The authors proposed a general way to find particular solutions for overdetermined systems of PDEs previously, where the number of equations is greater than the number of unknown functions. In this paper, we propose an algorithm for finding solutions for overdetermined PDE systems, where we use a method for finding an explicit solution for overdetermined algebraic (polynomial) equations. Using this algorithm, the solution of some overdetermined PDE systems can be obtained in explicit form. The main difficulty of this algorithm is the huge number of polynomial equations that arise, which need to be investigated and solved numerically or explicitly. For example, the overdetermined hydrodynamic equations obtained earlier by the authors give a minimum of 10 million such equations. However, if they are solved explicitly, then we can write out the solution of the hydrodynamic equations in a general form, which is of great scientific interest.
Deep Dive into Towards identification of explicit solutions to overdetermined systems of differential equations.
The authors proposed a general way to find particular solutions for overdetermined systems of PDEs previously, where the number of equations is greater than the number of unknown functions. In this paper, we propose an algorithm for finding solutions for overdetermined PDE systems, where we use a method for finding an explicit solution for overdetermined algebraic (polynomial) equations. Using this algorithm, the solution of some overdetermined PDE systems can be obtained in explicit form. The main difficulty of this algorithm is the huge number of polynomial equations that arise, which need to be investigated and solved numerically or explicitly. For example, the overdetermined hydrodynamic equations obtained earlier by the authors give a minimum of 10 million such equations. However, if they are solved explicitly, then we can write out the solution of the hydrodynamic equations in a general form, which is of great scientific interest.
Partial differential equations (equations of mathematical physics) are often found in various fields of mathematics, physics, mechanics, chemistry, biology, and in numerous applications [1,2]. The authors proposed a general way to reduce the dimension for arbitrary systems of partial differential equations (PDE), which allows reducing the PDE systems in volume to systems on the surface [3][4][5][6]. From the point of view of numerical methods, reduction is beneficial in the sense that it is not necessary to solve differential equations in the whole space. It is required to supplement the original PDE system with additional constraint equations and make transformations. On the basis of this idea, the authors of [5,6] also proposed a method for finding particular solutions for overdetermined PDE systems, where the number of equations is greater than the number of unknown functions, which are very important for numerical calculations. In this method, finding solutions reduces to solving systems of ordinary implicit equations. In the articles of the authors [3][4][5][6][7], overdetermined equations of hydrodynamics are given, as well as methods for overriding any systems of PDE. In this paper, we propose an algorithm for finding solutions for overdetermined PDE systems, where we use a method for finding an explicit solution for overdetermined algebraic equations. Using this algorithm, the solution of some overdetermined PDE systems can be obtained in explicit form.
It is required to find solutions for an overdetermined system of first-order pn partial differential equations with respect to unknowns
where … ,… ,…
. We apply the method described in articles [5,6]. We consider the following system of equations of the form
with respect to unknowns
Here,
11 0…
We also have
12 …
Consider the matrix
with respect to unknowns
But functions from multi-indices
, contrary to ( 6), ( 7) let be determined from the conditions:
We also have
Differentiating expression (12) with respect to the variable
where
Here, indices are taken such that, 1
that is, there are no terms with an index
. Therefore, the recurrence relation (18) is correct.
Solution method. We recursively determine implicit equations from (18) using ( 4), ( 5) and solve them. We calculate the rank (10) on these solutions. Calculate manually the number of variables real S N (11)! and compare with the rank of the matrix (10). If they coincide, then we have a solution to system (2) (see ( 4)). Obviously, we are only interested in the case:
We also have the following estimates [5,6]
where the minimum is realized at 12 …
The system of equations ( 12), if it is sufficiently “good”, can be somewhat simplified.
Consider an overdetermined system of pn partial differential equations of the first with respect to unknowns
A method for solving system of overdetermined algebraic equations by their reduction Equations ( 2) and ( 12) with respect to unknowns
algebraic. There are various methods for solving polynomial equations. In particular, the far nontrivial Bukhberger algorithm [8] is widespread. In our case, we can apply the following simple method to the solution of systems of overdetermined algebraic equations.
Consider an overdetermined system of two polynomials of order n 0 0
We substitute (24) into (23). We have
As a result, we obtain a polynomial of the form:
where
We multiply both sides of (26) by x and substitute instead of n x its expression from (24):
As a result, we obtain a polynomial of the form:
where
Thus, we have obtained two polynomials (26) and (31) of order no more than 1 n . We show that the sets of solutions of the system of equations ( 22), (23) and system (26), (31) coincide, under condition (27). We have shown that (26), (31) follows from ( 22), (23). Let us prove the opposite. Let x be some solution (26), (31). Multiply both sides of (26) by x . Then equation ( 29) holds. On the other hand, equation ( 30) is satisfied, since this is another form of notation (31). Subtract equation (30) term by term from (29). Then we find that
We substitute (35) into (34). Then 0 pb pc q
x r aa (37)
The system of equations ( 36), ( 37) is equivalent to the system (33), (34) under the condition
If condition (40) is not satisfied, but (38) is true, then system (33), (34) has no solutions.
In the case of an overdetermined system of more than two polynomials, one can prove a similar statement and find the condition when the original system is equivalent to a system of the same number of polynomials, but of an order less than that of the original system.
Consider the general case of an overdetermined system of
where
The system of equations (42) can be considered as an overdetermined system of 1 m algebraic equations with respect to a variable m x , but with variable coefficients (43). Under certain conditions, it can be equivalently reduced sequentially to a system of 1 m linear eq
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