On the solutions of three-point boundary value problems using variational-fixed point iteration method

摘要

Abstract Given a three-point fourth-order boundary value problems $$egin{aligned} y^{(iv)}+p(x)y^{prime prime prime }+q(x)y^{prime prime }+r(x)y^{prime }+s(x)y=f(x),a le x le b end{aligned}$$ y ( i v ) + p ( x ) y ″ ′ + q ( x ) y ″ + r ( x ) y ′ + s ( x ) y = f ( x ) , a ≤ x ≤ b such that $$egin{aligned} y(a)=y(b)=y^{prime prime }(b)=y^{prime prime }(lpha )=0,a le lpha le b; end{aligned}$$ y ( a ) = y ( b ) = y ″ ( b ) = y ″ ( α ) = 0 , a ≤ α ≤ b ? where $$p,q,r,s,f in C[a,b]$$ p , q , r , s , f ∈ C [ a , b ] , we combine the application of variational iteration method and fixed point iteration process to construct an iterative scheme called variational-fixed point iteration method that approximates the solution of three-point boundary value problems. The success of the variational or weighted residual method of approximation from a practical point of view depends on the suitable selection of the basis function. The method is self correcting one and leads to fast convergence. Problems were experimented to show the effectiveness and accuracy of the proposed method.