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- W2200672046 abstract "For solving a non-linear system of algebraic equations of the type: F(subscript i)(x(subscript j)) = 0, i, j = 1, …, n, a Newton-like algorithm is still the most popular one; however, it had some drawbacks as being locally convergent, sensitive to initial guess, and time consumption in finding the inversion of the Jacobian matrix ∂F(subscript i) /∂x(subscript j). Based-on a manifold defined in the space of (x(subscript i), t) we can derive a system of non-linear Ordinary Differential Equations (ODEs) in terms of the fictitious time-like variable t, and the residual error is exponentially decreased to zero along the path of x(t) by solving the resultant ODEs. We apply it to solve 2D non-linear PDEs, and the vector-form of the matrix-type non-linear algebraic equations (NAEs) is derived. Several numerical examples of non-linear PDEs show the efficiency and accuracy of the present algorithm. A scalar equation is derived to find the adjustive fictitious time stepsize, such that the irregular bursts appeared in the residual error curve can be overcome. We propose a future direction to construct a really exponentially convergent algorithm according to a manifold setting." @default.
- W2200672046 created "2016-06-24" @default.
- W2200672046 creator A5090276712 @default.
- W2200672046 date "2012-08-01" @default.
- W2200672046 modified "2023-09-27" @default.
- W2200672046 title "A MANIFOLD-BASED EXPONENTIALLY CONVERGENT ALGORITHM FOR SOLVING NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS" @default.
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- W2200672046 doi "https://doi.org/10.6119/jmst-011-0421-1" @default.
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