The main
purpose of the research is the fundamental question of the possibility of using
Functional-Voxel modeling tools in problems based on obtaining a solution to a
differential equation. The implementation of this problem is in demand due to
the progressive development of FV-modeling tools in design and control problems,
i.e. - it is proposed to model algorithms for solving such problems as: tracing
the path with obstacle avoidance [1], algorithms for calculating physical
characteristics [2,5], algorithms for solving mathematical programming problems
[6], differentiation and integration of the Functional-Voxel model of a function
[7], etc.
The
isocline method was chosen as the basis of the research not by chance -it is
based on the preliminary construction of a tangential field on a given area and
the further application of its data in the construction of integral curves. The
principle of FV-modeling is similar and requires the preliminary construction
of an appropriate FV model to organize an algorithm for solving a problem.
A lot of research has been devoted to
solving differential equations using various approaches, and each of these
approaches has the right to exist and to be applied according to their benefits
and advantages [8,9]
One of the graphical methods for solving an
ordinary differential equation
is
based on the construction of isoclines defined as lines along which the value
of first derivative is constant [10, 11, 15]. At the same time, automation of
this approach requires the generation of a tangential surface in the
space
and the construction of isolines on it, cut by unidirectional segments
(tangents to the integral curve). Figure 1 depicts an example of finding the integral
curves for a differential equation
,
which
is often demonstrated in various works:
|
|
(1)
|
Here the isoclines are
co-directed and parallel to the
axis,
so it makes no sense to display them. But the segments cutting them with a
fixed step, demonstrating the direction of the tangent to the integral curve at
a given point, visually represent the general solution of this differential
equation. The disadvantage of this graphical approach is the difficulty of
using these visual data obtained to build a fairly accurate picture of the
integral curves. It is also difficult to contribute such an approach in
automated calculations. To do this, it is necessary to express an indefinite
integral from the initial differential equation and calculate the corresponding
coefficient C, which is a traditional analytical approach.
The lack of
analytical expression in the obtained graphical result of the isocline method
does not allow the researcher to apply it with confidence in scientific calculations,
since it rather carries the visibility of the integral curves’ shape and, as a
rule, has the substantial loss of accuracy. Since the range of values of the
isoclines in this case is a continuous surface, then there exists an integral
curve passing through the origin at
C=0. Let's conduct a numerical
experiment and determine the remaining roots for such an integral curve.
Figure 1
–
An Example of the construction of
integral curves by isoclines
On integrating the expression (1) we
obtain:
|
|
(2)
|
Let's find the
roots of the resulting equation (2). Since here we deal with a cubic equation, then,
in addition to the origin point, the
axis
can be intersected twice more. Equating the right part to zero and rearranging we
obtain:
|
|
(3)
|
Obviously, the first root is
and
it remains to solve the quadratic equation in parentheses to get the remaining
roots:
|
|
(4)
|
In Figure 1, the integral curve (2) shown in red is located in the
middle and intersects the
axis at coordinates (4). It is clear that
the other two integral curves are constructed for
and
.
The functional
voxel method (FV-method) [13] provides - on a given area of the analytical
function - filling with the local functions describing a linear law for each
minimal neighborhood of a point on the area, which makes it possible to apply
in further calculations not just a number, but the corresponding analytical
expression with all the advantages that follow from this.
Let's try to
figure out how, applying the principles of functional voxel modeling, to
automate the isocline method for computer application.
In
order to solve equation (1), we will define a certain area (by analogy with the
selected example in Figure 1, we will choose the area
).
In the Functional
Voxel method, the partial derivative
is
considered as the ratio
,
where
and
are the angles of
deviation of the unit gradient vector from the axes
and
,
respectively. We can say that:
|
|
(5)
|
Taking
into account the following equalities:
and
,
we
obtain coefficients for the arguments of the tangent equation for the point
under consideration, but passing through the origin (local function):
|
|
(6)
|
Then we
display the specified range of values for each of the cosines as a separate
image. A legitimate question arises: why should one tangent surface be
converted into two cosine surfaces? In fact, the answer “lies on the surface”.
