Abstract
Authors
D.V. Ionescu
Institutul de Calcul
Keywords
?
Paper coordinates
D.V. Ionescu, L’application de la méthode des approximations successives à l’intégration numérique des équations différentielles. (French) Bull. Math. Soc. Sci. Math. Phys. R. P. Roumaine (N.S.) 3 (51) 1959 423–431.
About this paper
Journal
Bull. Math. Soc. Sci. Math. Phys. R. P. Roumaine
Publisher Name
DOI
Print ISSN
Online ISSN
Note: this work has been republished in Romanian:
D.V. Ionescu, Application of the method of successive approximations to the numerical integration of differential equations, Acad. R. P. Romîne Fil. Cluj Stud. Cerc. Mat., 11 1960, 273–286.
[Aplicarea metodei aproximărilor succesive la integrarea numerică a ecuațiilor diferențiale]
google scholar link
??
Paper (preprint) in HTML form
Original text
Rate this translation
Your feedback will be used to help improve Google Translate
BULL. MATH. of the Society. Sci. Math. Phys. of the RPR
Volume 3 (53)n^(@)4, 1959
Volume 3 (53)n^(@)4, 1959
APPLICATION OF THE METHOD OF SUCCESSIVE APPROXIMATIONS TO THE NUMERICAL INTEGRATION OF DIFFERENTIAL EQUATIONS
BY
Let us consider the differential equation
y^(')=f(x,y)
where the functionf(x,y)is defined and has continuous first- and second-order partial derivatives in the rectangleDdefined by inequalities
x_(0) <= x <= x_(0)+a,quad|y| <=b.
Let y(x) denote the integral of this equation that satisfies the conditiony(x_(0))=0and eitherepsia given positive number.
In this work, we will show, by applying the method of successive approximations, that we can determine on the interval[x_(0),x_(0)+a)a networkGammaknotsx_(0),x_(1),dots,x_(n)and a calculation algorithm for number calculationsy_(i)^((z))Ors=0,1,2,dots, v so as to have
|y(x_(i))-y^((y))| < 2epsi
on all network nodesGamma
To fix the number v, we take into account the method of successive approximations, and for the choice of the number of nodes n and the calculation algorithm for the numberssum_(i)^((s)),s=0,1,dots, v we will use the trapezoid quadrature formula.
To fix the number v, we take into account the method of successive approximations, and for the choice of the number of nodes n and the calculation algorithm for the numberssum_(i)^((s)),s=0,1,dots, v we will use the trapezoid quadrature formula.
This theorem was communicated at the Colloquium on Mechanics [1] held in Bucharest (October 25-29, 1959) and an extension to second-order hyperbolic partial differential equations was communicated at the Colloquium on the Theory of Partial Differential Equations [2] held in Bucharest (September 21-26, 1959)
§ 1. Dilential equations
- Consider the differential equation
(1)
y^(')=f(x,y)
where the function /(x,y)eal continue dane Ie metangleDdefined by frog infegiltes
(2)quadx_(b) <= x <= x_(b)+a.quad|y| <=b
and satisfies the condition of LIPSCHITZ
(3)quad|//(x,F)f(x,y)|cdots A|Fy|
OrHASis a constant.
Under these conditions, we know that the differential equation (1) has a unique integral of zero forx-x_(0)- kile cat definie wor l'intervalle[x_(0)x_(0)+h_(1)]where
(4)
(2)quadx_(b) <= x <= x_(b)+a.quad|y| <=b
and satisfies the condition of LIPSCHITZ
(3)quad|//(x,F)f(x,y)|cdots A|Fy|
OrHASis a constant.
