Friday, 24 June 2022

SOME APPLICATIONS OF VOLTERRA TYPE INTEGRO DIFFERENTIAL EQUATIONS

 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| < αΆ―), G1=(x0 ≤ s ≤ x≤ X,|y| ≤ αΆ―)

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(π‘₯,𝑠,𝑦)=Ξ£(π‘Žπ‘–(x)𝑏𝑖(s,y) )π‘šπ‘–=1

Then equation can be written as

y'(x) = f(x,y) + Ξ£π‘Žπ‘–(x)yπ‘šπ‘–=0∫𝑏𝑖(s,y(s))ds,yπ‘₯π‘₯0 y(x) = yo

Using the notation

Vi(x)=∫𝑏𝑖(s,y(s))π‘₯π‘₯0ds, (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)+ Ξ£(π‘Žπ‘–(x)𝑣𝑖(x))π‘šπ‘–=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

Ξ£π›Όπ‘˜π‘–=0i yn+i = hΞ£π›½π‘˜π‘–=0ifn+i + hΞ£π›½π‘˜π‘–=0ian+ivn+i (n=0,1,2,⋯ ⋯,𝑁−π‘˜)

Ξ£π›Όπ‘˜π‘–=0ivn+i = hΞ£π›½π‘˜π‘–=0ibn+i (n=0,1,2,⋯⋯,𝑁−π‘˜)

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 = Ξ£(π›Όπ‘˜π‘–=0iyn+i + hΞ²ifn+i + hΞ²ian+i)

yn+k = Ξ£(π‘˜−1𝑖=0Ξ±'iyn+i + hΞ²'ifn+i + hΞ²'ian+ivn+i) + hΞ²' k f (f ( xn+k+yn+k) +an+kvn+k)

Where the coefficients of prediction are method are denoted Ξ±i, Ξ²j (l = 0,1,2, k-i) and by a="/a

Bi - B/a (i = 0,1,2....k-1).

Ξ²' k=Ξ²'k /Ξ±k, 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 = Ξ£Ξ±π‘˜−1𝑖=0'ivn+i + h Ξ£Ξ²π‘˜−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) = Ξ£[π›Όπ‘˜−1𝑖=0iy(xn+i) + hΞ²if(xn+i,y(xn+i) + hΞ²ia(xn+i)v(xn+i)] +Řn

y(xn+k) = Ξ£[𝛼′π‘˜−1𝑖=0iy(xn+i) + hΞ²'if(xn+i,y(xn+i) + hΞ²'ia(xn+i)v(xn+i)] + hΞ²'kf(xn+k,y(xn+k) +

hΞ²'ka(xn+k)v(xn+i) + Rn

V(xn+k) = Ξ£Ξ±π‘˜−1𝑖=0'iv(xn+i) + h Ξ£Ξ²π‘˜π‘–=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(πœ†) = Ξ£π›Όπ‘˜π‘–=0iπœ†i

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

Ξ£[π›Όπ‘˜π‘–=0iy(x+i) - hΞ²iy'(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≤π‘š≤𝑁(β„°m,Êm) ≤ A exp (BX)(πœ‡max0≤π‘₯≤1| ℰ𝑖|+π‘šπ‘‘)

Here A, B and d are some bounded values and ΓŠπ‘š=𝑣(π‘₯π‘š)−π‘£π‘š,β„°π‘š=𝑦(π‘₯π‘š)−π‘¦π‘š(π‘š=0,1,2,⋯⋯,)

Proof:

ℰ𝑛+π‘˜=Ξ£(𝛼𝑖ℰ𝑛+𝑖+β„Žπ›½π‘–β„’π‘›+π‘˜β„‡π‘›+𝑖+β„Žπ›½π‘–π‘Žπ‘›+π‘–πœ€π‘›+𝑖 )+Ε˜π‘›π‘˜−1𝑖=0