Cosine values are normalized to the interval
,
which
makes it easy to convert them into a color palette
which
is suitable for computer representation in the form of raster images:
|
|
(7)
|
Such
information is not only illustrative, but also relatively compact compared to a
two-dimensional array of corresponding real values. Additionally, the problem
of representing infinitely large values for vertical tangents, etc. disappears.
Taking the function
as
the argument
on the
area, we get two raster images responsible for storing
and
on a given area
(Fig.2). Further, such images will be called M-images, as is customary in terms
of the FV method.
We show
that the obtained graphical information in the form of two M-images is
sufficient to automate the algorithm for constructing the integral curve shown
in Fig.1.
|
|
|
|
a)
|
b)
|
Figure
2 – An example of constructing M-images for surfaces: a)
and
b)
In
order to organize the sequence of actions of an algorithm for constructing an
integral curve using the M-images obtained in Figure 2, it is necessary to
determine the relation between the dimensions of the real area of the function
specified by real numbers and the M-image resolution, which is integer and
contain information about the number of points in two directions of the raster.
Following
the problem discussed earlier, the function area is set by the parameters:
The
dimensions of the M-image are
Let's
determine the scaling factor along the axes, which ensures the transition from
the function area to the image and back in the calculations:
|
|
(8)
|
Since
the M-image contains 400 points along the
axis, and the
origin for the region is located in the middle, and there is also information
that the desired integral curve passes through the origin, then it is proposed
to build such an integral curve first in the positive direction of the
semi-axis from the point
(0,0),
and then in the negative direction.
Let's
set the starting point to the origin, recalculating it to the M-image
coordinates:
|
|
(9)
|
It is obvious
that at coordinates
(0,0)
on the M-image, the point will be in the
middle of the window with coordinates
(200, 200).
For the resulting
point, there is a specific color on both M-images
,
which,
when converted back, becomes the cosine value:
|
|
(10)
|
To
obtain a local equation at a point with coordinates
(0,0),
we find the
third component of the gradient of the neighborhood of this point:
|
|
(11)
|
In the
case of the first point, when both coordinates are zero
,
which
means that the line described by this local function really passes through the
origin.
To
determine the value of the next point of the integral curve, we perform a shift
along the
axis
by a step
:
|
|
(12)
|
Having obtained
the solution of the integral curve at the next point, we proceed to it:
|
|
(13)
|
Now we
calculate the
coordinates
again using the formula (9), we get the color
on two
M-images at a new point
()
in
order to determine the new
and
for it using the
formula (10). We define
for
the obtained point using the formula (11) and proceed to the finding a new
point of the integral curve (12, 13) and then repeat the process until one of
the coordinates reaches the boundary of the region.
Similarly,
the algorithm is built in the opposite direction from a given point, with the
only difference that the
parameter
is determined for the point
,
since
we assume that the current point also belongs to the previous neighborhood.
Figure
3 shows a bundle of integral curves for
Ñ
=[-3,-1,-2,0,1,2,3],
superimposed on the M-image with
value
mapping for clarity.
On the
resulting integral curve, it can be seen that the roots of its function at
C=0
correspond to the required values (4).
Consider
the example
,
where
in addition to the argument
,
the
function
itself
is also present in the equation. To do this, we will also choose one of the
cases often considered in classical textbooks [16]:
|
|
(14)
|
An
image with the construction of isoclines and the resulting integral curves, as
well as the automatic construction of an integral curve by the FV method is
shown in Figure 4. It is not difficult to make sure that the integral curve
passing through the origin has a single root at the extreme point. We equate
the derivative to zero to construct the isocline of extremes:
|
|
(15)
|
Figure 3 – The result of the FV-constructing
an integral curve for (1)
The
resulting straight line of the isocline passes through the origin, which means
that the point of the extremum of the integral curve for
Ñ
=
0 will
be located at the origin. We will set the starting point for the algorithm to
work right there and start the process of constructing the integral curve. The
result confirming the correctness of the algorithm is shown in Figure 4a. In
Figure 4b, the integral curve passing through the origin is shown in red.
|
|
|
|
a)
|
b)
|
Figure 4
–
The
result of the FV-constructing integral curves for (14): a) the isocline method,
b) FV-method
For the
algorithm to work, it is necessary to pre-construct M-images of the mapping on
a given area of
and
as shown in
Figure 5.
