Homotopy Analysis Method For Several Problems In Mechanics And Finance

There exist many nonlinear equations in mechanics and finance.Obtaining the analytic approximation solutions of these equations are of great meaning.In this paper,we apply the homotopy analysis method(HAM)for following four kinds of problems: the limiting Stokes wave with arbitrary water depth,the large deflection of a circular plate under arbitrary uniform pressure,the post-buckling problems and the backward/forward-backward stochastic differential equations.Main results are shown as following:(1)In this paper,accurate series solutions of the limiting Stokes wave with arbitrary water depth are successfully obtained by means of the HAM.To the best of author’s knowledge,it is the first time to give accurate wave profile in extreme shallow water without using any extrapolation techniques.Therefore,in the frame of the HAM,the Stokes waves can be used as a unified theory for all kinds of waves,including periodic waves in deep and intermediate depth,cnoidal waves in shallow water and solitary waves in extremely shallow water.This enriches people’s understanding of the steady periodic waves.(2)In this paper,we employ the HAM to solve the nonlinear equations governed by a large deformed circular thin plate under uniform pressure to arbitrary magnitude.Accurate series solutions are successfully obtained.Besides,we successfully prove that the three classic methods–the perturbation method with arbitrary perturbation quantity,the modified iteration method and the interpolation iterative method–are all the special cases of the HAM.Furthermore,in the frame of the HAM,we systematically discuss the importance of the convergence-control parameter c0,the effectiveness of using iteration technique and expanding the uniform pressure Q into the series of the central deflection.(3)The HAM is employed to solve a non-linear differential equation governed by a large deformed elastic beam with non-convex total potential energy.Our results agree well with the numerical solutions.More importantly,our strategy is easy to implement,which demonstrates HAM’s validity for computational non-convex mechanics.(4)In this paper,the HAM is employed to solve some backward stochastic differential equations(BSDEs)and forward-backward stochastic differential equations(FBSDEs),including one with high dimensionality(up to 12 dimensions).By means of the HAM,convergent series solutions can be quickly obtained with high accuracy for a FBSDE in a 6-dimensional case,within less than 1/3000 CPU time used by a currently reported numerical method for the same case.Especially,as dimensionality enlarges,the increase of computational complexity for the HAM is not as dramatic as this numerical method.All of these demonstrate the validity and high efficiency of the HAM for the backward/forward-backward stochastic differential equations in science,engineering,and finance.