SOME APPLICATIONS OF VOLTERRA TYPE INTEGRO DIFFERENTIAL EQUATIONS
Dr,K.R.Salini[1] , M. Deivanai[2], S.Madhavan[3] and Dr.S.Bamini[4]
[1] Guest Lecturer, PG & Department of Mathematics, Government Arts & Science College, Kariyampatti, Tirupattur, Tamilnadu, India.
Email: shalinimanoj1985@gmail.com
[2],[3] II M.Sc, Mathematics, PG & Department of Mathematics Government Arts & Science College, Kariyampatti, Tirupattur, Tamilnadu, India.
[4] Research Coordinator, Assistant Professor, PG & Research Department of Mathematics,
Marudhar Kesari Jain College for Women, Vaniyambadi, Tamilnadu, India.
Email: saransham07@gmail.com
Abstract
We determine the numerical solution of the specific nonlinear Voltera-Fredholm Integro-Differential Equation is proposed. The method is based on new vector forms for representation of triangular functions and its operational matrix.
Introduction
The nonlinear Volterra type integro-differential equation of the first order
y'(x)=f(x,y)+K(x. s. y(s))ds, y(x)= yox € [x.X]
Here the functions f(x,y). K(x.s, y) are determined in the domains
G=(x0 < x < x , |y| < ᶯ), G1=(x0 ≤ s ≤ x≤ X,|y| ≤ ᶯ)
and respectively and equation (1.1) has a unique solution.
In case K(x.s, y) ≡ 0 equation (1.1) turns into the Cauchy problem for ordinar
differential equations of the first order.
Non-Linear Volterra Type Integro Differentia Equation Applying Multi Step Method
The aim is to apply the multistep method with constant coefficients to the
solution of equation
Assume that the kernel K(x.s, y) is degenerate that is 𝐾(𝑥,𝑠,𝑦)=Σ(𝑎𝑖(x)𝑏𝑖(s,y) )𝑚𝑖=1
Then equation can be written as
y'(x) = f(x,y) + Σ𝑎𝑖(x)y𝑚𝑖=0∫𝑏𝑖(s,y(s))ds,y𝑥𝑥0 y(x) = yo
Using the notation
Vi(x)=∫𝑏𝑖(s,y(s))𝑥𝑥0ds, (i=1.2,...m)
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We can rewrite equation in the form of a system of differential equations
y' = f(x,y)+ Σ(𝑎𝑖(x)𝑣𝑖(x))𝑚𝑖=1
v'i(x)=bi (x,y(x)), ( i=1,2,....,m )
The initial conditions for the equation (1.3) of differential equations have the
following form
y(x0) = y0, vi(x0) = 0 ( i =1.2 ,……., m)
Consider the simple case and assume m = 1 then we have
y' ≡ f(x,y)+a(x)v(x),y(x 0 ) ≡ y0
v' (x) = b(x,y(x)),v(x0 ) = 0
Let's divide the section [x0 , X] by means of the constant step h > 0 into N equal parts
We apply k - th step method with constant coefficients to the numerical solution of
equation Then we can write
Σ𝛼𝑘𝑖=0i yn+i = hΣ𝛽𝑘𝑖=0ifn+i + hΣ𝛽𝑘𝑖=0ian+ivn+i (n=0,1,2,⋯ ⋯,𝑁−𝑘)
Σ𝛼𝑘𝑖=0ivn+i = hΣ𝛽𝑘𝑖=0ibn+i (n=0,1,2,⋯⋯,𝑁−𝑘)
Where, am= a(xm)
vm = v(Xm)
bm = b(xm, ym)
fm = f(xm,ym)
xm = x0 + mh ( m = 0,1,2,..........)
If we assume that ak ≠ 0, then from equation (1.6) and equation (1.7) we can find yn+k and Vn+k respectively. However, relation (1.7) is the implicit nonlinear finite difference equation. Usually in these cases the different varients of prediction correction method is used
Non-Linear Volterra Type Integro Differential Equation Applying Prediction-Correction Method
Applying the prediction-correction method to equation (1.6) ,we have
ẏn+k = Σ(𝛼𝑘𝑖=0iyn+i + hβifn+i + hβian+i)
yn+k = Σ(𝑘−1𝑖=0α'iyn+i + hβ'ifn+i + hβ'ian+ivn+i) + hβ' k f (f ( xn+k+yn+k) +an+kvn+k)
Where the coefficients of prediction are method are denoted αi, βj (l = 0,1,2, k-i) and by a="/a
Bi - B/a (i = 0,1,2....k-1).
