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The paper presents a computational algorithm for solving the problem of pricing an American put-option described by the Black-Sholes equation. Firstly, two changes of variables are presented. First change is used to eliminate a discontinuity of solution derivatives. The second one transforms the equation on the domain with free boundary to the equation with an unknown coeffcient on a domain with fixed boundary. Secondly, the finite-difference approximation of the obtained equation is fulfilled including boundary condition at infinity. Finally, the computational algorithm is presented for solving discrete analogue of parabolic equation with iteration of the secant method at each time layer to found an unknown coeffcient with the help of the additional boundary condition. [ABSTRACT FROM AUTHOR]

In this paper we will investigate a family of five-stages high algebraic order embedded explicit six-step finite difference pairs, for the numerical solution of second order initial and/or boundary-value problems with periodical and/or oscillating solutions. Requiring the elimination of the phase-lag of the methods of the family of five-stages embedded explicit six-step algorithms and requiring also the elimination of the phase-lag's derivatives of the methods of the family of five-stages embedded explicit six-step schemes, we obtain a system of of equations. The solution of the previously obtained system of equations, leads to the determination of the free parameters of the new proposed finite difference pairs. Following the building of the new proposed finite difference pairs, we analyze theoretically the resulting schemes as follows: • We will study the local truncation error (LTE) of the resulting schems. • We will study the asymptotic form of the LTE which are produced using as model problem the radial Schrödinger equation. • We will study the comparison of the asymptotic forms of LTEs for several finite difference schemes of the same family. The comparison leads to the conclusions on the effectiveness of each member of the family of five-stages high algebraic order embedded explicit six-step finite difference pairs. • We will study the stability and the interval of periodicity of the resulting algorithms of the family of five-stages high algebraic order embedded explicit six-step finite difference pairs. • We will study the applications of the new builded family of five-stages high algebraic order embedded explicit six-step finite difference pairs to the numerical solution of several second order problems like the radial Schrödinger equation, astronomical problems etc. The previously described applications lead to the conclusions on the effectiveness of the members of the new proposed family of five-stages high algebraic order embedded explicit six-step finite difference pairs. [ABSTRACT FROM AUTHOR]

In this paper we propose an exponential fitting (EF) method with two fitting parameters for solving initial value problem x" = f(x(t), x'(t)) which has oscillatory solutions with varying amplitude. Local truncation error formula is used to determine these fitting parameters. Numerical examples are presented to show efficiency of the method for oscillatory problems in the long time integration. [ABSTRACT FROM AUTHOR]