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Abstract
The solution of differential equations, whether through analytical or numerical approaches, remains a fundamental problem in applied mathematics. Various numerical methods, including Euler’s method, the Runge-Kutta method, and Taylor series, have been widely used to solve first-order ordinary differential equations. Some researchers have explored these equations using Newton’s interpolation method, while others have combined Newton’s method with Lagrange’s interpolation technique. This study introduces a hybrid approach that integrates Newton’s interpolation method with Aitken’s method to solve Bernoulli differential equations. The efficiency and accuracy of this hybrid method are analyzed through a series of examples, demonstrating its effectiveness compared to conventional numerical techniques.
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References
- Aldin, N. (2020). Comparison of Newton's Interpolation and Aitken's Method with Runge-Kutta Method for Solving First Order Differential Equation. Middle-East Journal of Scientific Research, 28(5), 391-394. doi:10.5829/idosi.mejsr.2020.391.394
- Araz, S., & Abdon, A. (2021). New Numerical Scheme with Newton Polynomial: Theory, Methods, and Applications. Academic Press.
- Atangana, A., & Araz, S. (2020). New numerical method for ordinary differential equations: Newton polynomial. Journal of Computational and Applied Mathematics, 372, 112622. doi:https://doi.org/10.1016/j.cam.2019.112622
- Boukhelkhal, I., & Zeghdane, R. (2024). Lagrange interpolation polynomials for solving nonlinear stochastic integral equations. Numerical Algorithms, 96(2), 583-618. doi:10.1007/s11075-023-01659-x
- Chouhan, V., & Ray, S. (2021). Approximation using Lagrange and Hermite Form of Polynomial Interpolation: An Experimental Study. 2021 International Conference on Advances in Electrical, Computing, Communication and Sustainable Technologies (ICAECT) (S. 1-6). Bhilai, India: IEEE. doi:10.1109/ICAECT49130.2021.9392472
- Cuevas et al. (2024). Interpolation and Polynomials. In Computational Methods with MATLAB® (S. 77-101). Springer Nature Switzerland. doi:10.1007/978-3-031-40478-8_4
- de Camargo, A. (2020). On the numerical stability of Newton’s formula for Lagrange interpolation. Journal of Computational and Applied Mathematics, 365, 112369. doi:https://doi.org/10.1016/j.cam.2019.112369
- Denis, B. (Dec 2020). An Overview of Numerical and Analytical Methods for solving Ordinary Differential Equations. doi:10.13140/RG.2.2.11758.64329
- DinIde, N. (2020). Numerical study for Solving Bernoulli Differential Equations by using Runge-Kutta Method and “Newton's Interpolation and Aitken's Method". International Journal of Scientific and Innovative Mathematical Research(IJSIMR), 8(5), 9-13. doi:https://doi.org/10.20431/2347-3142.0805002
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- Essanhaji, A., & Errachid, M. (2022). Lagrange Multivariate Polynomial Interpolation: A Random Algorithmic Approach. Journal of Applied Mathematics, 2022(1), 8227086. doi:10.1155/2022/8227086
- Ide, N. A. (2020). Using Newton's Interpolation and Aitken's Method for Solving First Order Differential Equation. World Applied Sciences Journal, 38(3), 191-194.
- Junjua et al. (2020). CONSTRUCTION OF OPTIMAL DERIVATIVE FREE ITERATIVE METHODS FOR NONLINEAR EQUATIONS USING LAGRANGE INTERPOLATION. Journal of Prime Research in Mathematics, 16(1), 30-45.
- Karpfinger, C. (2022). Polynomial and Spline Interpolation. In Calculus and Linear Algebra in Recipes: Terms, phrases and numerous examples in short learning units (S. 311-320). Berlin: Springer Berlin Heidelberg. doi:10.1007/978-3-662-65458-3_29
- Khiar et al. (2024). On the accurate computation of the Newton form of the Lagrange interpolant. Numerical Algorithms. doi:https://doi.org/10.1007/s11075-024-01843-7
- lde, N. (2022). Modification on Euler-Cauchy Method for Solving First-order Differential Equations. Asian Journal of Pure and Applied Mathematics, 4(1), 56-62. Von https://jofmath.com/index.php/AJPAM/article/view/97 abgerufen
- lDE, N. A. (2020). Numerical Study for Solving Quadratic Riccati Differential Equations. Middle-East Journal of Scientific Research, 28(4), 348-356. doi:10.5829/idosi.mejsr.2020.348.356
- Lychagin, V. (2024). Semi-analytical methods for solving non-linear differential equations: A review. Journal of Mathematical Analysis and Applications, 531(1). doi:https://doi.org/10.1016/j.jmaa.2023.127821
- Mbagwu, J., & Ide, N. (December 2021). Comparison of Newton's Interpolation and Aitken's Methods with Some Numerical Methods for Solving System of First and Second Order Differential Equation. International Journal of Scientific World, 164, 108-121.
