Framework definitions of earlier informally used notions of effect and phenomenon are proposed, examples of their role in developing mathematics are given. Phenomena of singular cycle and deepening boundary layer in t...Framework definitions of earlier informally used notions of effect and phenomenon are proposed, examples of their role in developing mathematics are given. Phenomena of singular cycle and deepening boundary layer in the theory of singularly perturbed differential equations and ancient popular synergetic process described by system of random difference equations are presented.展开更多
In this paper, we study the fractional-order biological population models （FI3PMs） with Malthusian~ Verhulst, and porous media laws. The fractional derivative is defined in Caputo sense. The optimal homotopy asympto...In this paper, we study the fractional-order biological population models （FI3PMs） with Malthusian~ Verhulst, and porous media laws. The fractional derivative is defined in Caputo sense. The optimal homotopy asymptotic method （OHAM） for partial differ- ential equations （PDEs） is extended and successfully implemented to solve FBPMs. Third-order approximate solutions are obtained and compared with the exact solutions. The numerical results unveil that the proposed extension in the OHAM for fractional- order differential problems is very effective and simple in computation. The results reveal the effectiveness with high accuracy and extremely efficient to handle most complicated biological population models.展开更多
文摘Framework definitions of earlier informally used notions of effect and phenomenon are proposed, examples of their role in developing mathematics are given. Phenomena of singular cycle and deepening boundary layer in the theory of singularly perturbed differential equations and ancient popular synergetic process described by system of random difference equations are presented.
文摘In this paper, we study the fractional-order biological population models （FI3PMs） with Malthusian~ Verhulst, and porous media laws. The fractional derivative is defined in Caputo sense. The optimal homotopy asymptotic method （OHAM） for partial differ- ential equations （PDEs） is extended and successfully implemented to solve FBPMs. Third-order approximate solutions are obtained and compared with the exact solutions. The numerical results unveil that the proposed extension in the OHAM for fractional- order differential problems is very effective and simple in computation. The results reveal the effectiveness with high accuracy and extremely efficient to handle most complicated biological population models.