ℰ𝑛+π‘˜=Ξ£(𝛼′𝑖ℰ𝑛+𝑖+β„Žπ›½π‘–β„’π‘›+π‘˜β„‡π‘›+𝑖+β„Žπ›½′π‘–π‘Žπ‘›+π‘–πœ€π‘›+𝑖 )+β„Žπ›½′π‘˜β„’π‘›+π‘˜β„‡π‘›+π‘˜+π‘˜−1𝑖=0

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β„Žπ›½′π‘˜π‘Žπ‘›+π‘˜β„‡π‘›+π‘˜+Ε˜π‘›

Here β„’π‘š=𝑓′𝑦(π‘₯π‘š,πœ‰π‘š),β„’π‘š=𝑓′𝑦(π‘₯π‘š,πœ‰π‘š),π‘Žπ‘š=π‘Ž(π‘₯π‘š),ΓŠπ‘š=𝑦(π‘₯π‘š)−αΊπ‘š

,( m = 0,1,2,.....)

Where πœ‰π‘š is found between y(xm) and ym and πœ‰π‘š between y(xm) and ẏm.

We have πœ€π‘›+π‘˜=Ξ£π‘Žπ‘–π‘’π‘›+𝑖+β„ŽΞ£π‘π‘–π‘’π‘›+𝑖+β„ŽΞ£π‘π‘–π‘˜−1𝑖=0πœ€π‘›+𝑖+β„Žπ›½′π‘˜π‘Žπ‘›+π‘˜πœ€π‘›+π‘˜+β„Žπ›½′π‘˜β„’π‘›+π‘˜Ε˜π‘›+β„›π‘›π‘˜−1𝑖=0π‘˜−1𝑖=0

Where, π‘Žπ‘–=𝛼′𝑖+β„Žπ›½′𝑖ℒ𝑛+𝑖𝛼𝑖, 𝑏𝑖=𝛽′𝑖ℒ𝑛+𝑖+β„Žπ›½′π‘˜β„’π‘›+π‘˜π›½π‘–β„’π‘›+1, 𝑐𝑖=𝛽′π‘–π‘Žπ‘›+𝑖+β„Žπ›½′𝑖ℒ𝑛+π‘˜π‘Žπ‘›+𝑖

we will obtain, πœ€π‘›+π‘˜=Ξ£(π‘Ž′𝑖𝑒𝑛+𝑖+β„Žπ›½′𝑖ℒ𝑛+π‘–π‘˜−1𝑖=0πœ€π‘›+𝑖)+β„Žπ›½′π‘˜β„’π‘›+π‘˜πœ€π‘›+π‘˜+ℛ𝑛

Where β„’π‘š=𝑏′𝑦(π‘₯π‘š,π‘¦π‘š) (m= 0,1,2,⋯⋯⋯,)πœ‚π‘š are between y(xm) and ym for error estimation of the method applied to the solution

We will obtain, πœ€π‘›+π‘˜=Ξ£π‘Žπ‘–πœ€π‘›+π‘–π‘˜−1𝑖=0+β„ŽΞ£π‘π‘–π‘˜−1𝑖=0πœ€π‘›+𝑖+ℛ𝑛

πœ€π‘›+π‘˜=Ξ£(π‘˜−1𝑖=0π‘Ž′π‘–πœ€π‘›+𝑖+β„Žπ›½′𝑖ℒ𝑛+π‘˜πœ€π‘›+𝑖+𝛽′π‘˜β„’π‘›+π‘˜Ξ£(π‘˜−1𝑖=0π‘Žπ‘–πœ€π‘›+𝑖+ β„Ž2π‘π‘–πœ€π‘›+𝑖)+β„Žπ›½′π‘˜β„’π‘›+π‘˜β„›π‘›+ℛ𝑛

Where, π‘Žπ‘–=π‘Žπ‘–+β„Žπ‘π‘–1−β„Ž2𝑑𝑛+π‘˜ 𝑏𝑖=𝑐𝑖+π‘Žπ‘›+π‘˜π›½′π‘˜π›Ό′𝑖1−β„Ž2𝑑𝑛+π‘˜ 𝑏𝑖=𝑏𝑖+β„Žπ‘Žπ‘›+π‘˜π›½′π‘˜π›½′𝑖ℒ𝑛+𝑖 𝑑𝑛+π‘˜=π‘Žπ‘›+π‘˜(𝛽′π‘˜)2ℒ𝑛+π‘˜