- Neamvonk, A. (2023). Solving the First Order Differential Equations using Newton's Interpolation and Lagrange Polynomial. European Journal of Pure and Applied Mathematics, 16(2), 965-974. doi:https://doi.org/10.29020/nybg.ejpam.v16i2.4727
- Salisu, I. (2023). Application of Lagrange Interpolation Method to Solve First-Order Differential Equation Using Newton Interpolation Approach. EURASIAN JOURNAL OF SCIENCE AND ENGINEERING, 9(1), 89-98.
- Senu et al. (2022). Numerical solution of delay differential equation using two-derivative Runge-Kutta type method with Newton interpolation. Alexandria Engineering Journal, 61(8), 5819-5835. doi:https://doi.org/10.1016/j.aej.2021.11.009
- Sinthea et al. (Jan 2010). Numerical solution of logistic differential equations by using the Laplace decomposition method. World Applied Sciences Journal, 8, 1100-1105.
- Taylor, M. (2021). Introduction to Differential Equations: Second Edition (Second Ausg.). American Mathematical Society.
- Tunç et al. (2022). Solving second order ordinary differential equations by using Newton’s interpolation and Aitken’s methods. International Journal of Nonlinear Analysis and Applications, 13(1), 1057–1066. doi:http://dx.doi.org/10.22075/ijnaa.2021.24027.2657
- Udoh et al. (August 2024). On Newton and Lagrange Interpolation Method for first order ordinary differential equations. World Journal of Applied Science & Technology, 15(2). doi:10.4314/wojast.v15i2.34
- UWAEZUOKE, M. (2024). Numerical Methods for Solving First Order Ordinary Differential Equations. The International Journal of Engineering and Science (IJES), 13(14), 58-63. doi:DOI:10.9790/1813-13045863
- William et al. (2017). Elementary Differential Equations. John Wiley & Sons.
References
Aldin, N. (2020). Comparison of Newton's Interpolation and Aitken's Method with Runge-Kutta Method for Solving First Order Differential Equation. Middle-East Journal of Scientific Research, 28(5), 391-394. doi:10.5829/idosi.mejsr.2020.391.394
Araz, S., & Abdon, A. (2021). New Numerical Scheme with Newton Polynomial: Theory, Methods, and Applications. Academic Press.
Atangana, A., & Araz, S. (2020). New numerical method for ordinary differential equations: Newton polynomial. Journal of Computational and Applied Mathematics, 372, 112622. doi:https://doi.org/10.1016/j.cam.2019.112622
Boukhelkhal, I., & Zeghdane, R. (2024). Lagrange interpolation polynomials for solving nonlinear stochastic integral equations. Numerical Algorithms, 96(2), 583-618. doi:10.1007/s11075-023-01659-x
Chouhan, V., & Ray, S. (2021). Approximation using Lagrange and Hermite Form of Polynomial Interpolation: An Experimental Study. 2021 International Conference on Advances in Electrical, Computing, Communication and Sustainable Technologies (ICAECT) (S. 1-6). Bhilai, India: IEEE. doi:10.1109/ICAECT49130.2021.9392472
Cuevas et al. (2024). Interpolation and Polynomials. In Computational Methods with MATLAB® (S. 77-101). Springer Nature Switzerland. doi:10.1007/978-3-031-40478-8_4
de Camargo, A. (2020). On the numerical stability of Newton’s formula for Lagrange interpolation. Journal of Computational and Applied Mathematics, 365, 112369. doi:https://doi.org/10.1016/j.cam.2019.112369
Denis, B. (Dec 2020). An Overview of Numerical and Analytical Methods for solving Ordinary Differential Equations. doi:10.13140/RG.2.2.11758.64329
DinIde, N. (2020). Numerical study for Solving Bernoulli Differential Equations by using Runge-Kutta Method and “Newton's Interpolation and Aitken's Method". International Journal of Scientific and Innovative Mathematical Research(IJSIMR), 8(5), 9-13. doi:https://doi.org/10.20431/2347-3142.0805002
Dormand, J. (2017). Numerical Methods for Differential Equations: A Computational Approach (1st Ausg.). Boca Raton. doi:https://doi.org/10.1201/9781351075107
Essanhaji, A., & Errachid, M. (2022). Lagrange Multivariate Polynomial Interpolation: A Random Algorithmic Approach. Journal of Applied Mathematics, 2022(1), 8227086. doi:10.1155/2022/8227086
Ide, N. A. (2020). Using Newton's Interpolation and Aitken's Method for Solving First Order Differential Equation. World Applied Sciences Journal, 38(3), 191-194.
