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On the iterative methods of linearization, decrease of order and dimension of the Karman-type PDEs.

A V Krysko¹, J Awrejcewicz², S P Pavlov³

¹Engels Institute of Technology (Branch), Saratov State Technical University, Department of Higher Mathematics and Mechanics, Russian Federation, Ploschad Svobodi 17, Engels, Saratov 413100, Russia.

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Summary

This summary is machine-generated.

This study introduces iterative methods to simplify complex eighth-order nonlinear partial differential equations into biharmonic and Poisson-type equations. The developed computer programs validate the reliability of these novel linearization techniques.

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Area of Science:

Numerical Analysis
Computational Mathematics
Applied Mathematics

Background:

Nonlinear partial differential equations (PDEs) of high order pose significant challenges in mathematical modeling and simulation.
Existing methods for solving these equations often struggle with complexity and computational cost.
The biharmonic and Poisson-type equations are fundamental in various scientific and engineering fields.

Purpose of the Study:

To propose novel iterative methods for the linearization of eighth-order nonlinear PDEs.
To reduce the order and dimension of these complex equations.
To transform them into more tractable biharmonic and Poisson-type differential equations.

Main Methods:

Development of iterative algorithms for equation linearization.
Order and dimension reduction techniques for nonlinear PDEs.
Implementation of custom computer programs for validation.

Main Results:

Successful linearization of eighth-order nonlinear PDEs into biharmonic and Poisson-type equations.
Demonstration of reduced complexity and dimensionality.
Validation of the proposed methods' effectiveness and reliability through computational analysis.

Conclusions:

The proposed iterative methods offer an efficient approach to handle high-order nonlinear PDEs.
The transformation into biharmonic and Poisson-type equations simplifies analysis and computation.
The developed computational tools confirm the validity and reliability of the presented techniques.