An Improved Bat Algorithm For Solving Fredholm Integral Equations Of The First Kind

Fredholm integral equations of the first kind are an important branch in the research field of inverse problems,which are widely used in the fields of structural engineering,image processing,geological survey,etc.It is difficult to solve stably the problems because Fredholm integral equations of the first kind are ill-posed.From the existing research methods,the main method to solve the integral equations is to discretize them and then obtain their numerical solutions.With the expansion of the scale of the problems,the longer traditional methods take to solve the problems,the lower the computational efficiency and accuracy,which have great limitations.Therefore,it is still worth discussing to solve quickly and stably the inverse problem of integral equations.At present,the literatures of solving the inverse problem of integral equations based on intelligent algorithm are relatively few.So,this paper put forward bat algorithm and improved bat algorithm for solving Fredholm integral equations of the first kind,and studies the speed and stability of solving such problems.In order to study whether the basic bat algorithm can resolve the ill-posed property of Fredholm integral equations of the first kind,the basic bat algorithm is applied to solve the integral equations in this paper.The integral equations are discretized by using the compound trapezoid formula,which reduces the solution of the integral equations to the solution of linear algebraic equations.And then the objective function of the basic bat algorithm is constructed by using th e least square method to obtain the solution of the inverse problems by bat algorithm.The experimental results show that the basic bat algorithm can not effectively solve integral equations of this kind due to the ill-posed property,but it has faster convergence speed and better stability.Subsequently,the Tikhonov regularization method is combined with bat algorithm,and the Tikhonov regularization bat algorithm is constructed to solve Fredholm integral equations of the first kind.Tikhonov functional is used to correct the objective function of the basic bat algorithm,which transforms the ill-posed problems into the well-posed problems,and then they are solved by the bat algorithm.The experimental results show that the Tikhonov regularization bat algorithm can obtain quickly and stably the numerical solution of integral equations,and the fitting effect and solving accuracy are superior to the classical Tikhonov regularization method,but both of them have severe deviation points.Based on the Tikhonov regularization bat algorithm,a new improved bat algorithm is proposed to solve Fredholm integral equations of the first kind.The discretization method of the integral equations is improved to construct the objective function of the improved bat algorithm,and the convergence and stability of the improved discretization method are proved.The severe deviation points are corrected to enhance the fitting effect,the coefficient of velocity inertia is adjusted to increase the diversity of population,and Gauss disturbance is added to further optimize cluster.Experimental results show that the convergence speed of the improved bat algorithm is better than the Tikhonov regularization bat algorithm,which solves the problem of severe deviation points,and the fitting effect is generally better than Tikhonov regularization method and the Tikhonov regularization bat algorithm.