Collocation Method For Integro-differential Equations Based On The Fractional C~? Space

The fractional integro-differential equations arise when dealing with practical problems and extremely common in engineering models and physics,such as chemical reaction-diffusion?elasticity?heat conduction?population ecology model?control theory and biochemistry.The fractional integro-differential equations play an important role in explaining many phenomena with the rapid development of science and technology in recent years.On the one hand,the fractional integro-differential equations have been widely used in many disciplines.The numerical solution is more necessary than the analytical solution in practical applications.On the other hand,it is difficult to get the analytical solution.Because the study of the numerical solution of the integro-differential equations will inevitably promote the development of related theories in this discipline,such as operator theory,function approximation theory.Therefore,the numerical algorithm for exploring the equations becomes an important means for qualitatively studying such equations and how to construct a high-precision numerical algorithms need to be further explored.In the second chapter of this paper,we study the numerical solution of fractional integro-differential equations with the weakly singular kernel.It is worth noting that the smallest space containing the true solution of the equation,fractional space,is found.And a set of dense subsets of this space is given.We discussed further the existence of the approximate solution of the equation.Meanwhile,the proof of convergence and error analysis are also given.The accuracy and efficiency of our method are illustrated through numerical examples.In the third chapter of this paper,a new algorithm based on fractional space is proposed to solve fractional pantograph differential equation and a complete theoretical system is established.Finally,the results of numerical experiments show that the proposed algorithm can obtain high-precision numerical solutions with a small amount of calculation.In the fourth chapter of this paper,the numerical method of the time fractional convection-diffusion-reaction equation with variable coefficients is studied based on the fractional space.The concept of two-dimensional fractional space is presented and a set of dense subset is given.The collocation method can achieve the numerical accuracy required for practical applications with a small amount of calculation.This paper separately discusses three different fractional integro-differential equations.Based on the properties of fractional space,a new numerical algorithm is proposed and it is easy to operate and can obtain high-precision numerical solutions.