Abstract
AbstractSolutions to nonlinear boundary value problems modelling physical phenomena in engineering applications have traditionally been approximated using numerical methods. More recently, several semi-analytical methods have been developed and used extensively in diverse engineering applications. This work compares the method of variation of parameters to semi-analytical methods for solving nonlinear boundary value problems arising in engineering. The accuracy and efficiency of the method of variation of parameters are compared to those of two widely used semi-analytical methods, the Adomian decomposition method and the differential transformation method, for three practical engineering boundary value problems: (1) the deflection of a cantilevered beam with a concentrated load, (2) an adiabatic tubular chemical reactor which processes an irreversible exothermal reaction, and (3) the electrohydrodynamic flow of a fluid in an ion drag configuration in a circular cylinder conduit. The accuracy and convergence of each method is investigated using the error remainder function. The method of variation of parameters is significantly more efficient than both the semi-analytical methods and traditional numerical methods while maintaining comparable accuracy. Unlike the semi-analytical methods, the efficiency of variation of parameters is independent of the nonlinearity. Variation of parameters is shown to be an attractive alternative to semi-analytical methods and traditional numerical methods for solving boundary value problems encountered in engineering applications in which solution efficiency is important.
Highlights
Mathematical modelling of physical phenomena in all branches of science and engineering frequently results in boundary value problems governed by nonlinear differential equations
Variation of parameters is shown to be an attractive alternative to semianalytical methods and traditional numerical methods for solving boundary value problems encountered in engineering applications in which solution efficiency is important
The method of variation of parameters is shown to be an attractive alternative to semi-analytical methods and traditional numerical methods for solving boundary value problems encountered in engineering applications, especially for problems in which solution efficiency is an important consideration
Summary
Mathematical modelling of physical phenomena in all branches of science and engineering frequently results in boundary value problems governed by nonlinear differential equations. These problems generally do not have closed-form, exact analytical solutions. Methods developed to approximate solutions to nonlinear boundary value problems (BVPs) were strictly numerical. Used numerical methods for solving BVPs include shooting methods, Runge-Kutta methods, finite-difference methods, and integral equation methods. Extensive treatment of these and other numerical methods for solving BVPs are widely available [2, 3]
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