Abstract
Interest in the construction of efficient methods for solving initial value problems that have some peculiar properties with it or its solution is recently gaining wide popularity. Based on the assumption that the solution is representable by nonlinear trigonometric expressions, this work presents an explicit single-step nonlinear method for solving first order initial value problems whose solution possesses singularity. The stability and convergence properties of the constructed scheme are also presented. Implementation of the new method on some standard test problems compared with those discussed in the literature proved its accuracy and efficiency.
Highlights
Many of the numerical methods for obtaining the solution of the first order ordinary differential equation y′ = f ( x, y ( x)), x ∈[x0, X ], y ( x0 ) =η (1)are based on the assumption that the solution is locally representable by a polynomial
Since rational functions are more appropriate for the representation of functions close to singularities than polynomials, the limitation is overcome by a local representation of the theoretical solution with a rational expression
The new method NLM4 is suitable for solving initial value problems whose solution possesses singularities
Summary
The authors in [4] were the first to develop quadrature formulas based on rational interpolating functions. Since rational functions are more appropriate for the representation of functions close to singularities than polynomials, the limitation is overcome by a local representation of the theoretical solution with a rational expression. This approach appears to be promising as several methods are being constructed in this direction [6] [7]. An explicit single-step nonlinear method involving higher derivatives of the state function for solving (1) is presented. The local truncation error and absolute stability of the new method are discussed
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