Transformation Of Boundary Value Problem Research Articles

A note on the transformation of boundary value problems to initial value problems: The iterative transformation method

In this paper, a concise review is presented, of how the scaling transformation group can be employed to determine a modified boundary value problem [6,10,11], which is invariant under an extended two-parameter scaling group. The main contribution is then to show that since the non-physical scaling parameter in the extended group is unity, a simple iterative method which does not require an extended group can be deduced.

Modification Adomian Decomposition Method for solving Seventh OrderIntegro-Differential Equations

In this paper, a method based on modified adomian decomposition method for solving Seventh order integro-differential equations (MADM). The distinctive feature of the method is that it can be used to find the analytic solution without transformation of boundary value problems. To test the efficiency of the method presented two examples are solved by proposed method. Keyword: Adomian decomposition method; boundary-value problems; integro-differential equation

Powell-Eyring model flow near an accelerated plate

Powell-Eyring model flow near an accelerated plate is studied by three methods namely, (i) the orthogonal collocation, (ii) satisfaction of asymptotic boundary conditions, and (iii) transformation of boundary value problem to initial value problem. The results obtained in the three cases are compared with each other.

Transformation of boundary value problems into initial value problems

In a previous paper of the author (SIAM Review, 1962, pp. 43–47), the idea of Blasius for the transformation of the B.V.P., y‴ + yy′' = 0, y( o) = y′( o) = 0, y′(∞) = 2, into a pair of similar I.V.P.'s was extended to D.E.'s or systems of D.E.'s of any order which were invariant under certain groups of homogeneous linear transformations. The boundary conditions were specified at the origin and at infinity and were homogeneous at the initial point. Subsequently, T. Y. Na (SIAM Review, 1967, pp. 204–210; 1968, pp. 85–87) showed that the method was also applicable to finite intervals and also considered other groups of transformations. He noted that for the method to apply, the boundary conditions have to be homogeneous at the initial point. Here, however, we show that homogeneity is not necessary by treating equations subject to a variety of nonhomogeneous boundary conditions.

Transforming Boundary Conditions to Initial Conditions for Ordinary Differential Equations

One of the interesting and important classes of transformations in the solution of boundary value problems is the class of transformations from a boundary value to an initial value problem. A classical example is given in Blasius' solution of the steady, two-dimensional, incompressible boundary layer equations with uniform mainstream velocity [1]. This technique, when it applies, simplifies the process of obtaining numerical solutions. Otherwise a trial-and-error technique generally has to be used in order to match the boundary condition at the other point. A recent contribution by Klamkin [2] has greatly extended the range of application of the technique, and as a result, a much wider class of problems, including some simultaneous differential equations, can be solved in this manner. In the present paper, the technique is extended to cases where the boundary conditions are specified over finite intervals, i.e., from zero to a finite value of the independent variable, with one or more boundary conditions required to be specified at the initial point. It also shows that the transformation treated in previous works is not the only type of transformation possible. Similar to the class of problems treated by Blasius [1] and Klamkin [2], the boundary conditions at the initial point have to be homogeneous for the method to apply. It can be homogeneous or nonhomogeneous at the other point. The method given in this paper is developed in terms of a oneor two-parameter group of transformations, which is seen to be as simple here as in its application to the similarity analysis [3]. The concept can best be illustrated by an example. Consider the problem of steady heat conduction with linear heat generation and power-law thermal conductivity. The energy equation can be written as

On the Transformation of a Class of Boundary Value Problems into Initial Value Problems for Ordinary Differential Equations

On the Transformation of a Class of Boundary Value Problems into Initial Value Problems for Ordinary Differential Equations