Non-iterative Transformation Method Research Articles

A non-iterative transformation method for boundary-layer with power-law viscosity for non-Newtonian fluids

A non-iterative transformation method for boundary-layer with power-law viscosity for non-Newtonian fluids

A non-iterative transformation method for Blasius equation with moving wall or surface gasification

We define a non-iterative transformation method for Blasius equation with moving wall or surface gasification. The defined method allows us to deal with classes of problems in boundary layer theory that, depending on a parameter, admit multiple or no solutions. This approach is particularly convenient when the main interest is on the behaviour of the considered models with respect to the involved parameter. The obtained numerical results are found to be in good agreement with those available in the literature.

Free Boundary Formulation for BVPs on a Semi-infinite Interval and Non-iterative Transformation Methods

This paper is concerned with two examples on the application of the free boundary formulation to BVPs on a semi-infinite interval. In both cases we are able to provide the exact solution of both the BVP and its free boundary formulation. Therefore, these problems can be used as benchmarks for the numerical methods applied to BVPs on a semi-infinite interval and to free BVPs. Moreover, we emphasize how for two classes of free BVPs, we can define non-iterative initial value methods, whereas BVPs are usually solved iteratively. These non-iterative methods can be deduced within Lie's group invariance theory. Then, we show how to apply the non-iterative methods to the two introduced free boundary formulations in order to obtain meaningful numerical results. Finally, we indicate several problems from the literature where our non-iterative transformation methods can be applied.

A non-iterative transformation method for Newton's free boundary problem

In book II of Newton's Principia Mathematica of 1687 several applicative problems are introduced and solved. There, we can find the formulation of the first calculus of variations problem that leads to the first free boundary problem of history. The general calculus of variations problem is concerned with the optimal shape design for the motion of projectiles subject to air resistance. Here, for Newton's optimal nose cone free boundary problem, we define a non-iterative initial value method which is referred in the literature as a transformation method. To define this method we apply invariance properties of Newton's free boundary problem under a scaling group of point transformations. Finally, we compare our non-iterative numerical results with those available in the literature and obtained via an iterative shooting method. We emphasize that our non-iterative method is faster than shooting or collocation methods and does not need any preliminary computation to test the target function as the iterative method or even provide any initial iterate. Moreover, applying Buckingham Pi-Theorem we get the functional relation between the unknown free boundary and the nose cone radius and height.

Numerical transformation methods: Blasius problem and its variants

Blasius problem is the simplest nonlinear boundary-layer problem. We hope that any approach developed for this epitome can be extended to more difficult hydrodynamics problems. With this motivation we review the so called Töpfer transformation, which allows us to find a non-iterative numerical solution of the Blasius problem by solving a related initial value problem and applying a scaling transformation. The applicability of a non-iterative transformation method to the Blasius problem is a consequence of its partial invariance with respect to a scaling group. Several problems in boundary-layer theory lack this kind of invariance and cannot be solved by non-iterative transformation methods. To overcome this drawback, we can modify the problem under study by introducing a numerical parameter, and require the invariance of the modified problem with respect to an extended scaling group involving this parameter. Then we apply initial value methods to the most recent developments involving variants and extensions of the Blasius problem.

On the equivalence of non-iterative transformation methods based on scaling and spiral groups

The non-iterative numerical solution of nonlinear boundary value problems is a subject of great interest. The present paper is concerned with the theory of non-iterative transformation methods (TMs). These methods are defined within group invariance theory. Here we prove the equivalence between two non-iterative TMs defined by the scaling group and the spiral group, respectively. Then, we report on numerical results concerning the steady state temperature space distribution in a non-linear heat generation model. These results improve the ones, available in the literature, obtained by using the invariance with respect to a spiral group. Copyright © 2009 John Wiley & Sons, Ltd.

The Blasius problem formulated as a free boundary value problem

In the present paper we point out that the correct way to solve the Blasius problem by numerical means is to reformulate it as free boundary value problem. In the new formulation the truncated boundary (instead of infinity) is the unknown free boundary and it has to be determined as part of the numerical solution. Taking into account the “partial” inavariance of the mathematical model at hand with respect to a stretching group we define a non-iterative transformation method. Further, we compare the improved numerical results, obtained by the method in point, with analytical and numerical ones. Moreover, the numerical results confirm that the question of accuracy depends on the value of the free boundary. Therefore, this indicates that boundary value problems with a boundary condition at infinity should be numerically reformulated as free boundary value problems.

A noniterative transformation method applied to two-point boundary-value problems

Considering similarity properties of nonlinear two-point boundary-value problems with separated Dirichlet, Neumann, or Dirichlet-Neumann boundary conditions, it is possible to define a noniterative transformation method. We characterize classes of problems for which the method is applicable. Essentially we require invariance under the most general stretching or spiral group of transformation. The possibility of application of the method to a wide class problems of physical interest is shown, and some examples are treated. The main feature of the method is the possibility of determining the numerical solution after two integrations. Moreover the procedure is self-validating: the numerical solution, in order to be acceptable, has to give a good approximation of the final boundary value. We also make an extension of the method to free-boundary-value problems.

A nonlinear hyperbolic free boundary value problem

The present paper is concerned with the application of a non-iterative transformation method to the numerical solution of a nonlinear hyperbolic free boundary value problem.

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