β' k=β'k /αk, we denote the coefficients of the correction method.
Then the numerical method for solution of equation is obtained from methods
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vn+k = Σα𝑘−1𝑖=0'ivn+i + h Σβ𝑘−1𝑖=0'ibn+i (n=0,1,2, ⋯⋯)
t is obvious that the found value ym vm (m ≥ k) by methods aren't the exact values of equation at the points xm (m ≥0).
Therefore, if in methods instead of the approximate values ym,vm we put their exact values y(xm), v(xm), these methods will take the following form
ẏ(xn+k) = Σ[𝛼𝑘−1𝑖=0iy(xn+i) + hβif(xn+i,y(xn+i) + hβia(xn+i)v(xn+i)] +Řn
y(xn+k) = Σ[𝛼′𝑘−1𝑖=0iy(xn+i) + hβ'if(xn+i,y(xn+i) + hβ'ia(xn+i)v(xn+i)] + hβ'kf(xn+k,y(xn+k) +
hβ'ka(xn+k)v(xn+i) + Rn
V(xn+k) = Σα𝑘−1𝑖=0'iv(xn+i) + h Σβ𝑘𝑖=0'ib(xn+I,y(xn+i)) + Rn
Here the error of corresponding methods is denoted by Ř,Rn
Note that ẏ(xm) = y(xm) ( m = 0,1,2,⋯⋯)
Definition
Equation is stable if the roots of the polynomial
𝜌(𝜆) = Σ𝛼𝑘𝑖=0i𝜆i
lie inside a unique circle on whose boundary there are no multi roots.
Definition:
The integer-valued p is called of the degree of method (1.6),if the following
Σ[𝛼𝑘𝑖=0iy(x+i) - hβiy'(x+ih)) = o(hp),h→0
holds
Determination of Convergence of The Proposed Method
Let us prove the following theorem for determination of convergence of the
proposed method.
Theorem
Let method (1.6) be stable, satisfy the conditions A, B, C. The first derivatives
of the function b (x, y) and f(x, y) by y be bounded. Then it holds
max0≤𝑚≤𝑁(ℰm,Êm) ≤ A exp (BX)(𝜇max0≤𝑥≤1| ℰ𝑖|+𝑚𝑑)
Here A, B and d are some bounded values and Ê𝑚=𝑣(𝑥𝑚)−𝑣𝑚,ℰ𝑚=𝑦(𝑥𝑚)−𝑦𝑚(𝑚=0,1,2,⋯⋯,)
Proof:
ℰ𝑛+𝑘=Σ(𝛼𝑖ℰ𝑛+𝑖+ℎ𝛽𝑖ℒ𝑛+𝑘ℇ𝑛+𝑖+ℎ𝛽𝑖𝑎𝑛+𝑖𝜀𝑛+𝑖 )+Ř𝑛𝑘−1𝑖=0
ℰ𝑛+𝑘=Σ(𝛼′𝑖ℰ𝑛+𝑖+ℎ𝛽𝑖ℒ𝑛+𝑘ℇ𝑛+𝑖+ℎ𝛽′𝑖𝑎𝑛+𝑖𝜀𝑛+𝑖 )+ℎ𝛽′𝑘ℒ𝑛+𝑘ℇ𝑛+𝑘+𝑘−1𝑖=0
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ℎ𝛽′𝑘𝑎𝑛+𝑘ℇ𝑛+𝑘+Ř𝑛
Here ℒ𝑚=𝑓′𝑦(𝑥𝑚,𝜉𝑚),ℒ𝑚=𝑓′𝑦(𝑥𝑚,𝜉𝑚),𝑎𝑚=𝑎(𝑥𝑚),Ê𝑚=𝑦(𝑥𝑚)−ẏ𝑚
,( m = 0,1,2,.....)
Where 𝜉𝑚 is found between y(xm) and ym and 𝜉𝑚 between y(xm) and ẏm.