Under these conditions, we know that the differential equation (1) has a unique integral of zero forx-x_(0)- kile cat definie wor l'intervalle[x_(0)x_(0)+h_(1)]where
(4)
h_(1)=min(a,(b)/(M))
the numberMholding a saperiodic boundary of|//(x,y)|in the rectangleDThe
complete collectiony(x)can be obtained by the method of successive approximations. We construct the function map{y^((n))(x)}, where
(b)
complete collectiony(x)can be obtained by the method of successive approximations. We construct the function map{y^((n))(x)}, where
(b)
{:[y^((m))(x)-int_(-4)^(3)f[xi","0]d xi],[(6)y^((n))(x)-int_(alpha_(4))^(pi)f[xiy^((0-1))(xi)]d xi.]:}
We know that the mother
y^((1))(x)+sum_(i=1)^(oo)[y^((1))(x)-y^((0-1))(x)]
is absolutely and uniformly convergent on the interval[x_(0),x_(0)+h_(1)]and that the sum of this series represents the integral of the differential equation (1) which satisfied the conditiony(x_(0))-0We can read the complete text.y(x)in the form
y(x)=y^((n))(x)+sum_(n=1)^(oo)[y^((n))(x)-y^((n-1))(x)]
and we demonstrate the inequality
|y(x)-y^((1))(x)| < (M)/(A)e^(mu_(1))((Amu_(1))^(gamma+3))/((nu+2)!).
Given a positive number, we can then choose the smallest natural number v to have
{:(7)(M)/(A) < ((Ah_(1))^(r+1))/((v+2)T) < epsilon:}
and we will have
(8)
(8)
|y(x)-y^((y))(x)| < c.
the natural number y once chosen, remain fixed and play an important role in the numerical integration of the differential equation (1).
____
9. For the numerical integration of the differential equation (I), adjust the conditiony(x_(0))=0, we will make new hypotheses about the function / (x, y), which are related to the numerical interpretation procedure that we will now give in this work.
____
9. For the numerical integration of the differential equation (I), adjust the conditiony(x_(0))=0, we will make new hypotheses about the function / (x, y), which are related to the numerical interpretation procedure that we will now give in this work.
We will assume that the functionf(z,g)sit des derivatives participles par npport &xot ay of the first and of the round order, continuous in the triangle D. Under these conditions the number A of Lipachitz's inequality (3) is an upper bound of|(del f)/(del y)|in the rectangleD.
We demonstrate that the gos functions (x) donsers by the formals (6) and (6) cat of the first and second order derivatives continuous by the interval[x_(0),x_(0)+h_(1)]We can calculate upper bounds of
|(dy^((n))(x))/(dx)|*|(d^(n)(x))/(dx^(2))|,quad(x=0,1.2 dots.v)
over the interval[x_(0),x_(0)+h_(1)]by using the upper horns of
|//(x,y)|.quad|(del f)/(ix)|*|(del f)/(del y)|
in your rectangleDIt
follows that the functions
follows that the functions
p^(2n)(x)=//|x,y^(0-1)(x)|
an upper bound of|(P^(**)(x))/(P^(3))|
Fors=0,1,dots,v_(0)on the interval[x_(0),x_(0)+h_(1)]
The number N will play an important role in the approximate calculation of the integrals (0) and (6) fors=0,1,dots,v.
The number N will play an important role in the approximate calculation of the integrals (0) and (6) fors=0,1,dots,v.
We will denote by h a positive number of 6́finite by
(9)
(9)
A=min(a(b-delta)/(M))
where & is a positive number, domá, awes small. I eat tvident que nons avoush < h_(1).
With these hypotheses, everyone could proceed to the anmerical integration of the equation
over the interval[x_(6),x_(6)+h], in arithmetic programming and an algorithm for calculating numbers go for s=0,1,dots, v so as to have on the nodes of the network I
over the interval[x_(6),x_(6)+h], in arithmetic programming and an algorithm for calculating numbers go for s=0,1,dots, v so as to have on the nodes of the network I{:(10)|y^((1))(x_(1))-y^(p)| <c.:}
of your trapezoid quadrature formula{:(11)int_(0)^(0)f(x)dx=(beta-alpha)/(2)U(alpha)+//(beta))+R_(1).:}
on
(12)
(12)
R=int_(0)^(0)((x-alpha)(x-beta))/(2)f^(')(x)dx.
