A numerical scheme to solve unstable boundary value problems

A new iterative scheme for solving boundary value problems is presented. It consists of the introduction of an artificial time dependence into a modified version of the system of equations. Then explicit forward integrations in time are followed by explicit integrations backwards in time. The method converges under much more general conditions than schemes based in forward time integrations (false transient schemes). In particular it can attain a steady state solution of an elliptical system of equations even if the solution is unstable, in which case other iterative schemes fail to converge. The simplicity of its use makes it attractive for solving large systems of nonlinear equations

Solving nonlinear problems by Ostrowski Chun type parametric families

In this paper, by using a generalization of Ostrowski' and Chun's methods two bi-parametric families of predictor-corrector iterative schemes, with order of convergence four for solving system of nonlinear equations, are presented. The predictor of the first family is Newton's method, and the predictor of the second one is Steffensen's scheme. One of them is extended to the multidimensional case. Some numerical tests are performed to compare proposed methods with existing ones and to confirm the theoretical results. We check the obtained results by solving the molecular interaction problem.This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02 and FONDOCYT, Republica Dominicana.Cordero Barbero, A.; Maimo, J.; Torregrosa Sánchez, JR.; Vassileva, M. (2015). Solving nonlinear problems by Ostrowski Chun type parametric families. Journal of Mathematical Chemistry. 53(1):430-449. https://doi.org/10.1007/s10910-014-0432-zS430449531M.S. Petkovic̀, B. Neta, L.D. 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An approximation algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method

In this paper we propose a collocation method for solving some well-known classes of Lane-Emden type equations which are nonlinear ordinary differential equations on the semi-infinite domain. They are categorized as singular initial value problems. The proposed approach is based on a Hermite function collocation (HFC) method. To illustrate the reliability of the method, some special cases of the equations are solved as test examples. The new method reduces the solution of a problem to the solution of a system of algebraic equations. Hermite functions have prefect properties that make them useful to achieve this goal. We compare the present work with some well-known results and show that the new method is efficient and applicable.Comment: 34 pages, 13 figures, Published in "Computer Physics Communications

Some Convergence Properties of Broyden's Method

In 1965 Broyden introduced a family of algorithms called(rank-one) quasi—New-ton methods for iteratively solving systems of nonlinear equations. We show that when any member of this family is applied to an n x n nonsingular system of linear equations and direct-prediction steps are taken every second iteration, then the solution is found in at most 2n steps. Specializing to the particular family member known as Broyden’s (good) method, we use this result to show that Broyden's method enjoys local 2n-step Q-quadratic convergence on nonlinear problems.

Solution of the Least Squares Method problem of pairwise comparison matrices

The aim of the paper is to present a new global optimization method for determining all the optima of the Least Squares Method (LSM) problem of pairwise comparison matrices. Such matrices are used, e.g., in the Analytic Hierarchy Process (AHP). Unlike some other distance minimizing methods, LSM is usually hard to solve because of the corresponding nonlinear and non-convex objective function. It is found that the optimization problem can be reduced to solve a system of polynomial equations. Homotopy method is applied which is an efficient technique for solving nonlinear systems. The paper ends by two numerical example having multiple global and local minima

A New Approach for Solving System of Local Fractional Partial Differential Equations

In this paper, we apply a new method for solving system of partial differential equations within local fractional derivative operators. The approximate analytical solutions are obtained by using the local fractional Laplace variational iteration method, which is the coupling method of local fractional variational iteration method and Laplace transform. Illustrative examples are included to demonstrate the high accuracy and fast convergence of this new algorithm. The obtained results show that the introduced approach is a promising tool for solving system of linear and nonlinear local fractional differential equations. Furthermore, we show that local fractional Laplace variational iteration method is able to solve a large class of nonlinear problems involving local fractional operators effectively, more easily and accurately; and thus it has been widely applicable in physics and engineering

Efficiency of 2-order iterative methods

In this article the problem of solving a system of singular nonlinear equations will be discussed. New the iterative 2-order method for this problem is presented. The article includes the numerical results for the method

A family of parametric schemes of arbitrary even order for solving nonlinear models