We have 𝜀𝑛+𝑘=Σ𝑎𝑖𝑒𝑛+𝑖+ℎΣ𝑏𝑖𝑒𝑛+𝑖+ℎΣ𝑐𝑖𝑘−1𝑖=0𝜀𝑛+𝑖+ℎ𝛽′𝑘𝑎𝑛+𝑘𝜀𝑛+𝑘+ℎ𝛽′𝑘ℒ𝑛+𝑘Ř𝑛+ℛ𝑛𝑘−1𝑖=0𝑘−1𝑖=0
Where, 𝑎𝑖=𝛼′𝑖+ℎ𝛽′𝑖ℒ𝑛+𝑖𝛼𝑖, 𝑏𝑖=𝛽′𝑖ℒ𝑛+𝑖+ℎ𝛽′𝑘ℒ𝑛+𝑘𝛽𝑖ℒ𝑛+1, 𝑐𝑖=𝛽′𝑖𝑎𝑛+𝑖+ℎ𝛽′𝑖ℒ𝑛+𝑘𝑎𝑛+𝑖
we will obtain, 𝜀𝑛+𝑘=Σ(𝑎′𝑖𝑒𝑛+𝑖+ℎ𝛽′𝑖ℒ𝑛+𝑖𝑘−1𝑖=0𝜀𝑛+𝑖)+ℎ𝛽′𝑘ℒ𝑛+𝑘𝜀𝑛+𝑘+ℛ𝑛
Where ℒ𝑚=𝑏′𝑦(𝑥𝑚,𝑦𝑚) (m= 0,1,2,⋯⋯⋯,)𝜂𝑚 are between y(xm) and ym for error estimation of the method applied to the solution
We will obtain, 𝜀𝑛+𝑘=Σ𝑎𝑖𝜀𝑛+𝑖𝑘−1𝑖=0+ℎΣ𝑏𝑖𝑘−1𝑖=0𝜀𝑛+𝑖+ℛ𝑛
𝜀𝑛+𝑘=Σ(𝑘−1𝑖=0𝑎′𝑖𝜀𝑛+𝑖+ℎ𝛽′𝑖ℒ𝑛+𝑘𝜀𝑛+𝑖+𝛽′𝑘ℒ𝑛+𝑘Σ(𝑘−1𝑖=0𝑎𝑖𝜀𝑛+𝑖+ ℎ2𝑏𝑖𝜀𝑛+𝑖)+ℎ𝛽′𝑘ℒ𝑛+𝑘ℛ𝑛+ℛ𝑛
Where, 𝑎𝑖=𝑎𝑖+ℎ𝑏𝑖1−ℎ2𝑑𝑛+𝑘 𝑏𝑖=𝑐𝑖+𝑎𝑛+𝑘𝛽′𝑘𝛼′𝑖1−ℎ2𝑑𝑛+𝑘 𝑏𝑖=𝑏𝑖+ℎ𝑎𝑛+𝑘𝛽′𝑘𝛽′𝑖ℒ𝑛+𝑖 𝑑𝑛+𝑘=𝑎𝑛+𝑘(𝛽′𝑘)2ℒ𝑛+𝑘
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ℛ𝑛=(ℎ𝛽′𝑘ℒ𝑛+𝑘ℛ𝑛+ℛ𝑛+ℎ𝑎𝑛+𝑖ℛ𝑛)/1−ℎ2𝑑𝑛+𝑘
Consider the following relation
𝑎𝑖+ℎ𝑏𝑖/1−ℎ2𝑑𝑛+𝑘=𝑎𝑖+ℎ𝑙𝑖
Where 𝑙𝑖=(𝑏𝑖+ℎ𝑎𝑖𝑑𝑛+𝑘)/1−ℎ2𝑑𝑛+𝑘
For smallness of h we can assume |1−ℎ2𝑑𝑛+𝑘|≥12
Then we obtain that |𝑙𝑖|≤𝑙(i= 0,1,2, ⋯⋯-k-1)
It is easy to show that 𝑏𝑖=(𝑐𝑖+𝑎𝑛+𝑘𝛽′𝑖𝛼𝑖)/(1−ℎ2𝑑𝑛+𝑘) (𝑖= 0,1,2,⋯ ⋯,𝑘−1)
are also bounded, that is,|𝑏𝑖| ≤ b.
