American Journal of Computational Mathematics, 2013, 3, 205-210 Published Online September 2013 (
Some Remarks to Numerical Solutions of the Equations
of Mathematical Physics
Ludmila Petrova
Department of Computational Mathematics and Cybernetics, Moscow State University, Moscow, Russia
Received May 16, 2013; revised June 25, 2013; accepted July 7, 2013
Copyright © 2013 Ludmila Petrova. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The equations of mathematical physics, which describe some actual processes, are defined on manifolds (tangent, a
companying or others) that are not integrable. The derivatives on such manifolds turn out to be inconsistent, i.e. they
don’t form a differential. Therefore, the solutions to equations obtained in numerical modelling the derivatives on such
manifolds are not functions. They will depend on the commutator made up by noncommutative mixed derivatives, and
this fact relates to inconsistence of derivatives. (As it will be shown, such solutions have a physical meaning). The exact
solutions (functions) to the equations of mathematical physics are obtained only in the case when the integrable struc-
tures are realized. So called generalized solutions are solutions on integrable structures. They are functions (depend
only on variables) but are defined only on integrable structure, and, hence, the derivatives of functions or the functions
themselves have discontinuities in the direction normal to integrable structure. In numerical simulation of the deriva-
tives of differential equations, one cannot obtain such generalized solutions by continuous way, since this is connected
with going from initial nonintegrable manifold to integrable structures. In numerical solving the equations of mathe-
matical physics, it is possible to obtain exact solutions to differential equations only with the help of additional methods.
The analysis of the solutions to differential equations with the help of skew-symmetric forms [1,2] can give certain
recommendations for numerical solving the differential equations.
Keywords: Two Systems of Reference; Nonintegrable Manifolds and Integrable Structures; Solutions of Two Types;
Discrete Transitions; Observable Formations
1. Specific Features of Solutions to E qua tio ns
Describing Actual Processes
Let us take the simplest case: the first-order partial dif-
ferential equation
,, 0,
xuppu x (1)
The exact solution (the solution that depends only on
variables, i.e. it is a function) can be obtained in the case
when the derivatives obeying the equations made up the
Let us construct the differential made up of derivatives
that obey the differential equation
where (the summation over repeated indices
is implied). It should be noted that is a skew-
symmetric differential form of the first degree.
It appears that, in the general case when differential
Equation (1) describes any physical processes, the form
made up of derivatives of differential equa-
tion is not a differential.
For the form be a differential, its different-
tial has to be equal to zero.
The differential d
of the form can be
written as
ij j
xx, where
ij ji
px px
are the components of the differential form commutator.
From Equation (1) it does not follow (explicitly) that the
derivatives i
, which obey to the equation (and
to given boundary or initial conditions), are consistent,
that is, their mixed derivatives are commutative. The
components of the commutator ij
is nonzero. There-
fore, the differential form commutator and the different-
tial of the form
are nonzero. This points to the fact
that the differential expression made up of the derivatives
of differential equation is not a differential. That is, the
derivatives of differential equation do not made up a dif-
opyright © 2013 SciRes. AJCM
ferential (without additional conditions).
This means that the solution to Equation (1) ob-
tained from such derivatives is not a function of variables
only. This solution will depend on the commutator
with nonzero value related to inconsistence of de-
To obtain the solution that is a function (i.e., the de-
rivatives of this solution made up a differential), it is
necessary to add the closure condition (vanishing the
form differential) for the form and for the
relevant dual form (in the present case the functional
plays a role of the form dual to
dd 0.
If to expand the differentials, one gets a set of homo-
geneous equations with respect to di
and d (in the
-dimensional space):
dd dd0
px xp
 
It is well known that vanishing the determinant com-
posed of coefficients at di
, is the solvability con-
dition of the set of homogeneous differential equations.
