Improved Tikhonov Regularization Method Of The First Kind Volterra Integral Equations

The first kind Volterra integral equation which develops in the twentiethcentury is a very important part in integral equations. It has a wide range ofapplications in mathematical physics, astronomy, mechanics, engineering,biomedical and other disciplines. All kinds of problems in many areas can betransformed into the first kind Volterra integral equation models. But when thekernel function is continuous or weakly singular, it is usually difficult to get exactsolutions. Thus, numerical solutions of the first kind Volterra integral equationshave an important theoretical significance and practical value.By analyzing advantages and disadvantages of Tikhonov regularizationmethod and improved Tikhonov regularization method which was proposedby"The regularization of the first kind Fredholm integral equation", we present anew improved Tikhonov regularization method. This method not only removesimproved Tikhonov regular solutions' fluctuant points, but also weakens thesupression of Tikhonov regularization method to signals. We verify thesuperiority of the new improved Tikhonov regularization method by solving somespecific the first kind Fredholm integral equations and the first kind Volterraintegral equationsFor specific ill-posed integral equations, we can solve them by threeregularization methods under different disturbances and compare the truesolutions and the regular solutions. The numerical results show that the newimproved Tikhonov regular solutions are closer to the true solutions.Then based on different disturbances and different regularization parameterselection strategies, we obtain error of solution, convergence rate and CPU timeby different regularization methods. Numerical results reflect the high accuracyand high convergence rate and feasibility of the new improved Tikhonovregularization method.