We have found the constructive necessary and sufficient conditions for solvability and the scheme for constructing solutions of a nonlinear autonomous boundary value problem for a nondegenerate differential-algebraic system. The nonlinear boundary value problem for the autonomous system significantly differs from similar autonomous boundary value problems by its dependence on an arbitrary continuous vector function. We have also constructed a convergent iterative scheme for finding approximate solutions of the nonlinear autonomous boundary value problem for the nondegenerate differential-algebraic system in critical and noncritical cases. Proposed in this paper scheme for studying the nonlinear autonomous boundary value problem for the nondegenerate system of differential-algebraic equations can be transferred to degenerate systems of differential-algebraic equations in the same way. Differential-algebraic equations, nonlinear autonomous boundary value problems.
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2023 Copyright for this paper by its authors. problem of finding solution [1] of a differential-algebraic system which satisfy the boundary condition
Let and are (
× )dimensional matrices and ( , )is a dimensional vector function. We will call a weakly nonlinear autonomous periodic differential-algebraic boundary value problem the ( , ): (⋅, )∈ 1[ , ( )], ( ,⋅)∈ [0, 0], (0): = ∗ ′ = +
( , ), ℓ (⋅, )= .
ℓ (⋅, ): [ , ( )] → .
A z0 B z0 z0 () Rq 1[ , ∗] of the Noether ( ≠ )differential-algebraic generating boundary value problem The solution of the problem (1), (2) is found in a small neighbourhood of the solution 0( )∈ The vector function ( , )we assume to be continuously differentiable with respect to the unknown ( , ) in a small neighbourhood of the solution of the generating problem and continuously differentiable with respect to a small parameter in a small positive neighbourhood of zero. The matrix we assume, in general, to be rectangular: ≠ , or square, but degenerate [2, 3, 4]. Under the condition
∗ = 0 the generating system (3) is reduced to the traditional system of ordinary differential equations [5] here
rank : = < .
In addition, + is a pseudo-inverse (by Moore-Penrose) matrix, ∗ is a matrix-orthoprojector: ′0 = + 0 + 0 0( ); ∗: → ( ∗), 0 is a ( × 0) matrix formed from 0 linearly independent columns of ( × ) matrixorthoprojector
: → ( ), 0( )∈ 0 is an arbitrary continuous vector function. Under the condition (4) the system (1) we will call nondegenerate. In the critical case for a fixed vector function 0( )∈ [ , ∗] under the condition the generating problem (3) has parametric family of solutions [5]
∗ ≠ 0, : = ℓ 0(⋅) ∗ { − ℓ [ 0 0( )](⋅)} = 0 0( , )= ( ) + [ 0 0( )]( ), ∈ .
Here 0( )is a normal ( 0( )= )fundamental matrix of the homogeneous part of the differential system (5), and [ 0 0( )]( )is the generalized Green’s operator [3, 5] of the generating periodic differential-algebraic boundary value problem (3), [ 0 0( )]( ) is the generalised Green’s operator [3, 5] of the Cauchy problem ( )= 0 for the differential-algebraic system (3). The matrix ∗ is formed from linearly independent rows of the matrix-orthoprojector ∗, and the matrix is formed from linearly independent columns of the matrix-orthoprojector . Under the condition (4) the system (1) leads to the traditional system of ordinary differential equations
′ = + + 0 0( )+ + ( , ); The periodic boundary value problem for an autonomous system (6) significantly differs [1, 6] from similar autonomous boundary value problems by its dependence on an arbitrary vector function 0( )∈ [ , ∗]. In addition, only in exceptional cases, the autonomous boundary value problem (1), (2) is solvable on a segment of fixed length.
Example 1. Let us find a solution to the autonomous nonlinear differential-algebraic boundary value problem for equation
The condition (4) is satisfied, so the system (8) is nondegenerate. In this case, the matrix is rectangular, and (4) (5) (6) (7) (8) of nonlinear autonomous differential-algebraic boundary value problem (8) on the segment [0, ] of fixed length. 2. Necessary condition of solvability
An autonomous boundary value problem for the system (7) is significantly different from similar nonautonomous boundary value problems; in that the length of the interval [ , ( )], on which we determined the solution of a nonlinear boundary value problem for the system (7), in general, is unknown. We will use the technique [1, 6], which consists in representing the unknown function through a new unknown
Lemma. Assume that the autonomous differential-algebraic boundary value problem (1), (2) for a fixed constant 0 ∈ 0 under the conditions (4) and (6) represents the critical case ∗ ≠ 0 and has the solution (10).