This leads to relations:
 
Relations (4) specify the integrating direction, which
defines an integrable structure, that is, a pseudostructure
(in its metric properties) on which the form
turns out to be closed one, i.e. this form becomes a dif-
On the pseudostructure, which is defined by relation
(4), the derivatives of differential Equation (1) constitute
the differential (on the pseudostructure),
and this means that the solution to Equation (1) becomes
a function. Solutions, namely, functions on the pseu-
dostructures, are so-called generalized solutions. The char-
acteristics, characteristic surfaces, singular points, poten-
tial surfaces, and others are examples of pseudostructures
or their formations.
upx u
It should be underlined the following. The solutions,
which are functions, are obtained only under additional
condition that determines integrable structures. This ad-
ditional condition, as one can see, is vanishing the deter-
minant. Such an additional condition is a condition of
degenerate transformation. The degenerate transforma-
tion executes the transition from tangent nonintegrable
manifold of differential equations to integrable structures
(pseudostructures). (The Legendre transformations are ex-
amples of such a transformation.) The realization of ad-
ditional conditions (that can be caused by any degree of
freedom) leads to the realization of integrable structure
and the transition from the solutions, which are not func-
tions, to generalized solutions (functions).
One can see that the generalized solutions cannot be
obtained by numerical modelling the differential equa-
tion only on original tangent manifold.
The first-order partial differential equation has been ana-
lyzed. Similar functional properties have all differential
equations describing actual processes. Below it will be
shown that the sets of differential equations that describe
actual processes possess such properties.
2. The Properties and Peculiarities of
Solutions to the Equations of Mechanics
and Physics of Continuous Medium
The equations of mechanics and the physics of continu-
ous media (material systems such as gas-dinamical and
cosmological systems, the systems of charged particles
and others) are the equations that describe conservation
laws for energy, linear momentum, angular momentum,
and mass [3] (The set of Navier-Stokes equations is an
example [3]). Such conservation laws can be referred to
as the balance conservation laws since these laws estab-
lish a balance between variations of physical quantities
and appropriate external action.
The equations of conservation laws are differential (or
integral) equations that describe the variation of func-
tions corresponding to physical quantities like the parti-
cle velocity (of elements), temperature or energy, pres-
sure and density. Since these physical quantities relate to
one material system, the connection between them has to
exist. This connection is described by state functional
that specifies the material system state. The action func-
tional, entropy, the Pointing vector, Einstein’s tensor, wave
function and others can be regarded as examples of such
functionals [4]. From the equations of conservation law,
it follows the evolutionary relation for state functional,
which enables one to disclose the properties and peculi-
arities of the solutions to the equations of mechanics and
the physics of continuous media.
2.1. Evolutionary Relation
When studying the solutions to partial differential equa-
tions the conjugacy of derivatives with respect to various
variables was analyzed. When describing physical proc-
esses in continuous media (in material systems) one ob-
tains not one differential equation but a set of differential
equations. And in this case it is necessary to investigate
the conjugacy of not only derivatives but also the conju-
gacy (consistency) of the equations of this set.
The equations are consistent if they can be contracted
into identical relations for the differentials, i.e. for closed
Copyright © 2013 SciRes. AJCM
Let us now analyze the consistency of the equations
that describe the conservation laws for energy and linear
In the accompanying frame of reference, which is tied
to the manifold built by the trajectories of particles (ele-
ments of material system), the equation for energy is
written in the form (see example [5])
Here 1
are the coordinates along the trajectory,
is the functional of the state, 1
is the quantity that de-
pends on specific features of the material system and
external (with respect to the local domain) energy actions
onto the system.
Similarly, in the accompanying frame of reference, the
equation for linear momentum appears to be reduced to
the equation of the form
are the coordinates along the direction nor-
mal to the trajectory,
are the quantities that depend
on the specific features of material system and the exter-
nal force actions.
Equations (5) and (6) can be convoluted into the rela-
dd 1,A
 (7)
where d
is the differential expression
 .
Relation (7) can be written as
here dA
is a skew-symmetric differential form
of the first degree.
