GENERALIZED QUASILINEARIZATION FOR DEGENERATE SINGULAR PERTURBATION PROBLEMS
AHMED RAGAEY ABD-ELLATEEF OMAR AHMED KAMAR;
Abstract
The method of upper and lower solutions, coupled with the monotone iterative technique, manifests itself as an effective and flexible mechanism that offers theoretical, as well as constructive, existence results in a closed set, namely, the sector. The upper and lower bounds for solutions that generate the sector serve as upper and lower bounds for solutions which can be improved by monotone iterative procedure. Moreover, iteration schemes are also useful for the investigation of qualitative properties of solutions.
If we utilize the technique of lower and upper solutions coupled with the method of quasilinearization, it is possible to construct concurrently lower and upper bounding monotone sequences whose elements are the solutions of the corresponding linear problems. Both sequences converge rapidly. Moreover, this unification provides a mechanism to enlarge the class of nonlinear problems considerably to which the method is applicable. For example, it is not necessary to impose convexity assumption on the given function, since one can allow much weaker assumption. Also, the problem when the given function admits a decomposition of two functions, one of them is convex and the other is concave can be handled.
In fact, there are several possibilities that can be investigated with this unified methodology. We therefore call this, technique, the method of generalized quasilinearization. Furthermore, these ideas can be extended, generalized and refined to other types of nonlinear problems such as second order boundary value problems, hyperbolic partial differential equations of first order and reaction diffusion equations with fixed moments of impulses.
Throughout our work, we have employed two important techniques for the degenerate singular perturbation problems, namely, the monotone iterative technique and the generalized quasilinearization method.
Moreover, we have employed the generalized quasilinearization method for singular system of differential equations.
If we utilize the technique of lower and upper solutions coupled with the method of quasilinearization, it is possible to construct concurrently lower and upper bounding monotone sequences whose elements are the solutions of the corresponding linear problems. Both sequences converge rapidly. Moreover, this unification provides a mechanism to enlarge the class of nonlinear problems considerably to which the method is applicable. For example, it is not necessary to impose convexity assumption on the given function, since one can allow much weaker assumption. Also, the problem when the given function admits a decomposition of two functions, one of them is convex and the other is concave can be handled.
In fact, there are several possibilities that can be investigated with this unified methodology. We therefore call this, technique, the method of generalized quasilinearization. Furthermore, these ideas can be extended, generalized and refined to other types of nonlinear problems such as second order boundary value problems, hyperbolic partial differential equations of first order and reaction diffusion equations with fixed moments of impulses.
Throughout our work, we have employed two important techniques for the degenerate singular perturbation problems, namely, the monotone iterative technique and the generalized quasilinearization method.
Moreover, we have employed the generalized quasilinearization method for singular system of differential equations.
Other data
| Title | GENERALIZED QUASILINEARIZATION FOR DEGENERATE SINGULAR PERTURBATION PROBLEMS | Other Titles | الطريقة الشبه خطية العامة لمعادلات الاضطراب الفردية المنحلة | Authors | AHMED RAGAEY ABD-ELLATEEF OMAR AHMED KAMAR | Issue Date | 1999 |
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