[EN] Many problems related to gas dynamics, heat transfer or chemical reactions are modeled by means of partial differential equations that usually are solved by using approximation techniques. When they are transformed in nonlinear systems of equations via a discretization process, this system is big-sized and high-order iterative methods are specially useful. In this paper, we construct a new family of parametric iterative methods with arbitrary even order, based on the extension of Ostrowski' and Chun's methods for solving nonlinear systems. Some elements of the proposed class are known methods meanwhile others are new schemes with good properties. Some numerical tests confirm the theoretical results and allow us to compare the numerical results obtained by applying new methods and known ones on academical examples. In addition, we apply one of our methods for approximating the solution of a heat conduction problem described by a parabolic partial differential equation.This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C02-2-P and FONDOCYT 2014-1C1-088 Republica Dominicana.Cordero Barbero, A.; Torregrosa Sánchez, JR.; Vassileva, MP. (2017). A family of parametric schemes of arbitrary even order for solving nonlinear models. Journal of Mathematical Chemistry. 55(7):1443-1460. https://doi.org/10.1007/s10910-016-0723-7S14431460557R. Escobedo, L.L. Bonilla, Numerical methods for quantum drift-diffusion equation in semiconductor phisics. Math. Chem. 40(1), 3–13 (2006)S.J. Preece, J. Villingham, A.C. King, Chemical clock reactions: the effect of precursor consumtion. Math. Chem. 26, 47–73 (1999)H. Montazeri, F. Soleymani, S. Shateyi, S.S. Motsa, On a new method for computing the numerical solution of systems of nonlinear equations. J. Appl. Math. 2012 ID. 751975, 15 pages (2012)J.L. Hueso, E. Martínez, C. Teruel, Convergence, effiency and dinamimics of new fourth and sixth order families of iterative methods for nonlinear systems. J. Comput. Appl. Math. 275, 412–420 (2015)J.R. Sharma, H. Arora, Efficient Jarratt-like methods for solving systems of nonlinear equations. Calcolo 51, 193–210 (2014)X. Wang, T. Zhang, W. Qian, M. Teng, Seventh-order derivative-free iterative method for solving nonlinear systems. Numer. Algor. 70, 545–558 (2015)J.R. Sharma, H. Arora, On efficient weighted-Newton methods for solving systems of nonlinear equations. Appl. Math. Comput. 222, 497–506 (2013)A. Cordero, J.G. Maimó, J.R. Torregrosa, M.P. Vassileva, Solving nonlinear problems by Ostrowski-Chun type parametric families. J. Math. Chem. 53, 430–449 (2015)A.M. Ostrowski, Solution of equations and systems of equations (Prentice-Hall, Englewood Cliffs, New York, 1964)C. Chun, Construction of Newton-like iterative methods for solving nonlinear equations. Numer. Math. 104, 297–315 (2006)A. Cordero, J.L. Hueso, E. Martínez, J.R. Torregrosa, A modified Newton-Jarratt’s composition. Numer. Algor. 55, 87–99 (2010)J.M. Ortega, W.C. Rheinboldt, Iterative solution of nonlinear equations in several variables (Academic, New York, 1970)C. Hermite, Sur la formule dinterpolation de Lagrange. Reine Angew. Math. 84, 70–79 (1878)A. Cordero, J.R. Torregrosa, Variants of Newton’s method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007

Multidimensional Homeier's generalized class and its application to planar 1D Bratu problem

[EN] In this paper, a parametric family of iterative methods for solving nonlinear systems, including Homeier’s scheme is presented, proving its third-order of convergence. The numerical section is devoted to obtain an estimation of the solution of the classical Bratu problem by transforming it in a nonlinear system by using finite differences, and solving it with different elements of the iterative family.This research was supported by Ministerio de Economía y Competitividad MTM2014-52016-C02-02.Cordero Barbero, A.; Franqués García, AM.; Torregrosa Sánchez, JR. (2015). Multidimensional Homeier's generalized class and its application to planar 1D Bratu problem. Journal of the Spanish Society of Applied Mathematics. 70(1):1-10. https://doi.org/10.1007/s40324-015-0037-xS110701Abad, M. F., Cordero, A., Torregrosa, J. R.: Fourth-and fifth-order for solving nonlinear systems of equations: an application to the global positioning system, Abstr. Appl. Anal. (2013) (Article ID 586708)Andreu, C., Cambil, N., Cordero, A., Torregrosa, J.R.: Preliminary orbit determination of artificial satellites: a vectorial sixth-order approach, Abstr. Appl. Anal. (2013) (Article ID 960582)Awawdeh, F.: On new iterative method for solving systems of nonlinear equations. Numer. Algorithms 54, 395–409 (2010)Boyd, J.P.: One-point pseudospectral collocation for the one-dimensional Bratu equation. Appl. Math. Comput. 217, 5553–5565 (2011)Bratu, G.: Sur les equation integrals non-lineaires. Bull. Math. Soc. France 42, 113–142 (1914)Buckmire, R.: Applications of Mickens finite differences to several related boundary value problems. In: Mickens, R.E. (ed.) Advances in the Applications of Nonstandard Finite Difference Schemes, pp. 47–87. World Scientific Publishing, Singapore (2005)Cordero, A., Hueso, J.L., Martínez, E., Torregrosa, J.R.: A modified Newton-Jarratt’s composition. Numer. Algorithms 55, 87–99 (2010)Gelfand, I.M.: Some problems in the theory of quasi-linear equations. Trans. Am. Math. Soc. Ser. 2, 295–381 (1963)Homeier, H.H.H.: On Newton-tyoe methods with cubic convergence. J. Comput. Appl. Math. 176, 425–432 (2005)Jacobsen, J., Schmitt, K.: The Liouville-Bratu-Gelfand problem for radial operators. J. Differ. Equ. 184, 283–298 (2002)Jalilian, R.: Non-polynomial spline method for solving Bratu’s problem. Comput. Phys. Comm. 181, 1868–1872 (2010)Kanwar, V., Kumar, S., Behl, R.: Several new families of Jarratts method for solving systems of nonlinear equations. Appl. Appl. Math. 8(2), 701–716 (2013)Mohsen, A.: A simple solution of the Bratu problem. Comput. Math. with Appl. 67, 26–33 (2014)Petković, M., Neta, B., Petković, L., Džunić, J.: Multipoint Methods for Solving Nonlinear Equations. Academic Press, Amsterdam (2013)Sharma, J.R., Guna, R.K., Sharma, R.: An efficient fourth order weighted-Newton method for systems of nonlinear equations. Numer. Algorithms 62, 307–323 (2013)Sharma, J.R., Arora, H.: On efficient weighted-Newton methods for solving systems of nonlinear equations. Appl. Math. Comput. 222, 497–506 (2013)Traub, J.F.: Iterative Methods for the Solution of Equations. Chelsea Publishing Company, New York (1982)Wan, Y.Q., Guo, Q., Pan, N.: Thermo-electro-hydrodynamic model for electrospinning process. Int. J. Nonlinear Sci. Numer. Simul. 5, 5–8 (2004