Figure
5 – mapping the isocline region as a FV-model for the equation (1.14)
As the
next example, it is proposed to consider the quadratic expression of the
differential equation:
|
|
(16)
|
The
result of the construction of isoclines and obtaining the integral curves in
the traditional version is shown in Figure 6.
Figure 6 – The construction
of integral curves of the equation (16) by the isocline method
Let's
visually highlight for ourselves in Figure 6 the integral curve passing through
the origin. For the algorithm to work, we define the corresponding M-images
describing a given area of the tangential field decomposed into cosines. The
result of constructing such M-images is shown in Figure 7. Figure 8
demonstrates the solution of the problem by the FV-method.
Figure
7 – Representation of the isocline region in the form of a FV model for the
equation (16)
Figure
8 – Integral curves of the equation (16)
Let's
consider the last test example of a first-order differential equation involving
a periodic function:
|
|
(17)
|
We will
compare the result with the traditional image included in many textbooks
(Fig.9) [15].
Figure 9 –
Construction of integral curves of the equation (17)
Let's
construct M-images to solve equation (17) (Fig. 10) and apply the proposed
algorithm for constructing an integral curve to these two M-images. The result
of constructing integral curves for equation (17) is shown in Figure 11.
The
conducted research shows that the application of Functional Voxel modeling to
solving first-order ordinary differential equations makes it possible to
transfer the graphical isocline method to an automated basis, with the only
difference that the information field formed in this case does not display the
tangent value, but is based on information about the components of the unit
gradient vector in the investigated area of the differential equation.
Figure 10 – Representation
of the isocline region in the form of a FV model for the equation (17)
Figure 11 –
Construction of an integral curve for equation (17) passing through the origin
Since
the isocline method is also applied to solving some second-order equations, we
will demonstrate the solution of an example taken from [14] which is displayed
at (Fig.12) for comparison. Consider the equations in a reduced form:
|
|
(18)
|
A new
variable
is
introduced. Then we have:
|
|
(19)
|
and
equation (18) takes the form of a first-order equation:
|
|
(20)
|
Let's consider
an example of solving a differential equation of the form:
|
|
(21)
|
We
assume that
Then
equation (21) takes the form:
|
|
(22)
|
Figure 12 –
Construction of integral curves by the isocline method for the equation
(19)
We
apply the Functional Voxel approach to solving equation (19). Let us pay
attention to the fact that in all examples of such a reduction of the initial
differential equation to the first order, division by the function
is implied, which
leads along the
axis
to a discontinuity of the continuous surface (Fig.13), since in order to obtain
M-images, the function (22) is transformed into a function:
|
|
(23)
|
To
construct integral curves, we use the regions
and
alternately,
i.e. we divide the domain of definition of a function into two subdomains
before and after the
axis.
Figure 13
a
and
b
show the result of constructing integral curves
at
Ñ
=1,
Ñ
=2 and
Ñ
=3
for
the positive area, and at
Ñ=-1,
Ñ=-2
and
Ñ=-3
for
the negative.
|
|
|
à)
|
|
|
|
b)
|
Figure
13 – Construction of integral curves by the FV method for the equation (19):
a) the solution
in area
,
b) the
solution in area
It is
not difficult to combine these two images in the further calculations to obtain
the comprehensive picture.
The
conducted research has shown the possibility of using the Functional Voxel modeling
in solving ordinary differential equations of the form
.
The
solution of the equation of the form
is
demonstrated. In the future, it is proposed to consider the use of the FV modeling
in solving applied problems based on the application of the first and second
order ordinary differential equations, the development of the principles for
the algorithm for ordinary differential equations of the form:
and
.
Since
the Functional Voxel model allows us to analytically describe complex geometry
in a given area, then its application in problems of fluid or gas motion as a
basis for modeling the differential laws formed in this case seems to the
authors very promising and relevant.
The
research was carried out within the framework of the scientific program of the
National Center for Physics and Mathematics, direction No. 9 "Artificial
intelligence and big data in technical, industrial, natural and social
systems".
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