Journal
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
Volume -, Issue -, Pages -Publisher
WILEY
DOI: 10.1002/mma.9736
Keywords
iterative method; multi step; order of convergence; system of problems
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In this work, a family of fourth-order methods is proposed to solve nonlinear equations, which satisfy the Kung-Traub optimality conjecture. The efficiency indices of the methods are increased by developing them into memory methods. The methods are then extended to multi-step methods for solving systems of problems. Numerical examples are provided to confirm the theoretical results, and the methods are applied to solve nonlinear problems related to the numerical approximation of fractional differential equations.
In this work, first, a family of fourth-order methods is proposed to solve nonlinear equations. The methods satisfy the Kung-Traub optimality conjecture. By developing the methods into memory methods, their efficiency indices are increased. Then, the methods are extended to the multi-step methods for finding the solutions to systems of problems. The formula for the order of convergence of the multi-step iterative methods is 2p, where p is the step number of the methods. It is clear that computing the Jacobian matrix derivative evaluation and its inversion are expensive; therefore, we compute them only once in every cycle of the methods. The important feature of these multi-step methods is their high-efficiency index. Numerical examples that confirm the theoretical results are performed. In applications, some nonlinear problems related to the numerical approximation of fractional differential equations (FDEs) are constructed and solved by the proposed methods.
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