Abstract
The method of generalized quasilinearization for second-order boundary value problems has been extended when the forcing function is the sum of-hyperconvex and-hyperconcave functions. We develop two sequences under suitable conditions which converge to the unique solution of the boundary value problem. Furthermore, the convergence is of order. Finally, we provide numerical examples to show the application of the generalized quasilinearization method developed here for second-order boundary value problems.
Highlights
The method of quasilinearization [1, 2] combined with the technique of upper and lower solutions is an effective and fruitful technique for solving a wide variety of nonlinear problems
In [12], we have obtained the results of higher order of convergence for first order initial value problems when the forcing function is the sum of hyperconvex and hyperconcave functions with natural and coupled lower and upper solutions
In this paper we extend the result to the second-order boundary value problems when the forcing function is a sum of 2-hyperconvex and 2-hyperconcave functions
Summary
The method of quasilinearization [1, 2] combined with the technique of upper and lower solutions is an effective and fruitful technique for solving a wide variety of nonlinear problems. In [4, 13], the authors have obtained a higher order of convergence (an order more than 2) for initial value problems They have considered situations when the forcing function is either hyperconvex or hyperconcave. In [12], we have obtained the results of higher order of convergence for first order initial value problems when the forcing function is the sum of hyperconvex and hyperconcave functions with natural and coupled lower and upper solutions. In this paper we extend the result to the second-order boundary value problems when the forcing function is a sum of 2-hyperconvex and 2-hyperconcave functions. In view of natural upper and lower solutions of (1.1), we will develop results when f is 2-hyperconvex and g is 2-hyperconcave. We show that these iterates converge uniformly and monotonically to the unique solution of (1.1), and the convergence is of order 3
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