[In the case of the Euler and Navier-Stokes equations a
concrete form of relation (8) and its properties were con-
sidered in papers [5]].
Since the equations of balance conservation laws are
evolutionary ones, the relation obtained is also an evolu-
tionary relation.
Relation (8) was obtained from the balance conserva-
tion law equations for energy and linear momentum. In
this relation the form
is that of the first degree. If the
balance conservation law equation for angular momen-
tum be added to the equations for energy and linear mo-
mentum, this form in the evolutionary relation will be a
form of the second degree. And in combination with the
equation of the balance conservation law for mass this
form will be a form of degree 3.
Thus, in the general case, the evolutionary relation can
be written as
where the form degree takes the values
Evolutionary relation obtained from the equations of
the balance conservation laws possesses some peculiarity.
This relation proves to be nonidentical since the differen-
tial form in the right-hand side of this relation is not a
closed form, and, hence, this form cann’t be a differential
like the left-hand side.
Let us analyse the relation (8).
The evolutionary relation d
is a nonidentical
relation as it involves the unclosed skew-symmetric dif-
ferential form dA
. The form
isn’t a close
form since its differential d
is nonzero. The differen-
tial d
can be written as ddK
, where
 
are the components of the differential form commutator
built of the mixed derivatives (here the term connected
with the nonintegrability of the manifold has not yet been
taken into account). The coefficients
of the form
can be obtained either from the equations of the bal-
ance conservation law for energy or from that for linear
momentum. This means that in the first case the coeffi-
cients depend on the energetic action and in the second
case they depend on the force action. In actual processes
energetic and force actions have different nature and ap-
pear to be inconsistent. The commutator of the form
constructed from the derivatives of such coefficients is
nonzero. This means that the differential of the form
is nonzero as well. Thus, the form
proves to be un-
closed and cannot be a differential.
[The skew-symmetric form in evolutionary relation is
defined on the manifold made up by trajectories of the
material system elements. Such a manifold is a deform-
ing manifold. The commutator of the skew-symmetric
form defined on such manifold includes an additional
term connected with the differential of the basis. This
term specifies the manifold deformation and hence is
nonzero. Both terms in the commutator (obtained by dif-
ferentiating the basis and the form coefficients) have a
different nature and, therefore, cannot compensate one
another. This fact once more emphasize that the evolu-
tionary form commutator, and, hence, its differential, are
nonzero. That is, the evolutionary form remains to be
Hence, without the knowledge of a particular expres-
sion for the form
, one can argue that for actual proc-
esses the evolutionary relation proves to be nonidentical.
The nonidentity of the evolutionary relation means that
the initial equations of conservation laws turn out to be
inconsistent, and hence, they are not integrable. The so-
lutions to these equations will not be functions without
additional conditions. They will depend on the commu-
tator, which is nonzero due to inconsistence of the con-
servation law equations.
Copyright © 2013 SciRes. AJCM
The solutions that are functions can be obtained only
under additional conditions when the identical relation
can be obtained from nonidentical evolutionary relation.
This will point to the consistency (but only local, under
additional condition) of the conservation law equations
and the local integrability.
The identical relation can be obtained from nonidenti-
cal evolutionary relation only in the case when the closed
exterior form, which is a differential, is obtained from
unclosed evolutionary form. This is possible only under
degenerate transformation, namely, under the transfor-
mation that does not conserve the differential, since, the
evolutionary form differential is nonzero, whereas the
differential of closed form is equal to zero. The addi-
tional conditions (which are the conditions of local inte-
grability) are the conditions of degenerate transforma-
The additional conditions are caused by any degrees of
freedom. The vanishing of functional expressions such as
the determinant, Jacobian and so on corresponds to the
additional conditions. These conditions can be realized
under changing the evolutionary relation, which is self-
If the conditions of degenerate transformation are re-
alized, from the unclosed evolutionary form with non-
vanishing differential , one can obtain the dif-
ferential form closed on pseudostructure. The differential
of this form equals zero. That is, it is realized the transi-
d0degenerate transformationd
 
is the dual form (which is a metric form).