we can write 𝜀𝑛+𝑘=Σ𝑎′𝑖𝜀𝑛+𝑖𝑘−1𝑖=0+ℎΣ𝑣𝑖𝑘−1𝑖=0𝜀𝑛+𝑖+ℎΣ𝑏𝑖𝑘−1𝑖=0𝜀𝑛+𝑖+ℛ𝑛
𝜀𝑛+𝑘=Σ𝑎′𝑖𝜀𝑛+𝑖𝑘−1𝑖=0+ℎΣ𝑣𝑖𝑘−1𝑖=0𝜀𝑛+𝑖+ℎ2Σ𝑏𝑖𝛽′𝑘ℒ𝑛+𝑘𝑘−1𝑖=0𝜀𝑛+𝑖+Ř𝑛
Where, Ř𝑛=ℛ𝑛+ℎ𝛽′𝑘ℒ𝑛+𝑘+ℛ𝑛 𝑣𝑖=𝛽′𝑘ℒ𝑛+𝑘𝛼𝑘+𝑙𝑖 𝑣𝑖=𝛽′𝑖ℒ𝑛+𝑖+𝛼𝑖𝛽′𝑘ℒ𝑛+𝑘 (𝑖=0,1,2,…,𝑘−1)
Here we can also write that lvi| ≤v and |ῡi|≤ῡ
Consider the following vectors,
Yn+k-1 (𝜀 n+k-1, 𝜀 n+k-2, … … … , 𝜀 n)
Ẏn+k-1 (𝜀 n+k-1, 𝜀 n+k-2, … … … , 𝜀 n)
Then adding to equation the following identities respectively,
𝑌𝑛+𝑘=𝐴𝑌𝑛+𝑘−1+ℎ𝑉𝑛+𝑘𝑌𝑛+𝑘−1+ℎ𝐵𝑛+𝑘Ẏ𝑛+𝑘−1+Ŵ𝑛,𝑘
Ẏ𝑛+𝑘=𝐴Ẏ𝑛+𝑘−1+ℎṼ𝑛+𝑘𝑌𝑛+𝑘−1+ℎ2𝐵𝑛+𝑘Ẏ𝑛+𝑘−1+𝑊𝑛,𝑘
Where the matrices A, Vn+kῡn+ki En+k. Ên+k and the vectors Wnk Ŵnk are
Determined in the following form 𝐴=(𝛼′𝑘−1𝛼′𝑘−2.10.00..𝛼′1𝛼′0 ....10) 𝑣𝑛+𝑘=(𝑣𝑘−1𝑣𝑘−2.00.00..𝑣0...0)
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𝑩𝑛+𝑘=(𝐵𝑘−1𝐵𝑘−2.00.00..𝑏0.0.0) 𝑩𝑛+𝑘=(𝑏𝑘−1𝑏𝑘−2.00.00..𝑏0.000)(𝛽′𝑘𝐿𝑛+𝑘)−1 𝑣𝑛+𝑘=(𝑣𝑘−1𝑣𝑘−2.00.00. .𝑣0 .0 .0) 𝑊𝑛,𝑘=(𝑅𝑛00) 𝑊𝑛,𝑘=(Ř𝑛00)
Using the vector Zn+k =(𝑦𝑛+𝑘,Ẏ𝑛+𝑘)
We can rewrite equations in the following form 𝑍𝑛+𝑘=𝐴𝑍𝑛+𝑘−1ℎ𝑉𝑛+𝑘𝑍𝑛+𝑘−1+Ŵ𝑛,𝑘
Where, 𝐴=(𝐴00𝐴) 𝑉𝑛+𝑘=(𝑉𝑛+𝑘𝐵𝑛+𝑘𝑉𝑛+𝑘ℎ𝐵𝑛+𝑘) Ŵ𝑛,𝑘=(Ŵ𝑛,𝑘Ŵ𝑛,𝑘)
Note that Zn+k = Cyn+k
Then after the multiplication of equations (1.24) on the left hand side by C-1 we will obtain 𝑦𝑛+𝑘=𝐷𝑦𝑛+𝑘−1ℎ𝐸𝑛+𝑘𝑦𝑛+𝑘−1+Ŵ𝑛,𝑘
determined in the following form 𝐷=𝐶−1Ậ𝐶 𝐸𝑛+𝑘=𝐶−1Ṽ𝑛+𝑘𝐶 Ŵ𝑛,𝑘=𝐶−1Ŵ𝑛,𝑘
Passing to the norm in equation
we have, ‖𝑦𝑛+𝑘‖≤‖𝐷‖‖𝑦𝑛+𝑘‖+ℎ‖𝐸𝑛+𝑘‖‖𝑦𝑛+𝑘−1‖+‖Ŵ𝑛,𝑘‖
For the stability of equation (1.6) we can assert that the characteristic numbers of the matrix A satisfy the following condition: all characteristic numbers of the matrix by the modulus are less than unit, and the roots equal by the modulus of a unit are simple
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Since the matrix A is a Frobenius matrix smallness of h and 15,| bi |≤b,I = 0,1,2,...k-1 Then subject to the sufficient we will obtain that all characteristic numbers of the matrix A are situated in a unit circle on whose boundary that are no multi roots, to roots by the module equal to unit correspond the Jordan cells of dimension one.
