Kaminogo : Boundary value problems for ordinary differential equationsThe method of upper and lower solutions for ordinary differential equation was introduced in by G. Scorza Dragoni for a Dirichlet problem. Since then a large number of contributions enriched the theory. Among others, one has to point out the pioneer work of M. Nagumo who associated his name with derivative dependent right hand side. Unable to display preview. Download preview PDF.
Boundary value problem
Journal of Optimization Theory and Applications. A new quasilinearization algorithm is presented which essentially eliminates the necessity for computer storage. The representation theorem for the standard quasilinearization procedure is reformulated in terms of the initial value of the solution to a final-value problem, leading to a modification of the successive approximations. Several theorems establishing the convergence properties are proved; as in the original procedure, these convergence properties are both quadratic and monotonic. Finally, the modified approximation scheme is illustrated through several examples.
Content type: Research Article. This paper investigates the existence and uniqueness of smooth positive solutions to a class of singular m -point boundary value problems of second-order ordinary differential equations.
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Journal of Optimization Theory and Applications. The methods commonly employed for solving linear, two-point boundary-value problems require the use of two sets of differential equations: the original set and the derived set. This derived set is the adjoint set if the method of adjoint equations is used, the Green's functions set if the method of Green's functions is used, and the homogeneous set if the method of complementary functions is used. With particular regard to high-speed digital computing operations, this paper explores an alternate method, the method of particular solutions, in which only the original, nonhomogeneous set is used. A general theory is presented for a linear differential system of n th order. The boundary-value problem is solved by combining linearly several particular solutions of the original, nonhomogeneous set. Both the case of an uncontrolled system and the case of a controlled system are considered.