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ℛ𝑛=(β„Žπ›½′π‘˜β„’π‘›+π‘˜β„›π‘›+ℛ𝑛+β„Žπ‘Žπ‘›+𝑖ℛ𝑛)/1−β„Ž2𝑑𝑛+π‘˜

Consider the following relation

π‘Žπ‘–+β„Žπ‘π‘–/1−β„Ž2𝑑𝑛+π‘˜=π‘Žπ‘–+β„Žπ‘™π‘–

Where 𝑙𝑖=(𝑏𝑖+β„Žπ‘Žπ‘–π‘‘π‘›+π‘˜)/1−β„Ž2𝑑𝑛+π‘˜

For smallness of h we can assume |1−β„Ž2𝑑𝑛+π‘˜|≥12

Then we obtain that |𝑙𝑖|≤𝑙(i= 0,1,2, ⋯⋯-k-1)

It is easy to show that 𝑏𝑖=(𝑐𝑖+π‘Žπ‘›+π‘˜π›½′𝑖𝛼𝑖)/(1−β„Ž2𝑑𝑛+π‘˜) (𝑖= 0,1,2,⋯ ⋯,π‘˜−1)

are also bounded, that is,|𝑏𝑖| ≤ b.

we can write πœ€π‘›+π‘˜=Ξ£π‘Ž′π‘–πœ€π‘›+π‘–π‘˜−1𝑖=0+β„ŽΞ£π‘£π‘–π‘˜−1𝑖=0πœ€π‘›+𝑖+β„ŽΞ£π‘π‘–π‘˜−1𝑖=0πœ€π‘›+𝑖+ℛ𝑛

πœ€π‘›+π‘˜=Ξ£π‘Ž′π‘–πœ€π‘›+π‘–π‘˜−1𝑖=0+β„ŽΞ£π‘£π‘–π‘˜−1𝑖=0πœ€π‘›+𝑖+β„Ž2Σ𝑏𝑖𝛽′π‘˜β„’π‘›+π‘˜π‘˜−1𝑖=0πœ€π‘›+𝑖+Ε˜π‘›

Where, Ε˜π‘›=ℛ𝑛+β„Žπ›½′π‘˜β„’π‘›+π‘˜+ℛ𝑛 𝑣𝑖=𝛽′π‘˜β„’π‘›+π‘˜π›Όπ‘˜+𝑙𝑖 𝑣𝑖=𝛽′𝑖ℒ𝑛+𝑖+𝛼𝑖𝛽′π‘˜β„’π‘›+π‘˜ (𝑖=0,1,2,…,π‘˜−1)

Here we can also write that lvi| ≤v and |αΏ‘i|≤αΏ‘

Consider the following vectors,

Yn+k-1 (πœ€ n+k-1, πœ€ n+k-2, … … … , πœ€ n)

Ẏn+k-1 (πœ€ n+k-1, πœ€ n+k-2, … … … , πœ€ n)

Then adding to equation the following identities respectively,

π‘Œπ‘›+π‘˜=π΄π‘Œπ‘›+π‘˜−1+β„Žπ‘‰π‘›+π‘˜π‘Œπ‘›+π‘˜−1+β„Žπ΅π‘›+π‘˜αΊŽπ‘›+π‘˜−1+Ŵ𝑛,π‘˜

αΊŽπ‘›+π‘˜=π΄αΊŽπ‘›+π‘˜−1+β„ŽαΉΌπ‘›+π‘˜π‘Œπ‘›+π‘˜−1+β„Ž2𝐡𝑛+π‘˜αΊŽπ‘›+π‘˜−1+π‘Šπ‘›,π‘˜