The condition π is an equation for a certain
pseudostructure π on which the differential of evolution-
ary form vanishes: π. This points to the fact that
the pseudostructure is realized, and the closed (inexact)
exterior form
is obtained on pseudostructure.
On the pseudostructure, from evolutionary relation
it is obtained the identical relation ππ
since the closed exterior form π
is a differential of
some differential form. (This relation will be an identical
one as the left and right sides of the relation contain dif-
ferentials). The identity of the relation obtained from the
evolutionary relation means that on pseudostructures the
original equations for material system (the equations of
conservation laws) become consistent and integrable.
Pseudostructures constitute the integrable surfaces (such
as characteristics, singular points, potentials of simple
and double layers, and others) on which the quantities of
material system desired (such as the temperature, pres-
sure, density) become functions of only independent
variables and do not depend on the commutator (and on
the path of integrating). This are generalized solutions.
They may be found by means of integrating (on inte-
gradle structures) the equations of conservation laws for
material systems.
Since generalized solutions are defined only on real-
ized integrable structures (pseudostructures), they or their
derivatives have discontinuities in the direction normal to
integrable structure [6].
One can see that the integrable structures are obtained
from the condition of degenerate transformation of the
evolutionary relation. The conditions of degenerate trans-
formation (a vanishing of such functional expressions as
determinants, Jacobians, Poisson’s brackets, residues) are
connected with the symmetries, which can be due to the
degrees of freedom of the material systems under con-
sideration (for example, the translational, rotational and
oscillatory degrees of freedom of material system).
The degenerate transformation is realized as the transi-
tion from the noninertial frame of reference to the locally
inertial one, i.e. the transition from nonintegrable mani-
fold (for example, tangent or accompanying) to integrable
structures and surfaces.
Thus, one can see that the solutions to the set of equa-
tions, as well as in the case of a single equation, may be
of two types: the solutions that are not functions, that is,
they depend not only on independent variables, and gen-
eralized solutions, which are functions, and are obtained
only under realization of additional conditions (which de-
termine integrable structures or surfaces). The specific
feature is the fact that they are definded on different spa-
tial objects. Such solutions cannot be obtained by con-
tinuous modelling the differential equations only on a
single spatial odject.
Before turning back to the problems of numerical solv-
ing the differential equations, it should call attention to
the physical meaning of the solutions to these equations.
2.2. Physical Meaning of the Solutions to the
Mathematical Physics Equations
The physical meaning of the solutions to the mathemati-
cal physics equations can be understood by the analysis
of the evolutionary relation.
The evolutionary relation includes the functional that
specifies the system state. Sinse this relation is noniden-
tical, it is impossible to obtain the state functional from
this relation. This points to the absence of the state func-
tion and nonequilibrium state of the material system un-
der consideration. The solutions of the first type just de-
scribe such nonequilibrium state. In this case, the com-
mutator describes the internal force that induces the non-
equilibrium state of material system. The solutions of the
second type (genealized solutions, which are functions)
are obtained under realization of additional conditions
Copyright © 2013 SciRes. AJCM
when the closed exterior forms is obtained from the un-
closed evolutionary form, and the identical relation is
realized. From such relation one can get the state func-
tional and find the state function. This fact will point to
the transition of material system into the locally equilib-
rium state.
The transition of the material system from nonequilib-
rium state into the locally-equilibrium one means that the
unmeasurable quantity described by the nonzero com-
mutator of the unclosed evolutionary differential form,
which acts as an internal force, transforms into the meas-
urable quantity. In material system, this reveals as the
emergence of certain observable formations, which de-
velop spontaneously. Such formations and their manifes-
tations are fluctuations, turbulent pulsations, waves, vor-
tices, and others [7].
