Numerical Methods for Boundary Value Problems defined on Infinite Intervals

Selected List of Papers- The Blasius Problem Formulated as a Free Boundary Value Problem
- The Falkner-Skan Equation: Numerical Solutions within Group Invariance Theory
- A Novel Approach to the Numerical Solution of Boundary Value Problems on Infinite Intervals
- A Survey on Free Boundary Identification of the Truncated Boundary in Numerical BVPs on Infinite Intervals
- A Free Boundary Approach and the Keller's Box Scheme for Boundary Value Problems on Infinite Intervals
- Numerical Transformation Methods: Blasius Problem and its Variants
- Blasius Problem and Falkner-Skan model: Toepfer's Algorithm and its Extension
- Finite Difference Schemes on Quasi-Uniform Grids for BVPs on Infinite Intervals.
- Free Boundary Formulation for BVPs on a Semi-infinite Interval and Non-iterative Transformation Methods
- The Iterative Transformation Method for the Sakiadis Problem
- A Non-Iterative Transformation Method for Blasius Equation with Moving Wall or Surface Gasification
- BVPs on Infinite Intervals: a Test Problem, a Non-standard Finite Difference Scheme and a Posteriori Error Estimator
- Numerical study on gas flow through a micro-nano porous medium based on finite difference schemes on quasi-uniform grids
- Two finite difference methods for a nonlinear BVP arising in physical oceanography
- The Non-Iterative Transformation Method
- The Iterative Transformation Method
- A Non-Iterative Transformation Method for an Extended Blasius Problem
- A Non-Iterative Transformation Method for Boundary-Layer with Power-Law Viscosity for non-Newtonian Fluids

** Abstract:** In the present paper we
point out that the correct way to solve the Blasius problem by
numerical means is to reformulate it as a 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" invariance 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.

** Abstract:** The iterative transformation
method, defined within the framework of the group invariance theory, is
applied to the numerical solution of the Falkner-Skan equation with
relevant boundary conditions. In this problem a boundary condition at
infinity is imposed which is not suitable for a numerical use. In order
to overcome this difficulty we introduce a free boundary formulation of
the problem, and we define the iterative transformation method that
reduces the free boundary formulation to a sequences of initial value
problems. Moreover, as far as the value of the wall shear stress is
concerned we propose a numerical test of convergence. The usefulness of
our approach is illustrated by considering the wall shear stress for
the classical Homann and Hiemenz flows. In the Homann's case we apply
the proposed numerical test of convergence, and meaningful numerical
results are listed. Moreover, for both cases we compare our results
with those reported in literature.

** Abstract:** The classical numerical
treatment of two-point boundary value problems on infinite intervals is
based on the introduction of a truncated boundary (instead of infinity)
where appropriate boundary conditions are imposed. Then, the truncated
boundary allowing for a satisfactory accuracy is computed by trial.
Motivated by several problems of interest in boundary layer theory,
here we consider boundary value problems on infinite intervals governed
by a third order ordinary differential equation. We highlight a novel
approach in order to define the truncated boundary. The main result is
the convergence of the solution of our formulation to the solution of
the original problem as a suitable parameter goes to zero. In the
proposed formulation the truncated boundary is an unknown free boundary
and has to be determined as part of the solution. For the numerical
solution of the free boundary formulation a non-iterative and an
iterative transformation method are introduced. Furthermore, we
characterize the class of free boundary value problems that can be
solved non-iteratively. A nonlinear flow problem involving two physical
parameters and belonging to the characterized class of problems is then
solved. Moreover, the Falkner-Skan equation with relevant boundary
conditions is considered and representative results, obtained by the
iterative transformation method, are listed for the Homann flow. All
the obtained numerical results clearly indicate the effectiveness of
the proposed approach. Finally, we discuss some open problems and
possible extensions for our approach.

** Abstract:**
A free boundary formulation for the
numerical solution of boundary value problems on infinite
intervals was proposed recently in [R. Fazio,
SIAM J. Numer. Anal., 33 (1996) 1473-1483].
We consider here a survey on recent developments related to the
free boundary identification of the truncated boundary.
The goals of this survey are: to recall the reasoning for a
free boundary identification of the truncated boundary,
to report on a comparison of numerical
results obtained for a classical test problem
by three approaches available in the literature,
and to propose some possible ways to extend the free boundary
approach to the numerical solution of problems defined on the
whole real line.

