On the Application of Ill-Posed Problems of Equations of Mathematical Physics
DOI:
https://doi.org/10.22105/kmisj.v1i2.53Keywords:
The Cauchy problem, regularization, factorization, regular solution, fundamental solutionAbstract
Regarding the implementation of ill-posed problems in the context of mathemat-ical physics equations, it is essential to acknowledge their significance. Ill-posed problems arecharacterized by their sensitivity to changes in initial or boundary conditions, leading to non-unique or unstable solutions. These challenges frequently arise in areas such as fluid dynamics,heat transfer, and wave propagation, where classical methods may struggle to provide reliableanswers. Consequently, researchers seek innovative numerical techniques and regularizationmethods to stabilize these problems and obtain meaningful solutions. Among the approachesutilized are Tikhonov regularization, particle filtering, and machine learning methods, all aimingto mitigate the effects of ill-posedness. Addressing these issues is crucial, as they have pro-found implications for both theoretical understanding and practical applications in engineeringand physics. It is clear that further advancements in this area will enhance the effectiveness ofpredictive models and simulation tools, ultimately contributing to the broader field of mathemat-ical physics. Through ongoing investigation, the mathematical community continues to refinestrategies that accommodate the complexities posed by ill-posed problems, ensuring progress inthis vital discipline.
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