It appears that the transition from the solutions of the
first type to the generalized solution corresponds to the
transition of material system from the nonequilibrium
state to the locally equilibrium one that is accompanied
by the emergence of a certain (observable) formation in
material system. The discontinuous functions that corre-
spond to generalized solutions just describe such obser-
vable formations.
Thus we obtain that the discrete realization of general-
ized solution points to the emergence of a certain (ob-
servable) formation in material system that is described
by discontinuous functions corresponding to generalized
It may be also noted that the type of solutions to the
equations describing material systems is of great signifi-
cance for mechanics and physics of continuous media. In
mechanics and physics of continuous media the same
equations are considered (the equations of conservation
laws for energy, linear momentum, angular momentum,
and mass). The set of Navier-Stokes equations is an ex-
ample [5]. However, the approaches to solving these equa-
tions in mechanics and physics are different. In physics
the interest is expressed in only generalized solutions that
are invariant ones and describe measurable physical quan-
tities (but not the process itself), and noninvariant solu-
tions are ignored (even if they have a physical meaning).
The aim of mechanics of continuous media is to describe
the process of the continuous media evolution. And in
this case the numerical methods of solving differential
equations are commonly used without studying the inte-
grability conditions of these equations. The question of
searching for invariant solutions that are realized only
under additional conditions is commonly not posed. That
is, one considers the solutions that are not functions.
Such restricted approaches, both in physics and me-
chanics, lead to nonclosure of relevant theories and this
has some negative points. In mechanics without finding
the generalized solutions it is impossible to describe such
processes as the emergence of vorticity, turbulence and
others. The physical approach enables one to find allowed
invariant solutions, however, in this approach there is no
way to say at what time instant of evolutionary process
one or another exact solution was realized. This does not
also discloses the causality of phenomenon described by
these solutions. It is evident that in mechanics, as well in
physics, it is necessary to seek for solutions of both types.
In particular, in the case of gas-dynamic system such an
approach had been studied in paper [5].
3. On the Problem of Numerical Solving the
Differential Equations
As it was noted, the equations that describe actual proc-
esses are definded on manifolds (tangent, accompanying),
which are nonintegrable. If to model the equations on
such a manifold, one can obtain, without additional con-
ditions, the solutions of only first type, i.e. the solutions
that depend on the commutator with nonzero value caused
by inconsistency of derivatives or equations in the set of
equations. It should be emphasized once more that such
solutions have physical meaning, namely, they describe
the nonequilibrium state induced by the physical proc-
esses proceeded in the system. The generalized solutions,
which are functions and describe discrete formations,
cannot be obtained by modelling the equations only on
original manifold, since they are obtained on integrable
structures that do not belong to original nonintegrable
manifold. Therefore, to obtain the generalized solutions
by numerical simulation, one must use two systems of
reference. One more problem of obtaining the general-
ized solution relates to the fact that the integrable struc-
tures with generalized solution are not initially given, and
they are realized spontaneously in the process of inte-
grating under the realization of additional conditions,
namely, the integrability conditions. (As additional con-
ditions it may serve, for example, the characteristic con-
ditions, the dynamical conditions of the consistency of
equations in the set of equations [8] and so on). To obtain
the integrable structures, it is necessary to trace for the re-
alization of additional conditions, which define the inte-
grable structures, in the process of numerical integrating
the equations on the original manifold. This gives a pos-
sibility to obtain the instant of realization the generalized
In this case, the transitions from inexact solutions to
generalized ones describe the process of emergence of
any observable formations (in particular, such as waves,
vorticity and others), which intensity is definded by gen-
eralized solution.
As it was noted, in mechanics and physics the interest
is expressed in various types of the solutions to equations.
The methods of numerical solving the equations relate to
Copyright © 2013 SciRes. AJCM
Copyright © 2013 SciRes. AJCM
this fact.
In mechanics this is the method of direct numerical
simulating the equations, which is fulfilled on tangent
manifold (being nonintegrable) and enables one to obtain
only inexact solutions.
