In this work we
apply the differential transformation method or DTM for solving some classes of
Lane-Emden type equations as a model for the dimensionless density distribution
in an isothermal gas sphere and as a
study of the gravitational potential of (white-dwarf) stars , which are nonlinear ordinary differential equations
on the semi-infinite domain [1][2]. The efficiency of the DTM is
illustrated by investigating the convergence results for this type of the
Lane-Emden equations. The numerical results show the reliability and accuracy
of this method.

The interpolation method in a semi-Lagrangian scheme is decisive to its
performance. Given the number of grid points one is considering to use for the
interpolation, it does not necessarily follow that maximum formal accuracy
should give the best results. For the advection equation, the driving force of
this method is the method of the characteristics, which accounts for the flow
of information in the model equation. This leads naturally to an interpolation
problem since the foot point is not in general located on a grid point. We use
another interpolation scheme that will allow achieving the high order for the
box initial condition.

Abstract:
We establish the conditions for the compute of the Global Truncation
Error (GTE), stability restriction on the time step and we prove the
consistency using forward Euler in time and a fourth order discretization in
space for Heat Equation with smooth initial conditions and Dirichlet boundary
conditions.

Abstract:
We consider the nonlinear boundary value problems for elliptic partial differential equations and using a maximum principle for this problem we show uniqueness and continuous dependence on data. We use the strong version of the maximum principle to prove that all solutions of two-point BVP are positives and we also show a numerical example by applying finite difference method for a two-point BVP in one dimension based on discrete version of the maximum principle.

In this work
we apply the differential transformation method (Zhou’s method) or DTM for solving white-dwarfs equation which
Chandrasekhar [1] introduced in his study of the gravitational potential of
these degenerate (white-dwarf) stars.
DTM may be considered as alternative and efficient for finding the approximate
solutions of the initial values problems. We prove superiority of this method
by applying them on the some Lane-Emden type equation, in this case.The power series solution of the
reduced equation transforms into an approximate implicit solution of the
original equation.

Abstract:
This paper describes a numerical solution for a two-point boundary value problem. It includes an algorithm for discretization by mixed finite element method. The discrete scheme allows the utilization a finite element method based on piecewise linear approximating functions and we also use the barycentric quadrature rule to compute the stiffness matrix and the L_{2}-norm.

Abstract:
A semi-linear second order ODE under a nonlinear two-point boundary condition is considered. Under appropriate conditions on the nonlinear term of the equation, we define a two-dimensional shooting argument which allows to obtain solutions for some specific situations by the use of Poincaré-Miranda's theorem. Finally, we apply this result combined with the method of upper and lower solutions and develop an iterative sequence that converges to a solution of the problem.

Abstract:
We establish the conditions for the compute of the stability restriction and local accuracy on the time step and we prove the consistency and local truncation error by using θ-scheme and 3-level scheme for Heat Equation with smooth initial conditions and for some parameter θ∈[0,1].

Abstract:
In this work, we apply the Zhou’s method
[1] or differential transformation method (DTM) for solving the Euler
equidimensional equation. The Zhou’s method may be considered as alternative
and efficient for finding the approximate solutions of initial values problems.
We prove superiority of this method by applying them on the some Euler type
equation, in this case of order 2 and 3 [2]. The power series solution of the
reduced equation transforms into an approximate implicit solution of the
original equations. The results agreed with the exact solution obtained via
transformation to a constant coefficient equation.