If N, ext an upper bound of|//^(@)(x)|because the interval[alpha,beta]we will have
(13)
(13)
|R| <= ((beta-alpha)^(3))/(12)N_(1).
4 Let's divide the interculle [x_(0),x+b] that are equal parts by pointsx_(1),x_(2),dots,x_(n-1). Northern islandsx_(3),x_(1)dots dotsx_(1)Orx_(1)=x_(6)+hform a rewasGammaNow let's calculate the values ​​of the functionsg^((4))(x), Fors∼0,1,dotsver les meads do réwer Γ.
Name we have first
5^(oo)(x_(1))=int_(z_(1))^(z_(1))//∣xi,0]d xi.
È applying the quadrature formula (II) to each interval [x_(a).x_(1)][x_(2),x_(2)]dots,[x_(i-1).x_(i)]and by adding, we have the formula
{:(HA)y^((n)){:(x_(i)):}=y_(i)^((n))+R_(i)^((n)).:}
ભ્યે
(Ib)
and
(Ib)
and
n^(p)=(h)/(2n)[f(x_(0),0)+f(x_(i),0)+2sum_(i=1)^(i-1)f(x_(i),0)]
|R^(p)| < (H^(3))/(12pi^(3))iN < (n^(3))/(12pi^(3))N
because(i)/(n) < ILet e
​​be a positive number, which we will determine later. We will choose the number n, the smallest natural number, such that
​​be a positive number, which we will determine later. We will choose the number n, the smallest natural number, such that
{:(16)(h^(3))/(12n^(3))N < c_(1):}
Since the number n is chosen in this way, it will remain fixed in the following, and we are sure of it in formula (14)
(17)
(17)
|R^((n))| < c
i. For In function y(1)(x). non nome
y^(11)(x_(1))-int_(x_(0))^(J_(1))∣xiy_(1)(E_(1))E_(0)
ei pur lea neuidnx_(1)we have
y^((1))(x_(1))-int_(x_(0))^(x_(4))f[xiy_(5)(xi)]epsi_(6).
We proceed as in no. 4, and apply the quadrature formula of the intervals[x_(0),x_(1)].[x_(1),x_(1)]dots,[x_(1),dots,x_(1)]none nurons
(18)
where
(10)
(18)
where
(10)
g^((1))(x_(1))-[g^((1))]+eta^((1))
(II)
|r_(1)^((n))| < c_(1)
and where
(20)
(20)
[y^((1))]-(h)/(2n)|f[x_(0),y^((n))(x_(0))]+f[x_(1),y^((n))(x_(1))]+2sum_(i=1)^(L-1)|[x_(1),y^(n)(x_(1))]|
Introduce the numbers yp^((1))by formula
(21)quady^((1))-(h)/(2n){f(x_(0),0)+f(x_(1),y_(1)^((n)))+2sum_(i=1)^(i-1)f(x_(1),y_(i)^((n)))}.
(21)quady^((1))-(h)/(2n){f(x_(0),0)+f(x_(1),y_(1)^((n)))+2sum_(i=1)^(i-1)f(x_(1),y_(i)^((n)))}.
