94,325 research outputs found
A Parameterized multi-step Newton method for solving systems of nonlinear equations
We construct a novel multi-step iterative method for solving systems of nonlinear equations by introducing a parameter. to generalize the multi-step Newton method while keeping its order of convergence and computational cost. By an appropriate selection of theta, the new method can both have faster convergence and have larger radius of convergence. The new iterative method only requires one Jacobian inversion per iteration, and therefore, can be efficiently implemented using Krylov subspace methods. The new method can be used to solve nonlinear systems of partial differential equations, such as complex generalized Zakharov systems of partial differential equations, by transforming them into systems of nonlinear equations by discretizing approaches in both spatial and temporal independent variables such as, for instance, the Chebyshev pseudo-spectral discretizing method. Quite extensive tests show that the new method can have significantly faster convergence and significantly larger radius of convergence than the multi-step Newton method.Peer ReviewedPostprint (author's final draft
A Novel Third Order Numerical Method for Solving Volterra Integro-Differential Equations
In this paper we introduce a numerical method for solving nonlinear Volterra
integro-differential equations. In the first step, we apply implicit trapezium
rule to discretize the integral in given equation. Further, the Daftardar-Gejji
and Jafari technique (DJM) is used to find the unknown term on the right side.
We derive existence-uniqueness theorem for such equations by using Lipschitz
condition. We further present the error, convergence, stability and bifurcation
analysis of the proposed method. We solve various types of equations using this
method and compare the error with other numerical methods. It is observed that
our method is more efficient than other numerical methods
A dynamic convergence control scheme for the solution of the radial equilibrium equation in through-flow analyses
One of the most frequently encountered numerical problems in scientific analyses
is the solution of non-linear equations. Often the analysis of complex phenomena
falls beyond the range of applicability of the numerical methods available in
the public domain, and demands the design of dedicated algorithms that will
approximate, to a specified precision, the mathematical solution of specific
problems. These algorithms can be developed from scratch or through the
amalgamation of existing techniques. The accurate solution of the full radial
equilibrium equation (REE) in streamline curvature (SLC) through-flow analyses
presents such a case. This article discusses the development, validation, and
application of an 'intelligent' dynamic convergence control (DCC) algorithm for
the fast, accurate, and robust numerical solution of the non-linear equations of
motion for two-dimensional flow fields. The algorithm was developed to eliminate
the large extent of user intervention, usually required by standard numerical
methods. The DCC algorithm was integrated into a turbomachinery design and
performance simulation software tool and was tested rigorously, particularly at
compressor operating regimes traditionally exhibiting convergence difficulties
(i.e. far off-design conditions). Typical error histories and comparisons of
simulated results against experimental are presented in this article for a
particular case study. For all case studies examined, it was found that the
algorithm could successfully 'guide' the solution down to the specified error
tolerance, at the expense of a slightly slower iteration process (compared to a
conventional Newton-Raphson scheme). This hybrid DCC algorithm can also find use
in many other engineering and scientific applications that require the robust
solution of mathematical problems by numerical instead of analytical means
Transformation Method for Solving Hamilton-Jacobi-Bellman Equation for Constrained Dynamic Stochastic Optimal Allocation Problem
In this paper we propose and analyze a method based on the Riccati
transformation for solving the evolutionary Hamilton-Jacobi-Bellman equation
arising from the stochastic dynamic optimal allocation problem. We show how the
fully nonlinear Hamilton-Jacobi-Bellman equation can be transformed into a
quasi-linear parabolic equation whose diffusion function is obtained as the
value function of certain parametric convex optimization problem. Although the
diffusion function need not be sufficiently smooth, we are able to prove
existence, uniqueness and derive useful bounds of classical H\"older smooth
solutions. We furthermore construct a fully implicit iterative numerical scheme
based on finite volume approximation of the governing equation. A numerical
solution is compared to a semi-explicit traveling wave solution by means of the
convergence ratio of the method. We compute optimal strategies for a portfolio
investment problem motivated by the German DAX 30 Index as an example of
application of the method
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