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Collocation method

A Simple Local Variational Iteration Method and Related Algorithm for Nonlinear Science and Engineering

A very simple and efficient local variational iteration method for solving problems of nonlinear science is proposed in this paper. The analytical iteration formula of this method is derived first using a general form of first order nonlinear differential equations, followed by straightforward discretization using Chebyshev polynomials and collocation method. The resulting numerical algorithm is very concise and easy to use, only involving highly sparse matrix operations of addition and multiplication, and no inversion of the Jacobian in nonlinear problems.

Bifurcation & Chaos in Nonlinear Structural Dynamics: Novel & Highly Efficient Optimal-Feedback Accelerated Picard Iteration Algorithms

A new class of algorithms for solving nonlinear structural dynamical problems are derived in the present paper, as being based on optimal-feedback-accelerated Picard iteration, wherein the solution vectors for the displacements and velocities at any time in a finitely large time interval are corrected by a weighted (with a matrix) integral of the error. We present 3 approximations to solve the Euler-Lagrange equations for the optimal weighting functions; thus we present 3 algorithms denoted as Optimal-Feedback-Accelerated Picard Iteration (OFAPI) algorithms-1, 2, 3.

A novel class of highly efficient and accurate time-integrators in nonlinear computational mechanics

A new class of time-integrators is presented for strongly nonlinear dynamical systems. These algorithms are far superior to the currently common time integrators in computational efficiency and accuracy. These three algorithms are based on a local variational iteration method applied over a finite interval of time. By using Chebyshev polynomials as trial functions and Dirac–Delta functions as the test functions over the finite time interval, the three algorithms are developed into three different discrete time-integrators through the collocation method.

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