We will have
(22)
(22)
[y^((1))]-y^((1))+p^((1))
Or
p^((1))-(h)/(2n){f(x_(1),y(x_(1)^((n))))-f(x_(1),y_(1)^((n)))+2sum_(-1)^(4)U(x_(1),y^((n))(x_(1)))-l(x_(1),y^(n))}
Finally, taking into account Lipsciirz inequality (3) and the integlities (17), we have
|p_(i)^((1))| < (1)/(2n)(2i-1)dc_(1)
that is to say
(29)
(29)
|p^((1))| < Delta hc_(1)
Returning to formula (18) we can berire
(24)
(24)
y^((1))(x_(1))=y^(v)+B^(v)
where, according to the inequalities (19) and (23) we have
that is to say
(26)
that is to say
(26)
{:[|R^((1))| <= epsi_(1)+dAepsi_(1)],[|R^((1))| <= (1+R)epsi_(1)]:}
od nons avons note
(26)
(26)
K-Ah
- Let us move on to the general case. Suppose that nons avone demonstrates that
(27)
|g^((t-1))(x_(j))-y_(j)^(f-4))| < (1+K+cdots+K^(-1))epsi_(1)
for jI. 2.... n, od̀
y^(f-1))-(1)/(2n)|f(x_(0),0)+f(x_(1),y_(1)^(f-m))+2sum_(m=1)^(i-1)|(x_(1),y_(0)^(f-m_(1)))|
and we demonstrate that by introducing the number yso by the formula
(28)
(28)
y^((1))-(h)/(2n){f(x_(0),0)+f(x_(0),x^((0-1)))+2sum_(j=1)^(j-1)f(x_(1),y_(j)^((0-1)))}
ones avons aresi
(29)
(29)
|y^((1))(x_(1))-y^((2))| <= (1+K+dots+K^(+))r_(1),
ED, indeed, mons have
y^((1))(x_(1))=int_(x_(0))^(oo)int[xiy^((0-1))(xi)]d xi.
By proceeding as in No. O, we have
{:(30)y^((0)){:(x_(i)):}=[y^((0))]+eta^((0)):}
where
(31)
(31)
|gamma^((0))| < r_(1)
and
(32)[y^((i))]=(h)/(2pi){[x_(0),y^((k-1))(x_(0))]+//[x_(1),y^((0-1))(x_(1))]+2sum_(j=1)^(j-1)//[x_(1),y^((0-1))(x_(j))]}
(32)[y^((i))]=(h)/(2pi){[x_(0),y^((k-1))(x_(0))]+//[x_(1),y^((0-1))(x_(1))]+2sum_(j=1)^(j-1)//[x_(1),y^((0-1))(x_(j))]}
Formulas (28) and (32) showed that
{:[[y^((i))]=y^((i))+p^((0))],[ hat(p)^((i))=(1)/(2n){//[x_(i),y^((o-1))(x_(i))]-//[x_(i),y_(i)^((o-1))]+2sum_(j=1)^(i=1)U(x_(i),y^((o-1))(x_(i)))-//(x_(i),y_(i)^((o-1)))]}.]:}
Taking into account the lapschitz inequality and inequalities (27), we have
|p(p)| < (h)/(2n)(2i-1)A(1+K+dots+K^(2-1))r_(1)
"
13)
13)
| hat(p)^((i))| <= K+K^(2)+dots+K^(0))c_(1).
II rAulte Alon que nons pouvais Ariur
y^((n))(x_(1))-y_(0)^(n)+k_(1)^(n).
Or
R^((n))-r^(n)+ hat(p)^(n).
Tonant compo des integaliten (31) et (33). il réwulte que nous avons
|R^((n))| <= (1+K+dots+K^('))_(n_(1))
This proves that inequality (29) is demonstrated.
7. We have therefore found an algorithm for calculating the numbers s₀ using formulas (16) and (28). It now remains for us to determine the number ₀. We will first take the numberepsilon_(1)in such a way that the middle member of Pinégalité (20) for s - y is smaller than c. We will therefore prove
7. We have therefore found an algorithm for calculating the numbers s₀ using formulas (16) and (28). It now remains for us to determine the number ₀. We will first take the numberepsilon_(1)in such a way that the middle member of Pinégalité (20) for s - y is smaller than c. We will therefore prove
c_(1) <= (epsi)/(1+K+dots+K).
