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Пермский национальный исследовательский политехнический университет

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Методы решения жестких и нежестких краевых задач

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}

mat_on_vect(expo_from_plus_step, Y_vspom, Y_rezult); for(int j=0;j<8;j++){

Y[0+i*10+2][j]=Y_rezult[j];//заполнение 2-ой точки

интервала

Y_vspom[j]=Y_rezult[j];//для следующего шага

}

mat_on_vect(expo_from_plus_step, Y_vspom, Y_rezult); for(int j=0;j<8;j++){

Y[0+i*10+3][j]=Y_rezult[j];//заполнение 3-ой точки

интервала

Y_vspom[j]=Y_rezult[j];//для следующего шага

}

mat_on_vect(expo_from_plus_step, Y_vspom, Y_rezult); for(int j=0;j<8;j++){

Y[0+i*10+4][j]=Y_rezult[j];//заполнение 4-ой точки

интервала

Y_vspom[j]=Y_rezult[j];//для следующего шага

}

mat_on_vect(expo_from_plus_step, Y_vspom, Y_rezult); for(int j=0;j<8;j++){

Y[0+i*10+5][j]=Y_rezult[j];//заполнение 5-ой точки

интервала

Y_vspom[j]=Y_rezult[j];//для следующего шага

}

mat_on_vect(expo_from_plus_step, Y_vspom, Y_rezult); for(int j=0;j<8;j++){

Y[0+i*10+6][j]=Y_rezult[j];//заполнение 6-ой точки

интервала

Y_vspom[j]=Y_rezult[j];//для следующего шага

}

mat_on_vect(expo_from_plus_step, Y_vspom, Y_rezult); for(int j=0;j<8;j++){

101

Y[0+i*10+7][j]=Y_rezult[j];//заполнение 7-ой точки

интервала

Y_vspom[j]=Y_rezult[j];//для следующего шага

}

mat_on_vect(expo_from_plus_step, Y_vspom, Y_rezult); for(int j=0;j<8;j++){

Y[0+i*10+8][j]=Y_rezult[j];//заполнение 8-ой точки

интервала

Y_vspom[j]=Y_rezult[j];//для следующего шага

}

mat_on_vect(expo_from_plus_step, Y_vspom, Y_rezult); for(int j=0;j<8;j++){

Y[0+i*10+9][j]=Y_rezult[j];//заполнение 9-ой точки

интервала

Y_vspom[j]=Y_rezult[j];//для следующего шага

}

}

//Вычисление момента во всех точках между краями for(int ii=0;ii<=100;ii++){

Moment[ii]+=Y[ii][4]*(-nju*nn2)+Y[ii][6]*1.0;//Момент

M1 в точке [ii]

//U[2][4]=-nju*nn2; U[2][6]=1.0; u_[2]=0;//Момент M1

}

}//ЦИКЛ ПО ГАРМОНИКАМ ЗДЕСЬ ЗАКАНЧИВАЕТСЯ

for(int ii=0;ii<=100;ii++){

102

fprintf(fp,"%f\n",Moment[ii]);

}

fclose(fp);

printf( "PRESS any key to continue...\n" ); _getch();

return 0;

}

103

Приложение 6. Метод главы 7 (лучший метод из предложенных) и программа на С++ на английском языке

The simplest method of solution of the stiff boundary value problems – the method of solution of the stiff boundary value problems without

orthonormalization

offered by Y. I. Vinogradov and A. Y. Vinogradov.

Alexey Yurievich Vinogradov (Candidate of science /Physics and Mathematics/)

(14 November 2011, Moscow, Russia)

The authors: Yuri Ivanovich Vinogradov (1938 year of birth), doctor of science (Physics and Mathematics) and Alexey Yurievich Vinogradov (1970 year of birth), candidate of science (Physics and Mathematics).

The idea of overcoming the problems of computing by dividing an integration interval into matching sectors belongs to Y. I. Vinogradov (doctor of science /Physics and Mathematics/). His doctorate thesis was based on that material (including that idea).

Proposed realization of this idea (division and matching) through matrix theory formulas i.e. by means of the matrix exponents (or by other words by means of a matrizant – the Cauchy matrix, Cauchy-Krylov functions) belongs to A. Y. Vinogradov (candidate of science /Physics and Mathematics/).

AT PRESENT MOMENT THIS METHOD IS THE BEST ONE. The method was finalized in the mid October 2011 and a C++ software program (that is given in this paper as well) was finalized and checked by comparative computations on 14 November 2011.

1. The simplest method of solution of the stiff boundary value problems.

Let’s divide a boundary value problem integration interval e.g. for three segments. We will have points (nodes) including the boundaries:

x0 , x1 , x2 , x3 .