** Abstract:**
A free boundary approach for the numerical solution of boundary value problems (BVPs) governed by a third-order
differential equation and defined on infinite intervals was proposed recently [SIAM J. Numer. Anal., 33 (1996),
pp. 1473–1483]. In that approach, the free boundary (that can be considered as the truncated boundary) is unknown
and has to be found as part of the solution. This eliminates the uncertainty related to the choice of the truncated
boundary in the classical treatment of BVPs defined on infinite intervals. In this article, we investigate some open
questions related to the free boundary approach. We recall the extension of that approach to problems governed by a
system of first-order differential equations, and for the solution of the related free boundary problem we consider now
the reliable Keller’s box difference scheme. Moreover, by solving a challenging test problem of interest in foundation
engineering, we verify that the proposed approach is applicable to problems where none of the solution components
is a monotone function.

** Abstract:**
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.

** Abstract:**
In this paper, we review the so-called Töpfer algorithm that 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. Moreover, we remark that the applicability of this algorithm can be extended to
any given problem, provided that the governing equation and the initial conditions are invariant under a scaling group of point transformations and that
the asymptotic boundary condition is non-homogeneous. Then, we describe
an iterative extension of Töpfer’s algorithm that can be applied to a general
class of problems. Finally, we solve the Falkner-Skan model, for values of
the parameter where multiple solutions are admitted, and report original numerical results, in particular data related to the famous reverse flow solutions
by Stewartson. The numerical data obtained by the extended algorithm are
in good agreement with those obtained in previous studies.

** Abstract:**
The classical numerical treatment of boundary value problems defined
on infinite intervals is to replace the boundary conditions at infinity by suitable boundary conditions at a finite point, the so-called truncated boundary.
A truncated boundary allowing for a satisfactory accuracy of the numerical solution has to be determined by trial and errors and this seems to be
the weakest point of the classical approach. On the other hand, the free
boundary approach overcomes the need for a priori definition of the truncated boundary. In fact, in a free boundary formulation the unknown free
boundary can be identified with a truncated boundary and being unknown it
has to be found as part of the solution.
In this paper we consider a different way to overcome the introduction
of a truncated boundary, namely non-standard finite difference schemes defined on quasi-uniform grids. A quasi-uniform grid allows us to describe the
infinite domain by a finite number of intervals. The last node of such grid
is placed on infinity so that right boundary conditions are taken into account
exactly. We apply the proposed approach to the Falkner-Skan model and
to a problem of interest in foundation engineering. The obtained numerical results are found in good agreement with those available in literature.
Moreover, we provide a simple way to improve the accuracy of the numerical results using Richardson’s extrapolation. Finally, we indicate a possible
way to extend the proposed approach to boundary value problems defined
on the whole real line.

** Abstract:**
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.

** Abstract:**
In a transformation method the numerical solution of a given boundary value problem is obtained by solving one or more related initial value problems.
This paper is concerned with the application of the iterative transformation method to the Sakiadis problem.
This method is an extension of the Toepfer's non-iterative algorithm developed as a simple way to solve the celebrated Blasius problem.
As shown by this author [Appl. Anal., 66 (1997) pp. 89-100] the method provides a simple numerical test for the existence and uniqueness of solutions.
Here we show how the method can be applied to problems with a homogeneous boundary conditions at infinity and in particular we solve the Sakiadis problem of boundary layer theory.
Moreover, we show how to couple our method with Newton's root-finder.
The obtained numerical results compare well with those available in literature.
The main aim here is that any method developed for the Blasius, or the Sakiadis, problem might be extended to more challenging or interesting problems.
In this context, the iterative transformation method has been recently applied to compute the normal and reverse flow solutions of Stewartson for the Falkner-Skan model [Comput. Fluids, 73 (2013) pp. 202-209].

** Abstract:**
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 literature.

** Abstract:**
As far as the numerical solution of boundary value problems defined on an infinite interval is concerned,
in this paper, we present a test problem for which the exact solution is known.
Then we study an a posteriori estimator for the global error of a non-standard finite difference scheme previously introduced by the authors.
In particular, we show how Richardson extrapolation can be used to improve the numerical solution using the order of accuracy and numerical solutions from two nested quasi-uniform grids. We observe that if the grids are sufficiently fine the Richardson error estimate gives an upper bound of the global error.

** Abstract:**
In this paper, the unsteady isothermal flow of a gas through a semi-infinite micro–nano porous medium described
by a non-linear two-point boundary value problem on a semi-infinite interval has been considered. We solve
this problem by a nonstandard finite difference method defined on quasi-uniform grids in order to derive a new
numerical approximation. By introducing a stencil that is constructed in such a way that the boundary conditions
at infinity are exactly assigned, the proposed method is effectively used to determine the numerical solution. In
addition, a mesh refinement and the Richardson’s extrapolation allow to improve the accuracy of the numerical
solution and to define a posteriori estimator for the global error of the proposed numerical scheme. We determine
the accurate initial slope du/dx(0) = -1.1917906497194208 calculated for alpha = 0.5 with good capturing the essential
behavior of u(x). This clearly demonstrates that the numerical solutions presented in this paper result highly
accurate and in excellent agreement with the existing solutions available in the literature.

