materials and Coordination Complexes, FMIPA Universitas Sriwijaya

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.26554/sti.2025.10.3.802-816

Abstract

Solving initial value problems (IVPs) in ordinary differential equations (ODEs) often requires numerical methods, with the fifth-order Runge-Kutta method being a widely used approach due to its balance between accuracy and computational efficiency. A novel and straight forward formula for the fifth order Runge-Kutta method is proposed, aiming to simplify calculations while maintaining high accuracy and stability. The method is derived using an optimized Taylor series expansion, leading to a more efficient formulation. Numerical experiments are conducted to compare the proposed method with existing fifth-order Runge-Kutta methods. The results showthat the proposed formula out performs existing methods in terms of accuracy, stability, and computational efficiency. This new formula provides a practical alternative for solving IVPs in ODEs with improved performance.
Comparison Study between Shooting and Finite Difference Methods for Solving Linear Boundary Value Problem with Dirichlet, Neumann, and Robin Boundary Conditions Ardiana, Dita; Rachman, Alifira Meliana; Nurkarimah, Dwi; Habibah, Ummu
Indonesian Journal of Mathematics and Applications Vol. 3 No. 1 (2025): Indonesian Journal of Mathematics and Applications
Publisher : Universitas Brawijaya

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.21776/ub.ijma.2025.003.01.2

Abstract

This study conducts a comparative analysis of the Shooting and Finite Difference methods for solving boundary value problems (BVPs) in ordinary differential equations (ODEs). The findings indicate that the Shooting method offers superior accuracy, particularly for smaller step sizes, whereas the Finite Difference method is more straightforward to implement and exhibits greater computational efficiency. The results further demonstrate that the Shooting method is particularly highly appropriate for problems with Dirichlet boundary conditions, as it achieves the lowest mean absolute error (MAE) across various step sizes. Conversely, the Finite Difference method attains higher computational efficiency for the same problem type but performs less advantageously in cases involving other boundary conditions. In contrast, the Shooting method demonstrates greater efficiency in solving problems with Neumann and Robin boundary conditions. The selection of an appropriate numerical method depends on the specific characteristics of the problem, necessitating a balance between accuracy and computational cost. This study provides a comprehensive evaluation of these numerical approaches to support the selection of the most suitable method for efficiently and accurately solving BVPs.
Higher-Order Numerical Solution of the KdV-BBM Equation: A Comparative Analysis of Temporal Integration Schemes in the Method of Lines Framework Ardiana, Dita; Habibah, Ummu; Trisilowati, Trisilowati; Ranom, Rahifa Binti
CAUCHY: Jurnal Matematika Murni dan Aplikasi Vol 11, No 1 (2026): CAUCHY: JURNAL MATEMATIKA MURNI DAN APLIKASI
Publisher : Mathematics Department, Universitas Islam Negeri Maulana Malik Ibrahim Malang

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.18860/cauchy.v11i1.41525

Abstract

This study investigates the numerical simulation of solitary wave propagation governed by the KdV-BBM equation using a robust Method of Lines (MOL) framework. The governing nonlinear equation is transformed into a system of ordinary differential equations through spatial discretization, and the performance of three temporal integration schemes is evaluated: the first-order Euler method, the fourth-order Runge-Kutta (RK-4), and the fifth-order iRK-5 method. Quantitative analysis using Mean Absolute Error (MAE) for various time steps (t = 0.2, 0.1, 0.05, and 0.01) reveals that the iRK-5 scheme provides enhanced temporal precision, achieving error magnitudes as low as 106 and consistently aligning with the exact traveling-wave solution. Notably, the iRK-5 method demonstrates greater algorithmic efficiency, achieving an accuracy level of 4.55 106 at a coarser time step of t = 0.1, whereas the RK-4 scheme requires a finer time step of t = 0.05 to reach the same precision. Both high-order methods eventually reach a spatial error floor where further temporal refinement yields no significant reduction in MAE, emphasizing that high-order temporal integration, particularly the iRK-5 scheme, is essential for preserving the physical integrity of complex nonlinear wave phenomena while maintaining optimal computational effort.
Co-Authors Medrano, Fermin Franco Nurkarimah, Dwi Permana, Adith Chandra Rachman, Alifira Meliana Ranom, Rahifa Binti Trisilowati Trisilowati Trisilowati, Trisilowati Ummu Habibah, Ummu