Consequently,
||D||≤1
We can show that the matrices Vn+k. En+kiῡn+k have the bounded norms.
Then we can write
||En+k|| ≤ B
Subject to the above metrices Vn+k,En+k,ῡn+k have the bounded norms
There we can write
||En+k||≤ B
Subject to the above mentioned in (1.26) we will obtain
||ym|| ≤ (1+hB)||ym-1||+d
Where.
d=||Ŵn,k||
we will obtain that ‖𝑦𝑚‖≤(1+ℎ𝐵)𝑚−𝑘‖𝑦𝑘‖+Σ(1+ℎ𝐵)𝑖𝑚−𝑘−1𝑖=0𝑑
It is known that (1+hB) m exp (mhB)
For h ≤ X-xo≤ X, We can rewrite in the following form ‖𝑦𝑚‖≤exp(𝐵𝑋)(‖𝑦𝑘‖+𝑑ℎ−1) ‖𝑦𝑧‖≤𝐴exp(𝐵𝑋)(‖𝜇‖‖𝑧𝑘‖+𝑑ℎ−1)
Where, 𝐴=‖𝐶‖ 𝜇=‖𝐶−1‖
From the last it follows that max𝑘≤𝑚≤𝑁(𝜀𝑚,𝜀𝑚)≤𝑎exp(𝐵𝑋)(𝜇max0≤𝑖≤𝑘−1|𝜀𝑚|+𝑑𝑛)
If we assume that equation (1.6) is of degree p the initial values are calculated with
accuracy p.
That is,
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max0≤𝑖≤𝑘−1𝜀𝑖=𝑂(ℎ𝑝)
Then from the assertions of the theorem it follows that 𝜀𝑚=𝑂(ℎ𝑝),ℎ→0
As we see from the proof of the theorem by using equation (1.7) it was assumed that
𝑦𝑛+𝑘 is known. However, for finding 𝑦𝑛+𝑘 is to be known 𝑣𝑛+𝑘
Note that in real calculations for finding the values𝑣𝑛+𝑘 - it is used approximate values of 𝑦𝑛+𝑘 calculated by the prediction method.
In this case, the proposed method for the solution of equation (1.5) is written in the
following form Ẏ𝑛+𝑘=Σ(𝑎𝑖𝑦𝑛+𝑖𝑘𝑖=0+ℎ𝛽𝑖𝑓𝑛+𝑖+ℎ𝛽𝑖𝑎𝑛+𝑖,𝑣𝑛+𝑖) 𝑣𝑛+𝑘=Σ𝑎′𝑖𝑣𝑛+𝑖𝑘−1𝑖=0+ℎΣ𝛽′𝑖𝑏𝑛+𝑖+𝑘−1𝑖=0ℎ𝛽′𝑖𝑏(𝑥𝑛+𝑘,Ẏ𝑛+𝑘) 𝑦𝑛+𝑘=Σ𝑎′𝑖𝑦𝑛+𝑖𝑘−1𝑖=0+ℎΣ𝛽′𝑖(𝑓𝑛+𝑖+𝑘−1𝑖=0𝑎𝑛+𝑖𝑣𝑛+𝑖)+ℎ𝛽′𝑘(𝑥𝑛+𝑘,Ẏ𝑛+𝑘)+𝑎𝑛+𝑘𝑣𝑛+𝑘
Reference
1. G.Yu.Mekhtieva, V.R.Ibrahimov “On one numerical method for solution of integro-differential equations”, Vestnik BSU, ser.phys.-math. Scien., 2003, No3, pp.21-27.(Russian).
2. H.Brunner, “Collection method for Volterra integral and Relation Functional Equations”, Cambridge University Press, Cambridge,2004.
3. P.Darania, E.Abadian, “A method for the numerical solution of the integro-differential equations”, Appl.Math. and Comput. 188(2007) 657-668.
4. B. Ahmad, S. Siva Sundaram, Existence Results For Nonlinear Impulsive Hybrid Boundary Value Problems involving Fractional Differential Equations, Non linear Analysis, hybrid System, 3(2009).
5. Z.B. Bai, H.S. Lii, Positive Solutions Of Boundary Valuve problems Of Nonlinear Fractional Differential Equations, J. Math Anal.Appl.311(2005)\
6. v. Lakshmikantham, D.D. Bainov And P.S. Simeonov, Theory Of Impulsive Differential Equations, modern Applied Mathematics (1989).
7. K.S. Miller, B.Ross, An Introduction To The Fractional Caculus And Fractional Differential Equations, Wiley, New York, 1993.