We have the boundary conditions in the following form:

104

UY (x	) u,
0
VY (x	) v.
3

We can write sector matching matrix equations as follows:

Y (x	) K(x		x )Y (x ) Y				(x		x )

0		0	1	1			0		1
Y (x ) K(x			x	)Y (x	) Y		(x		x	)

1	1		2	2			1		2
Y (x	) K(x		x	)Y (x	) Y		(x		x	)
		2						2
2			3	3					3

It is possible to rewrite these formulas in the more convenient form:

EY (x	) K(x		x )Y (x ) Y				(x		x )

0		0	1	1			0		1
EY (x ) K(x			x )Y (x		) Y		(x		x	)

1	1		2	2			1		2
EY (x	) K(x		x )Y (x ) Y				(x		x	)
		2						2
2			3	3					3

where E - unit matrix.

Then in the combined matrix form we obtain a system of the linear algebraic equations in the following form:

U		0		0					0				Y (x	)				u
													Y (x	)
E	K (x		x )	0					0				0		Y		(x		x )
E	K (x	0	x )	0					0						Y		(x	0	x )
		0	1										Y (x	)				0	1
													Y (x	)
0		E	K (x		x		)		0				1		Y		(x		x	)
0		E	K (x		x		)		0						Y		(x		x	)
			1			2							Y (x	)				1	2
													Y (x	)
0		0		E				K (x		x	)		2		Y		(x		x	)
0		0		E				K (x	2	x	)				Y		(x	2	x	)
									2	3			Y (x	)				2	3
													Y (x	)
0		0		0				V					3					v
0		0		0				V										v

This system is solved by the Gauss method by discrimination of the basic element.

Solution in the points located between the nodes is obtained by solving the Cauchy problem with initial conditions in the ith node:

Y (x) K(x xi )Y (xi ) Y (x xi )

It appears to be that there is no need to apply orthonormalization for the boundary value problems for stiff ordinary differential equations.

2. Detailed description of the designations used above.

Let’s use as an example a system of differential equations of the cylindrical shell of the rocket – a system of the ordinary differential equations of 8th order (after partial derivatives partitioning by the Fourier method).

105

A system of the linear ordinary differential equations has the following

form:

Y (x) AY (x) F (x)

where		Y (x)	-	- an unknown vector-function of the 8x1 dimensionality
problem,	Y (x)		-	a derivative of the	unknown vector-function of 8x1

dimensionality,			A	- a square matrix of	the coefficients of the differential
equation	of	8x8	dimensionality, F (x) -		a vector-function of the external

influence on the system of 8x1 dimensionality.

The boundary conditions have the following form:

UY (0) u,

VY (1) v,

where Y (0) - is the value of the unknown vector-function at the left boundary x=0 of the 8x1 dimensionality, U - a square horizontal matrix of the coefficients of the boundary conditions of the left boundary of the 4x8 dimensionality, and u - an external influence on the left boundary of the 4x1 dimensionality.

Y (1) is the value of the unknown vector-function at the right boundary x=1 of the 8x1 dimensionality, V - a square horizontal matrix of the coefficients of the boundary conditions of the right boundary of the 4x8 dimensionality, and v - an external influence on the left boundary of the 4x1 dimensionality.

In case when a system of differential equations has a matrix with constant coefficients A = const the solution of the Cauchy problem has the following form:

Y (x) e	A( x x

where e	A( x x0)	E A(x x0 ) A	(x
	A( x x0)	2

where E is a unit matrix. Matrix exponent can be named

denoted in the following form:

					x
0)	Y (x0 ) e			Ax	e	At	F (t)dt
	Y (x0 ) e				e		F (t)dt
					x0
x0 )		2			3			3	/ 3! ...,
x0 )			/ 2! A (x x0 )						/ 3! ...,

as the Cauchy matrix or matrizant and be

K(x x0 ) K(x x0 ) eA( x x0)

106

Then Cauchy problem solution can be written in the following form:

where

			Y (x) K(
				x
Y		(x x0 ) e	Ax	e	At	F (t)
Y		(x x0 ) e		e		F (t)
				x0

x x		)Y (x	) Y		(x x	)

	0	0			0
dt	is a vector of the partial solution of the

heterogeneous system of the differential equations.

A matrix exponent (the Cauchy matrix) multiplication feature is known from the matrix theory [1]:

K(x	x ) K(x		x	) K(x	x	) ... K(x	x ) K(x		x )
i	0	i	i 1	i 1	i 2	2	1	1	0

In case when a differential equation system has a matrix with variable coefficients A A(x) it is offered to find a solution of the Cauchy problem by means of the Cauchy matrix multiplication feature. That is the integration integral is split into small segments and the Cauchy matrixes are computed approximately in these small segments by means of the formula for a constant matrix in the exponent. Then the Cauchy matrixes computed in these small sections are multiplied:

K(x	x ) K(x		x	) K(x	x	) ... K(x	x ) K(x		x )
i	0	i	i 1	i 1	i 2	2	1	1	0

where the Cauchy matrixes are computed approximately by the following formula:

K(x	x ) e	A( xi) xi	exp(A(x ) x )

i 1	i		i	i

where xi xi 1 xi

Instead of the formula for computation of the vector of the partial solution of the heterogeneous system of the differential equations in the form [1]:

		x
Y (x x	) e Ax		e At F (t)dt
0
		x0

it is proposed to use the following formula for each individual segment of the integration interval:

107

							x	j
								j
Y		(x j	xi ) Y		(x j	xi ) K (x j	xi ) K(xi		t)F (t)dt
Y		(x j	xi ) Y		(x j	xi ) K (x j	xi ) K(xi		t)F (t)dt
							x	i
								i

Correctness of the above mentioned formula is justified by the following:

x

j

Y

(x j

xi ) exp(A(x j

xi )) exp(A(xi

t))F (t)dt

x

i

x

j

Y

(x j

xi )

exp(A(x j xi ))exp(A(xi

t))F (t)dt

x

i

x

j

Y

(x j

xi )

exp(A(x j xi

xi t))F (t)dt

x

i

x

j

Y

(x j

xi

) exp(A(x j

t))F (t)dt

x

i

x

j

Y

(x j xi ) exp(Axj ) exp( At)F (t)dt

x

i

x

Y

(x xi )

exp(Ax) exp( At)F (t)dt

x

i

as was to be justified.

Computation of the vector of the partial solution of the heterogeneous system of differential equations is carried out by representing the Cauchy matrix under the integral sign in the form of the series and integrating this series element by element:

108

x

j

Y

(x j

xi ) Y

(x j xi ) K (x j xi ) K (xi t)F (t)dt

x

i

x

j

K (x j xi ) (E A(xi t) A

2

(xi t)

2

/ 2! ...)F (t)dt

x

i

x

j

x

j

x

j

K (x j xi )(E F (t)dt A (xi

t)F (t)dt A

2

/ 2! (xi

t)

2

F (t)dt ...).

x

i

x

i

x

i

This formula is correct for a system of differential equations with the

constant coefficient matrix

A =const.

F (t) vector can be considered in the segment

(x j xi ) approximately in the

form of	the constant value F(xi ) constant	that allows its taking off from the
integral	sign that results in very simple series for computation in the
considered segment.

In case of the differential equations with variable coefficients the averaged matrix Ai A(xi ) of the coefficients of the differential equation system

can be used in the above mentioned formula for each segment.

Let’s consider an option when the integration interval steps are selected

as sufficiently small that results in considering

F (t)

vector in the segment

(x j

xi ) approximately in the form of the constant value

F(xi ) constantresulting

in taking off this vector from the integral signs:

x

j

x

j

x

j

Y

(x j xi ) K (x j xi )(E dt A (xi t)dt A

2

/ 2!

(xi t)

2

dt ...)F (t).

x

i

x

i

x

i

It is known that at T=(at+b) we have:

n	dt	1	T
T
		a(n 1)

n 1

const (at n -1).

In our case we have:

	(b - t)n dt	1	(b - t)n 1	const	(at n -1).

		(-1)(n 1)

109

Then we obtain

	x	j					1
		j
			(xi	t)	n	dt		(xi
					n

	x						n 1
	x	i
		i

x	j

)

n 1

.

Then we obtain the series for computation of the vector of the partial solution of the heterogeneous system of differential equations in the small

segment (x j

xi ) :

Y

(x

x ) K(x

x ) (E A(x

x

2

(x

x

)

2

/ 3! ...) (x

x ) F(x ).

j

j

) / 2! A

j

i

i

i

j

i

j

i

i

If the segment (x j

xi

)

is not a small one it is possible to divide it by sub-

segments and then it would be possible to propose the following recurrent (iteration) formulas for particular vector computing:

We have Y (x) K(x x0 )Y (x0 ) Y	(x x0 ) .

We have the formula for an individual sub-segment as well:

							x	j
								j
Y		(x j	xi ) Y		(x j	xi ) K (x j	xi ) K(xi		t)F (t)dt
Y		(x j	xi ) Y		(x j	xi ) K (x j	xi ) K(xi		t)F (t)dt
							x	i
								i

We can write:

Y (x ) K(x		x	)Y (x	) Y		(x

1	1	0	0			1
Y (x	) K(x	x )Y (x ) Y				(x

2	2	1	1			2



x	0	)
	0

x1 )

,

.

Let’s substitute Y (x1 ) in Y (x2 ) and obtain the following:

Y (x	) K(x				x )[K(x		x	)Y (x		) Y		(x	x )] Y			(x		x )
				2													2
2					1	1	0	0				1		0						1
K(x		2	x )K(x x )Y (x ) K(x						2	x )Y (x x ) Y							(x		2	x )
			1		1	0	0				1		1	0						1

Let’s compare the obtained expression with the formula:

Y (x2 ) K(x2 x0 )Y (x0 ) Y (x2 x0 )

and we will obtain evidently that:

K(x2 x0 ) K(x2 x1 )K(x1 x0 )

so, we obtain the formula for a particular vector:

110

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