In physics this is the method of solving equations
when the equations are provided with the integrability
conditions (the conditions of consistency) and this en-
ables one to obtain integrable structures or surfaces, that
is, to go out onto cotangent integrable manifold and ob-
tain exact solutions (the methods of characteristics, sym-
metries, eigen-functions and others are examples of such
methods). The analytical methods may be such methods.
It is possible that the integration of these both methods
will allow solve some problems on numerical integrating
the mathematical physics equations.
It should be emphasized that on integrable structures
the variables of the function desired do not coincide with
the variables of original manifold (since they belong to
different spatial objects). Thus, the coordinates of the
equations for characteristics are not identical to the inde-
pendent coordinates of the initial manifold, on which the
initial equation is defined.
It should be emphasized once more that the existence
of two types of solutions has a deep physical sense. This
peculiarity of the equations enables one to describe the
process of origin of discrete formations such as waves,
vorticities, turbulent pulsations [7] and so on.
It has been shown that the origin of discrete formations
is described by the transition from the solutions of the
first type, which depend on a certain commutator, to the
generalized solution that is a function. To describe the
process of origin of discrete formations, it is necessary,
firstly, have the solutions of the first type, which can be
obtained only by numerical modelling of the equation on
the original nonintegrable manifold (it is impossible to
find such a solution by analytical method), and, secondly,
have the solution of the second type (generalized solu-
tion). This solution can be obtained only on integrable
structure that is definded by the integrability conditions
being realized. Here there is a delicate point. The allow-
able generalized solutions can be obtained by analytical
methods if the integrability conditions are imposed on
the equations. However, in this case it is impossible to
define the instant of realization of generalized solution
and thereby to describe the process of the discrete forma-
tion emergence. The description of evolutionary proc-
esses is possible only either by numerical methods, but
with two frames of reference, or by using simultaneously
numerical and analytical methods.
4. Conclusions
It has been shown that the equations, which describe ac-
tual processes, have the solutions of two types, which are
defined on different spatial objects, and, therefore, can-
not be obtained by continuous numerical simulations of
the equations. The existence of two spatial objects on
which the solutions are defined gives rise to the problems
on which the attention must be focused while numerical
solving the mathematical physics equations.
It should be also emphasized that the methods of nu-
merical solving the equations with account for the exis-
tence of two types of solutions can allow describing the
evolutionary processes such as the emergence of any
discrete formations.
[1] L. I. Petrova, “Exterior and Evolutionary Differential
Forms in Mathematical Physics: Theory and Applica-
tions,”, 2008, 157 p.
[2] L. I. Petrova, “Role of Skew-Symmetric Differential Forms
in Mathematics,” 2010.
[3] J. F. Clarke and M. Machesney, “The Dynamics of Real
Gases,” Butterworths, London, 1964.
[4] L. I. Petrova, “Physical Meaning and a Duality of Con-
cepts of Wave Function, Action Functional, Entropy, the
Pointing Vector, the Einstein Tensor,” Journal of Mathe-
matics Research, Vol. 4, No. 3, 2012, pp. 78-88.
[5] L. I. Petrova, “Integrability and the Properties of Solu-
tions to Euler and Navier-Stokes Equations,” Journal of
Mathematics Research, Vol. 4, No. 3, 2012, pp. 19-22.
[6] L. I. Petrova, “Relationships between Discontinuities of
Derivatives on Characteristics and Trajectories,” Journal
of Computational Mathematics and Modeling, Vol. 20,
No. 4, 2009, pp. 367-372.
[7] L. I. Petrova, “The Noncommutativity of the Conserva-
tion Laws: Mechanism of Origination of Vorticity and
Turbulence,” International Journal of Theoretical and
Mathematical Physics, Vol. 2, No. 4, 2012, pp. 84-90.
[8] V. I. Smirnov, “A Course of Higher Mathematics, V. 4,”
Technology and Theory in the Literature, Moscow, 1957.
(in Russian)