Mnis there is still one condition for the number a. For formula (28) to make sense for s - v, the point with coordinates (x_(i).hat(y)^(-1)) is found in the rectangular D. Of identity
y^((n-1))--[y^((n-1))(x_(k))-y^((-1))]+y^((n-1))(x_(l))
and from the definition of the number h, we do not deduce that
|ℜ^((-1))| <= |R^((-1))|+b-delta < (1+K+dots+K^(-)}c_(1)+b-delta
or even
郎
郎
|m^((p-1))| < (1+K+dots+K)c_(1)+b-c
To have|y^((p-1))| <= b, it is necessary thatc_(1)checks the condition
c_(1) <= (3)/(1+K+dots+K)
So, we choose the number ca using the formula
{:(34)epsi_(1)-min((epsi)/(1+K+dots+K)","(lambda)/(1+K+dots+K)):}
lo nombroc_(1)building ainai prevish, the coordinate point(x_(1),y^((n)))Or3=0,1,dots,v-1is located in the rectangleDIndeed, by proceeding as above, we have
|y^((j))| < |k^((k))|+b-delta < (1+K+dots+K^('))c_(1)+b-b
or enooro
|y^((0))| < (1+K+dots+K^('))epsi_(1)+b-delta < b.
- Tranat accounts for inequalities and identity
!g(x)-y^((n))(x) <= e.quady^((n))(x_(1))-y^((n))∣<0
(38)
eta(x_(1))-xi^('')! < 2epsi.
pine priser pue ic. 9. Dansua auter uravil [3] nons avons montró que si la fonctionf(x,y)a des deviver partiedes par rapport ìxand to the left, of order greater than two, continuous in the rectangle D. It is practical to choose the network I in another way, using another quadrature formula. We have already treated in detail the integration and differential equation (I), using the method of mechanical approximations of the quadrature formula of K. Petr. [4,6]. We have also treated the quadrature formula of K. Petr. [4,6].
Buntim eax partial didvies of the secend exdre of hyperbelic type
IR. Now aroa makes an extension of the previous method of numerical integration of differential equations (1), to partial differential equations of the second order of hyperbolic type [2]. This work will soon appear in Marbrastica Toure ? (5). We have first established by an extension of the method of J. Radon [6]. the cubature formula
(36)quad∬_(D)f(x,y)dzdy=(((x_(1)-x_(1)))((y_(1)-y_(1))))/(2)U(x_(1),y_(1))+f(y_(2),y_(0))]+R,
odDent the rectangle defined by the inequalities
(36)quad∬_(D)f(x,y)dzdy=(((x_(1)-x_(1)))((y_(1)-y_(1))))/(2)U(x_(1),y_(1))+f(y_(2),y_(0))]+R,
odDent the rectangle defined by the inequalities
x_(1) <= x <= x_(2),quady_(1) <= y <= y_(2)
and where the revteRis done by the formula
{:(37)R-∬_(D)(rho(del^(2)f)/(delx^(2))+psi(del^(2)f)/(del x del y)+theta(del^(2)f)/(dely^(2)))dxdy:}
arec
{:[varphi(x","y)-(1)/(2){:(x-x_(1)):}{:(x-x_(1)):}","],[psi^(˙)(x","y)--(1)/(2)[(x-x_(1))(y-y_(2))+(x-x_(2))(y-y_(1))+(x_(3)-x_(1))(y_(2)-y_(1)))","],[0(x","y)-(1)/(2){:(y-y_(1)):}{:(y-y_(1)):}.]:}
We now apply the mucoxiver approximation method and the cubature formula (3si) to the numerical integration of the equation anx derivative parlinles
(del^(a)s)/(del x del y)-f(x,y,z,p,q).
Orp-(del z)/(del z)*q cdots(del z)/(del y), with the conditionsz(x,0)-0,z(0,y)-0, in the rectangleDeltaformed by your rightsx-0,x-x,y-0,y-mu
We have shown that we can determine us rewan T tormé by the linesx-x_(1),y-y_(2)where the pointsx_(1)Andy_(2)share the intervals (0,lambda) And (0,mu) in not m equal parts and seek a calculation algorithm for the numbers: Sy, P&. (') such that c being a positive number doan6, the values ​​sbolues of the difference
epsi(x_(1),y_(2))-dy_(p)^(')p(x_(i),y_(n))-p(y_(n)^('),q(x_(i),y_(n))-4y_(n)^(')
on the noculs of the rescauGamma, are smaller than 2:
BIBLIOGRAPHY