** Abstract:**
In this paper we define two finite difference methods for a nonlinear boundary
value problem on infinite interval. In particular, we report and compare the numerical results
for an ocean circulation model obtained by the free boundary approach and a treatment of
the problem on the original semi-infinite domain by introducing a quasi-uniform grid. In the
first case we apply finite difference formulae on a uniform grid and in the second case we use
non-standard finite differences on a quasi-uniform grid. We point out how both approaches
represent reliable ways to solve boundary value problems defined on semi-infinite intervals.
In fact, both approaches overcome the need to define a priori, or find by trials, a suitable
truncated boundary used by the classical numerical treatment of boundary value problems
defined on a semi-infinite interval. Finally, the reported numerical results allow to point out
how the finite difference method with a quasi-uniform grid is the least demanding approach
between the two and that the free boundary approach provides a more reliable formulation
than the classical truncated boundary one.

** Abstract:** The Blasius flow is the idealized flow of a viscous fluid past an infinitesimally thick, semi-infinite flat plate.
The definition of a non-iterative transformation method for the celebrated Blasius problem is due to T{\"o}pfer and dates more than a century ago.
Here we define a non-iterative transformation method for Blasius equation with a moving wall, a slip flow condition or a 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 literature.

** Abstract:** In a transformation method, the numerical solution of a given boundary value problem is obtained by solving
one or more related initial value problems. Therefore, a transformation method, like a shooting method, is
an initial value method. The main difference between a transformation and a shooting method is that the
former is conceived and derive its formulation from the scaling invariance theory. This paper is concerned
with the application of the iterative transformation method to several problems in the boundary layer theory.
The iterative method is an extension of the Töpfer’s non-iterative algorithm developed as a simple way to
solve the celebrated Blasius problem. This iterative method provides a simple numerical test for the existence
and uniqueness of solutions. Here we show how the method can be applied to problems with a homogeneous
boundary conditions at infinity and in particular we solve the Sakiadis problem of boundary layer theory.
Moreover, we show how to couple our method with Newton’s root-finder. The obtained numerical results
compare well with those available in the literature. The main aim here is that any method developed for
the Blasius, or the Sakiadis, problem might be extended to more challenging or interesting problems. In this
context, the iterative transformation method has been recently applied to compute the normal and reverse
flow solutions of Stewartson for the Falkner–Skan model (Fazio, 2013).

** Abstract:** In this paper, we define a non-iterative transformation method for an Extended Blasius Problem.
The original non-iterative transformation method, which is based on scaling invariance properties, was defined for the classical Blasius problem by T\"opfer in 1912.
This method allows us to solve numerically a boundary value problem by solving a related initial value problem and then rescaling the obtained numerical solution.
In recent years, we have seen applications of the non-iterative transformation method to several problems of interest.
The obtained numerical results are improved by both a mesh refinement strategy and Richardson's extrapolation technique.
In this way, we can be confident that the computed six decimal place are correct.

** Abstract:In this paper, we have defined and applied a non-iterative transformation method to an extended Blasius problem describing a 2D laminar boundary-layer with power-law viscosity for non-Newtonian fluids.
Let us notice that by using our method we are able to solve the boundary value problem defined on a semi-infinite interval, for each chosen value of the parameter involved, by solving a related initial value problem once and then rescaling the obtained numerical solution.
This is, of course, much more convenient than using an iterative method because it reduces greatly the computational cost of the solution.
Furthermore, by using Richardson's extrapolation we define a posteriori error estimator and show how to deal with the accuracy question.
For a particular value of the parameter involved, our problem reduces to the celebrated Blasius problem and in this particular case, our method reduces to the T\"opfer non-iterative algorithm.
In this case, we are able to compare favourably the obtained numerical result for the so-called missing initial condition with those available in the literature.
Moreover, we have listed the computed values of the missing initial condition for a large range of the parameter involved, and for illustrative purposes, we have plotted, for two values of the related parameter, the numerical solution computed rescaling the computed solution.
Finally, we have indicated the limitations of the proposed method as it seems not be suitable, for values of $n>1$, to compute the values of the independent variable where the second derivative of the solution becomes zero or goes to infinity.**