A Boundary Value Problem with Strong Degeneracy and Local Splines
I. G. BUROVA, G. O. ALCYBEEV
The Department of Computational Mathematics,
St. Petersburg State University,
7-9 Universitetskaya Embankment, St.Petersburg,
RUSSIA
Abstract: - A new method for solving the singular one-dimensional boundary value problem with a strong
degeneracy is proposed in this paper. In the case of the strong degeneration of the differential equation, the
boundary condition is set only at one end of the interval. This method is based on the use of the polynomial and
non-polynomial Lagrangian and Hermitian type local splines and the variational method. The use of splines of
Hermitian type with the first level is convenient if it is needed to obtain simultaneously a solution and the first
derivative of the solution at the grid nodes. Next, it is possible to construct the solution between the grid nodes
using the same spline approximation formulas. The non-polynomial splines help us to construct a more accurate
solution. The results of solving a one-dimensional boundary value problem with strong degeneracy are
presented in this paper.
Key-Words: - Boundary value problem, strong degeneracy, degenerate differential equation, variational
method, local splines, polynomial splines, trigonometrical splines, Hermitian type splines.
Received: July 11, 2024. Revised: November 14, 2024. Accepted: December 9, 2024. Published: December 31, 2024.
1 Introduction
Degenerate equations arise when solving many
applied problems, for example, in gas dynamics [1],
in modeling the motion of viscous droplets
spreading over a solid surface [2], [3], the study of
isoperimetric problems for probability measures,
[4]. Results for a class of nonlinear degenerate
Navier problems associated with the degenerate
nonlinear elliptic equations are given in [5].
In paper [6], the nonlinear degenerate elliptic
differential problem: , in
(0, 1), is discussed. This equation
is a simplified version of the nonlinear Wheeler
DeWitt equation. The Wheeler DeWitt equation
appears in quantum cosmological models and it is
used to model quantum states of the universe and
study the qualitative behavior of the universe wave
function.
The question of the existence of solutions for
degenerate equations was studied in detail in [7],
[8].
Among the papers published recently, we note
[9], [10].
The novelty of the paper [9] is that the authors
find proper weights under which the existence,
uniqueness, and regularity of solutions in Sobolev
spaces are established.
The main result of the paper [10] is the
establishment of the conditions for the existence or
not of eigenvalues of the linearization.
Boundary value problems for a degenerate
elliptic equation are often studied in the theory of
partial differential equations. The study of
differential equations with a coefficient at the
highest derivative that vanishes has been carried out
in many works. Such equations are obtained by
studying variable-type partial differential equations,
as well as by establishing asymptotic expansions of
bisingular problems. In the paper [11], the authors
consider a degenerate parabolic problem occurring
in the spatial diffusion of a biological population. In
paper [12], the authors consider a degenerate
parabolic equation occurring in the gas filtration
problems.
When modeling physical processes, we often
come to the need to numerically solve boundary
value problems and initial boundary value problems
with degeneracy.
When solving boundary value problems, the B-
splines or piecewise linear functions are often used,
[13], [14], [15].
In paper [14] the solution to the one-
dimensional Stationary Transport Equation is given.
Here, for the construction of the approximate
solution, a piecewise linear function and
trigonometrical polynomials are used.
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The [15], is devoted to the construction of the
Hermitian- type-splines.
Paper [16] is devoted to the solution of the
nonstationary integro-differential equation with a
degenerate elliptic differential operator.
Special difficulties arise in the case when
solving degenerate equations. Problems connected
with solving degenerate equations are discussed in
[17], [18], [19], [20].
In paper [21], a new approach to the local
improvement of an approximate solution which has
been obtained with the finite element method is
developed.
In paper [22], an algorithm of the adaptive-grid
for one-dimensional boundary value problems of the
second order is constructed and the corresponding
approximation theorems are established.
The cases are known when the difference
approximation of an elliptic differential equation
turns out to be non-elliptic. When applying
projection and variational methods, orthogonal
polynomials are often used as the basis functions. In
this case, it is necessary to distinguish between the
main and natural boundary conditions. The basis
functions must satisfy the main boundary conditions
but may not satisfy the natural ones.
We consider the approximation of the solution
in the degenerate Sobolev spaces. This means that
the weight function can vanish or go to infinity at
one of the ends of the interval, [15]. In our work, we
assume that the weight function can vanish at the
left end of the interval.
In this paper, we discuss the numerical solving
the problem:
In this equation, the function vanishes only
at . Thus, this equation degenerates at the
point 0. If , then the degeneration is
called the strong degeneration and the boundary
condition must be set only at one end of the interval.