Junjua et al. (2020). CONSTRUCTION OF OPTIMAL DERIVATIVE FREE ITERATIVE METHODS FOR NONLINEAR EQUATIONS USING LAGRANGE INTERPOLATION. Journal of Prime Research in Mathematics, 16(1), 30-45.
Karpfinger, C. (2022). Polynomial and Spline Interpolation. In Calculus and Linear Algebra in Recipes: Terms, phrases and numerous examples in short learning units (S. 311-320). Berlin: Springer Berlin Heidelberg. doi:10.1007/978-3-662-65458-3_29
Khiar et al. (2024). On the accurate computation of the Newton form of the Lagrange interpolant. Numerical Algorithms. doi:https://doi.org/10.1007/s11075-024-01843-7
lde, N. (2022). Modification on Euler-Cauchy Method for Solving First-order Differential Equations. Asian Journal of Pure and Applied Mathematics, 4(1), 56-62. Von https://jofmath.com/index.php/AJPAM/article/view/97 abgerufen
lDE, N. A. (2020). Numerical Study for Solving Quadratic Riccati Differential Equations. Middle-East Journal of Scientific Research, 28(4), 348-356. doi:10.5829/idosi.mejsr.2020.348.356
Lychagin, V. (2024). Semi-analytical methods for solving non-linear differential equations: A review. Journal of Mathematical Analysis and Applications, 531(1). doi:https://doi.org/10.1016/j.jmaa.2023.127821
Mbagwu, J., & Ide, N. (December 2021). Comparison of Newton's Interpolation and Aitken's Methods with Some Numerical Methods for Solving System of First and Second Order Differential Equation. International Journal of Scientific World, 164, 108-121.
Neamvonk, A. (2023). Solving the First Order Differential Equations using Newton's Interpolation and Lagrange Polynomial. European Journal of Pure and Applied Mathematics, 16(2), 965-974. doi:https://doi.org/10.29020/nybg.ejpam.v16i2.4727
Salisu, I. (2023). Application of Lagrange Interpolation Method to Solve First-Order Differential Equation Using Newton Interpolation Approach. EURASIAN JOURNAL OF SCIENCE AND ENGINEERING, 9(1), 89-98.
Senu et al. (2022). Numerical solution of delay differential equation using two-derivative Runge-Kutta type method with Newton interpolation. Alexandria Engineering Journal, 61(8), 5819-5835. doi:https://doi.org/10.1016/j.aej.2021.11.009
Sinthea et al. (Jan 2010). Numerical solution of logistic differential equations by using the Laplace decomposition method. World Applied Sciences Journal, 8, 1100-1105.
Taylor, M. (2021). Introduction to Differential Equations: Second Edition (Second Ausg.). American Mathematical Society.
Tunç et al. (2022). Solving second order ordinary differential equations by using Newton’s interpolation and Aitken’s methods. International Journal of Nonlinear Analysis and Applications, 13(1), 1057–1066. doi:http://dx.doi.org/10.22075/ijnaa.2021.24027.2657
Udoh et al. (August 2024). On Newton and Lagrange Interpolation Method for first order ordinary differential equations. World Journal of Applied Science & Technology, 15(2). doi:10.4314/wojast.v15i2.34
UWAEZUOKE, M. (2024). Numerical Methods for Solving First Order Ordinary Differential Equations. The International Journal of Engineering and Science (IJES), 13(14), 58-63. doi:DOI:10.9790/1813-13045863
William et al. (2017). Elementary Differential Equations. John Wiley & Sons.