Many predominant numerical algorithms used to approximate solutions of nonlinear boundary-value problems (BVPs) have a Runge-Kutta foundation. A shooting algorithm using a foundation of our Algebraic-Maclaurin-Pade’ method can potentially produce remarkable accuracy in significantly less time near a singularity. This also produces an effective numerical technique for BVPs, as it numerically generates and stores the coefficients of the Taylor polynomial of the solution at each step for each term of the series, using a simple progression of Cauchy products. The Taylor coefficients are then used to numerically create the coefficients of a rational polynomial Pade’ approximation to the solution at each step for singular BVPs. Our Algebraic-Maclaurin-Pade’ algorithm allows for a smaller number of steps as the solution marches toward the singularity and provides a simple manner in which to increase (or decrease) the order of the algorithm during the computation, resulting in general in a more accurate solution nearby and at the singularity. The method is first developed theoretically and then applied to several nonlinear and singular BVPs (cavitation, stretching, torsion, azimuthal shear) arising in modeling deformations of compressible nonlinearly elastic solids. Results are compared with analytic solutions developed in several papers by C. O. Horgan, as well as against standard Runge-Kutta and Taylor series methods. Significant improvements in both accuracy and efficiency are shown for the Algebraic-Maclaurin-Pade’ numerical method developed here. Finally, the numerical data is used to create animations and visualizations of solids undergoing the various deformations.
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