So we put .
In our paper in Section 1.1 we give general
remarks about the variational method, Section 2 is
devoted to the local splines. Section 3 is about the
construction of the solution of the boundary value
problem with the variational method.
1.1 General Remarks
Consider the equation
Here , the function is
measurable, bounded and non-negative. Suppose
that for . Of particular
interest is the case ,
where:
Let us denote by the operator:
The domain of the definition of this operator is
taken to be a set of functions satisfying the
condition: and
are absolutely
continuous on any segment where .
If , then these functions are continuous on
We will need the following facts, [16].
If the integral
converges, then we take .
If the integral
diverges, then we take .
Let us denote by the operator .
If the integral
converges, then the operator (and the operator )
is positive definite, [16].
If the integral
diverges, then the operator is positive, [16].
It is known that if the operator A is a positive
definite, then the equation Au=f has a unique
generalized solution. This solution belongs to the
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energy space . This solution also belongs to
In our case, consists of functions that
are absolutely continuous on any segment
and satisfy the condition . The
energy scalar production and norm are calculated
using the formula
On the interval (0,1) we will solve the following
boundary value problem with strong degeneracy:
where
To solve this problem, we construct a uniform
grid Divide the interval [0,1] into
parts. Thus, we have constructed a grid of nodes ,
,
.
We will look for an approximate solution
of problem (1) as shown below:
where are the basis splines, and are some
parameters determined from the condition of the
minimum of the functional:
The problem of minimizing this functional (3)
on the space of functions of the form (2) leads to the
system of equations:
It can be written in the short form:
In more detail, this system can be written as:
The coefficients and the right side of which are
calculated using the formulas:
2 Local Spline Application
Next, we consider the application of Lagrangian-
type splines and Hermitian-type splines to the
solution of a boundary value problem with the
strong degeneracy.
2.1 The Application of Splines of the Second
Order of Approximation
First, we apply the polynomial basis splines of the
second order of approximation.
2.1.1 The Polynomial Basis Splines
The support of the basis spline consists of two
parts: and :
and
Obviously, the derivatives of the basis splines
have the form:
and
The solution of the equation on the grid interval
we take in the form:
where the polynomial basis splines of the second
order of approximation on every grid
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interval have the following form
2.1.2 The Application of the Polynomial Basis
Splines to the Boundary Value Problem
Let us take the equation:
We construct the right part of the equation
in accordance with the exact solution
We are interested in the
values of the function in the internal nodes ,
. We know the value of the function in
the node . Let's consider the case,
We need to solve the system , where
the matrix has elements . The matrix turns
out to be a symmetric one and it has a banded
tridiagonal form.
In order to solve this problem with the
variational-difference method, we have to calculate
the integrals:
The right side of the system has the
elements as followed:
Next, we need to solve the system of equations
2.1.3 The Trigonometrical Basis Splines
The support of the basis spline consists of the
two parts: and :
and
Also, we have the relation:
The trigonometric splines are convenient for solving
equations of the form:
Next, we consider the application of Hermitian-
type splines to the solution of a boundary value
problem with strong degeneracy.
In the next section we will discuss the use of
local splines of a non-zero level for solving
boundary value problems with degeneracy. We
would like to recall the construction of the fourth-
order splines of the first level. By the level we mean
the number of derivatives of a function used to
construct the approximation. The peculiarity of
using these local splines is that we obtain a
continuously differentiable approximation to the
solution to boundary value problems. In addition, it
is easy to construct the first and the second
derivatives of this approximate solution.
2.2 The Application of Splines of the Fourth
Order of Approximation of the First
Level
Let us recall how the approximation is constructed
using local splines of a non-zero level. We call the
approximation level the number of derivatives that
are used to construct the approximation.
The approximation using local splines of a non-zero
level is constructed on every grid interval in the
following form:
Here are the basis splines. If
then the level of the approximation is
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We assume that the support of each of the basis
splines consists of two grid intervals. Let us
assume that at each grid node the values of the
function and its first derivative are known.
In this case, we construct an approximation of
function , in the form:
We assume that supp =supp
. Let the basis functions be
determined from the conditions:
From these conditions we obtain a system of
equations for determining the basis functions on the
interval :
Having solved the system of equations, we find
the basis splines on the grid interval .
The basis splines of the first level have the form:
The plots of basis splines on the interval
are presented in Figure 1, Figure 2, Figure
3 and Figure 4.