( , )= col( (1)( , ), . . . , ( )( , )), ( )(⋅, )∈ 1[ , ( )],
( )( ,⋅)∈ [0, 0], = 1, 2, . . . , , which for = 0 turns into the generating ( , 0)= 0( , ∗). Then the vector č 0 satisfies equation
The first components of ∗ ∈ of the root of the equation (10) determine the amplitude of the generating solution 0( , ∗), in small neighborhood of which the desired solution of the initial problem (1), (2) can exist. In addition, from the equation (10) the value ∗, which determines the first approximation to the unknown function Function ( )is to be determined in the process of finding a solution of the boundary value problem for the system (7). The technique consists in replacing the of the independent variable and finding the solution of the nonlinear boundary value problem (2), (7) and the function ( ), as a the condition of solvability of the nonlinear boundary value problem (2), (7) takes the form function of a small parameter. In the critical case, under the condition (6) for a fixed function 0( ) ∗ {(1 + ( )) − ℓ [ ( )( +
( , )+ 0 0( )+ +(1 + ( )) + ( ( , ), )](⋅)} = 0.
Using the continuity of the nonlinear vector function ( ( , ), ), on in a small positive neighborhood of zero, we find the limit for → 0 in equality (9) and obtain the necessary condition F (č 0) P { K[ ( A B z0 (s cr ) PA0 0 (s) AZ (z0 (s cr ) 0)]()} 0
Qd of existence of a solution to the boundary value problem (1), (2) in the critical case; here č 0 cr Rr1 ( )= ∗ + ( ∗ − ) ( ) ( )∈ [0, 0], (0): = ∗. = + ( − )(1 +
( )) 1( )= ∗ + ( ∗ − ) ∗, can be found. If the equation (10) has no real roots, then the initial differential algebraic problem (1), (2) has no desired solutions.
The equation (10) will be further called the equation for the generating constants of the autonomous nonlinear differential-algebraic boundary value problem (1), (2). The statement of the lemma generalizes the corresponding results of [1, 6, 7, 9] on the case of an autonomous nonlinear differential-algebraic boundary value problem (1), (2), namely, on the case ≠ . Similarly to [1, 6, 9] we demonstrate that the periodic problem for the system (7) is solvable provided that the roots of the equation for the generating constants (10) are simple. 3. Sufficient condition of solvability
Assume that the equation for the generating constants (10) has real roots. Fixing one of the solutions č 0 ∈ +1 of the equation (10), we come to the problem of finding solutions of the problem (1), (2) in the neighbourhood of the generating solution
( , )= 0( , ∗)+ ( , ) 0( , ∗)= ( ) ∗ + [ 0 0( )]( ).
The deviation of ( , )from the generating solution is determined by the boundary value problem ′ = + + { + + 0 0( )+ (1 + ) + ( , )}, ℓ (⋅, )= 0. (11) Using the continuous differentiability of the function ( , )with respect to both the first and second arguments in the neighbourhood of the generating solution 0( , ∗)and the point = 0, we get the expansion of this function
( 0( , ∗)+ ( , ), )= ( 0( , ∗),0)+ + 1( ) ( , )+ 2( )+ ( 0( , ∗)+ ( , ), ), where
( , ) ( , ) A1( )= | , A2( )= | ∂z = 0( , ∗), ∂ = 0( , ∗),
=0, =0, The residual ( ( , ), )of the expansion of the function ( ( , ), )of higher order of smallness on and in the neighbourhood of the points = 0 and = 0, than the first three terms of the expansion, so ( , )| = 0( , ∗), =0, = 0( , ∗),
=0, Let us denote by ( × ( + 1))the dimensional matrix ≡ 0, ( , )
| ≡ 0, ( , )
| = 0( , ∗), =0, ≡ 0 0 = − ∗ ℓ { +[ ∗ + 1( )] ( ); 0 0( )+ + 0( , ∗)}(⋅).
