Decomposition–Linearization–Sequential Homotopy Methods for Nonlinear Differential/Integral Equations

Chein-Shan Liu, Chung-Lun Kuo and Chih-Wen Chang ()
Additional contact information
Chein-Shan Liu: Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 202301, Taiwan
Chung-Lun Kuo: Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 202301, Taiwan
Chih-Wen Chang: Department of Mechanical Engineering, National United University, Miaoli 36063, Taiwan

Mathematics, 2024, vol. 12, issue 22, 1-29

Abstract: In the paper, two new analytic methods using the decomposition and linearization technique on nonlinear differential/integral equations are developed, namely, the decomposition–linearization–sequential method (DLSM) and the linearized homotopy perturbation method (LHPM). The DLSM is realized by an integrating factor and the integral of certain function obtained at the previous step for obtaining a sequential analytic solution of nonlinear differential equation, which provides quite accurate analytic solution. Some first- and second-order nonlinear differential equations display the fast convergence and accuracy of the DLSM. An analytic approximation for the Volterra differential–integral equation model of the population growth of a species is obtained by using the LHPM. In addition, the LHPM is also applied to the first-, second-, and third-order nonlinear ordinary differential equations. To reduce the cost of computation of He’s homotopy perturbation method and enhance the accuracy for solving cubically nonlinear jerk equations, the LHPM is implemented by invoking a linearization technique in advance is developed. A generalization of the LHPM to the n th-order nonlinear differential equation is involved, which can greatly simplify the work to find an analytic solution by solving a set of second-order linear differential equations. A remarkable feature of those new analytic methods is that just a few steps and lower-order approximations are sufficient for producing reasonably accurate analytic solutions. For all examples, the second-order analytic solution x 2 ( t ) is found to be a good approximation of the real solution. The accuracy of the obtained approximate solutions are identified by the fourth-order Runge–Kutta method. The major objection is to unify the analytic solution methods of different nonlinear differential equations by simply solving a set of first-order or second-order linear differential equations. It is clear that the new technique considerably saves computational costs and converges faster than other analytical solution techniques existing in the literature, including the Picard iteration method. Moreover, the accuracy of the obtained analytic solution is raised.

Keywords: nonlinear Volterra differential–integral equation; analytic solution; linearized homotopy perturbation method; He’s homotopy perturbation method; nonlinear jerk equations (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/12/22/3557/pdf (application/pdf)
https://www.mdpi.com/2227-7390/12/22/3557/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:12:y:2024:i:22:p:3557-:d:1520993

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().