Fig. 1: The plot of the basis spline ,
Fig. 2: The plot of the basis spline
Fig. 3: The plot of the basis spline ,
Fig. 4: The plot of the basis spline ,
Now we get formulas for the approximation
function ,
, (5)
Similarly, we construct basis splines on the
adjacent grid interval .
To construct a solution, we also need formulas
for basis splines on the interval
We assume that the support of the basis spline is
supp
The superlative point of the basis spline
will be called the point with coordinates
.
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Fig. 5: The plot of the basis spline ,
Fig. 6: The plot of the basis spline ,
We have the formulas:
By combining basis splines with a common
superlative point and taking into account the support
of the basis spline, we obtain local basis splines
shown in Figure 5 and Figure 6.
We use the constructed splines to solve the
boundary value problems.
If the function , then the next
estimation of the approximation of function is
valid when
Here
Splines of the fourth order of approximation and
the first level were applied in the Least Squares
Method, [23]. In the next section we discuss the
application the splines of the fourth order of
approximation and the first level for the solving the
boundary value problem.
3 The Construction of the Solution of
the Boundary Value Problem
Let us consider the following problem:
Note that to solve this problem using splines of
the first level, we will need the value of the first
derivative at the right end of the interval [0, 1]: here
we have.
On the interval [0, 1] we construct an ordered
grid of nodes. First, let the grid nodes be equally
spaced with step . The matrix of the system of
equations (according to (5)) will have a
block form:
where every has a band structure. Non-zero
elements are those lying on the main diagonal,
supradiagonal and subdiagonal.
We take The elements of the
matrix have the following form:
Note that the minimum number of nodes to solve
our problem is . If then we have:
In this case we have:
If then we have:
.
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Similarly we have,
and
We calculate the right side of the system of
equations using the formulas:
The elements of vector have
the form:
The elements of vector have
the form:
Next, we have to solve the system
,
where
To construct a continuously differentiable
smoothing solution of the boundary value problem,
we use the solution , , of
and the splines of the fourth order of approximation
of the first level
4 The Numerical Experiments
Example 1. We solve the boundary value problem
where the exact solution is
We apply the splines of the fourth order of
approximation and the first level. We take
The plot of the error of the solution and its first
derivative are given in Figure 7 and Figure 8. The
numbers of the grid nodes are plotted along the axis
.
Fig. 7: The plot of the error of the solution
Fig. 8: The plot of the error of the first derivative of
the solution
Example 2. We solve the boundary value problem:
where =1.5 and the exact solution
We apply the splines of the fourth order
of approximation and the first level. After we solve
the system of equations and obtain the values
we can connect the points of the grid solution using
the rule:
,
The plot of the error of the solution and its first
derivative are given in Figure 9 and Figure 10. The
numbers of the grid nodes are plotted along the axis
.
Fig. 9: The plot of the error of the solution
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Fig. 10: The plot of the error of the first derivative
of the solution
Example 3. We solve the boundary value problem
The exact solution is . We
take =1.5 and we apply the trigonometric splines.
The plot of the error of the solution are given in
Figure 11 and Figure 12. The numbers of the grid
nodes are plotted along the axis .
Fig. 11: The plot of the error of the solution
We obtained an approximate solution at the grid
nodes (grid solution). Next, we can connect the
points of the grid solution using the rule:
Here we used the trigonometric splines.
The plot of the error of the solution is shown in
Figure 12.
Fig. 12: The plot of the error of the solution
Example 4. We solve the boundary value problem
The exact solution is . We
apply the trigonometric splines. The plot of the error
of the solution is given in Figure 13. The numbers
of the grid nodes are plotted along the axis .
Fig. 13: The plot of the error of the solution
5 Conclusion
In the work, polynomial and trigonometric splines
of the second order of approximation were used to
solve a boundary value problem with a strong
degeneracy. Experiments have shown the advantage
of trigonometric splines if the equation has a
trigonometric right part and trigonometrical
coefficients. In the case of using splines of non-zero
level, we simultaneously find a continuously
differentiable solution and its derivative.
The next work will consider the use of spline
approximation with wide support and a non-uniform
grid for solving a boundary value problem with a
strong degeneracy.
Acknowledgement:
The authors are gratefully indebted to a resource
center of St. Petersburg University for providing the
Maple package (Pure ID: 119548249).
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
- I. G.Burova developed algorithms and conducted
numerical experiments,
- G. O. Alcybeev executed the numerical
experiments.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
The authors are gratefully indebted to a resource
center of St. Petersburg University for providing the
Maple package (Pure ID: 119548249).
Conflict of Interest
The authors have no conflicts of interest to declare.
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