We apply singularity theory to study bifurcation problems with trivial solutions. The approach is based on a new equivalence relation called t-equivalence which preserves the trivial solutions. We obtain a sufficient condition for recognizing such bifurcation problems to be t-equivalent and discuss the properties of the bifurcation problems with trivial solutions. Under the action of t-equivalent group, we classify all bifurcation problems with trivial solutions of codimension three or less.

This paper investigates the problem of identifying unknown coefficient of time dependent in heat conduction equation by new iteration method. In order to use new iteration method, we should convert the parabolic heat conductive equation into an integral equation by integral calculus and initial condition. This method constructs a convergent sequence of function, which approximates the exact solution with a few iterations and does not need complex calculation. Illustrative examples are given to demonstrate the efficiency and validity.

This paper investigates the boundary value in the heat conduction problem by a variational iteration method. Applying the iteration method, a sequence of convergent functions is constructed, the limit approximates the exact solution of the heat conduction equation in a few iterations using only the initial condition. This method does not require discretization of the variables. Numerical results show that this method is quite simple and straightforward for models that are currently under research.

In this paper, the boundedness of certain multilinear operator related to the multiplier operator from Lebesgue spaces to Orlicz spaces is obtained.