In the critical case, under the condition (6) for a fixed function 0( )and the solution č 0 of equation (10) the solution of the nonlinear boundary value problem (11) has the form here, and
( , )= ( ) ( )+ (1)( , ); (1)( , ): = [ +
+ 0 0( )+ (1 + ) + ( , )]( ).
( ): = ∗ + ( ), ( ): = col( ( ), ( ))∈ [0, 0], while the condition of solvability of the nonlinear boundary value problem (11) leads to the equation 0 ( )= − ∗ ℓ { ∗ + (1)( , )+ +[ 2( )+ ( ( , ), )]}(⋅), which solvable under the condition [1, 10, 11] 0∗ ∗ ℓ { ∗ + (1)( , )+ +[ 2( )+ ( ( , ), )]}(⋅)= 0. In particular, the condition of solvability of the nonlinear boundary value problem (11) is satisfied in the case of Thus, under the condition (12) we get an operator system that is equivalent to the problem of finding solutions of the boundary value problem (1), (2)
0∗ ∗ = 0. ( , )= 0( , ∗)+ ( , ), ( , )= ( ) ( )+ (1)( , ), (1)( , ): = [ + + 0 0( )+ (1 + ) + ( , )]( ), ( )= − 0+
∗ ℓ { ∗ + (1)( , )+ +[ 2( )+ ( ( , ), )]}(⋅).
The operator system (13) belongs to the class of systems for which the method of simple iterations is applicable [1]. Thus, the following theorem is proved.
Theorem. Assume that the autonomous differential-algebraic boundary value problem (1), (2) for a fixed constant 0 ∈ 0 under the conditions (4) and (6) represents the critical case ∗ ≠ 0. Suppose also that the equation for the generating constants (10) has real roots. Under the conditions (4), (6) and (12) for the fixed function 0( ) and for the solution č 0 of equation (10) the operator system (13) is equivalent to the problem of finding solutions of the boundary value problem (1), (2) and has at least one solution. To find the solution to the operator system (13) the method of simple iteration is applicable.
For finding the solution of the autonomous boundary value problem (1), (2) in the neighbourhood of the generating solution, the Newton-Kantorovich method can also be used [8].
Example 2. Let’s find a solution to the autonomous nonlinear differential-algebraic boundary value problem for the equation In this case, the condition (6) is satisfied, so the linear part of the problem (14) has a solution that depends on the continuous function 0( ): =
3 : The equation for the generating constants (10) has a real root 0( , )= ( ) + [ 0 0( )]( ), ( ): = 0( ); [ 0 0( )]( )= [ 0 0( )]( ), ∈ 3.
č 0 cr R4 cr 1 1 0 R3 0 here and where here ( , ) ( , ) ( , ) ( , ) ( ( , ), ): = (
), ℓ (⋅, ): = (0, )− ( , ).
Since the condition (4) is satisfied, the system (14) is nondegenerate. In this case, the matrix is rectangular, and 0 = 1 ≠ 0, 0 = (1), 0 0 so the homogeneous part of the system (14) has a solution that depends on an arbitrary continuous function; we put 0( ): = 3 : 0( , )= 0( ) + [ 0 0( )]( ), ∈ 3, (13) which corresponds to the full rank matrix thus, the condition of solvability (12) of the nonlinear boundary value problem (14) is satisfied.
Proposed in this paper scheme for studying a nonlinear autonomous boundary value problem for a nondegenerate system of differential-algebraic equations [12,13,14,15,16,17,18] can be, analogously [5], transferred to degenerate systems of differential-algebraic equations in the same way. For finding the solution of an autonomous boundary value problem for a nondegenerate system of differentialalgebraic equations (1), (2) the main requirement is the solvability requirement (12), which is equivalent to the condition of simplicity of the roots of the equation for the generating constants (10) [1, 6]. The scheme for studying a nonlinear autonomous boundary value problem for a nondegenerate system of differential-algebraic equations proposed in this paper can be similarly transferred to systems of differential-algebraic equations with a variable rank of derivative matrix. 4. References solvability analysis of differential algebraic equations, Computational 2012.