Where the matrices A, Vn+kῑn+ki En+k. Ên+k and the vectors Wnk Ŵnk are

Determined in the following form 𝐴=(𝛼′π‘˜−1𝛼′π‘˜−2.10.00..𝛼′1𝛼′0 ....10) 𝑣𝑛+π‘˜=(π‘£π‘˜−1π‘£π‘˜−2.00.00..𝑣0...0)

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𝑩𝑛+π‘˜=(π΅π‘˜−1π΅π‘˜−2.00.00..𝑏0.0.0) 𝑩𝑛+π‘˜=(π‘π‘˜−1π‘π‘˜−2.00.00..𝑏0.000)(𝛽′π‘˜πΏπ‘›+π‘˜)−1 𝑣𝑛+π‘˜=(π‘£π‘˜−1π‘£π‘˜−2.00.00. .𝑣0 .0 .0) π‘Šπ‘›,π‘˜=(𝑅𝑛00) π‘Šπ‘›,π‘˜=(Ε˜π‘›00)

Using the vector Zn+k =(𝑦𝑛+π‘˜,αΊŽπ‘›+π‘˜)

We can rewrite equations in the following form 𝑍𝑛+π‘˜=𝐴𝑍𝑛+π‘˜−1β„Žπ‘‰π‘›+π‘˜π‘π‘›+π‘˜−1+Ŵ𝑛,π‘˜

Where, 𝐴=(𝐴00𝐴) 𝑉𝑛+π‘˜=(𝑉𝑛+π‘˜π΅π‘›+π‘˜π‘‰π‘›+π‘˜β„Žπ΅π‘›+π‘˜) Ŵ𝑛,π‘˜=(Ŵ𝑛,π‘˜Ε΄π‘›,π‘˜)

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+π‘˜=𝐷𝑦𝑛+π‘˜−1β„ŽπΈπ‘›+π‘˜π‘¦π‘›+π‘˜−1+Ŵ𝑛,π‘˜

determined in the following form 𝐷=𝐢−1Ậ𝐢 𝐸𝑛+π‘˜=𝐢−1Ṽ𝑛+π‘˜πΆ Ŵ𝑛,π‘˜=𝐢−1Ŵ𝑛,π‘˜

Passing to the norm in equation

we have, ‖𝑦𝑛+π‘˜‖≤‖𝐷‖‖𝑦𝑛+π‘˜‖+β„Ž‖𝐸𝑛+π‘˜‖‖𝑦𝑛+π‘˜−1‖+‖Ŵ𝑛,π‘˜‖

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+kiαΏ‘n+k have the bounded norms.

Then we can write

||En+k|| ≤ B

Subject to the above metrices Vn+k,En+k,αΏ‘n+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=||Ε΄n,k||

we will obtain that ‖π‘¦π‘š‖≤(1+β„Žπ΅)π‘š−π‘˜‖π‘¦π‘˜‖+Ξ£(1+β„Žπ΅)π‘–π‘š−π‘˜−1𝑖=0𝑑

It is known that (1+hB) m exp (mhB)

For h ≤ X-xo≤ X, We can rewrite in the following form ‖π‘¦π‘š‖≤exp(𝐡𝑋)(‖π‘¦π‘˜‖+π‘‘β„Ž−1) ‖𝑦𝑧‖≤𝐴exp(𝐡𝑋)(‖πœ‡‖‖π‘§π‘˜‖+π‘‘β„Ž−1)

Where, 𝐴=‖𝐢‖ πœ‡=‖𝐢−1‖

From the last it follows that maxπ‘˜≤π‘š≤𝑁(πœ€π‘š,πœ€π‘š)≤π‘Žexp(𝐡𝑋)(πœ‡max0≤𝑖≤π‘˜−1|πœ€π‘š|+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≤π‘˜−1πœ€π‘–=𝑂(β„Žπ‘)

Then from the assertions of the theorem it follows that πœ€π‘š=𝑂(β„Žπ‘),β„Ž→0

As we see from the proof of the theorem by using equation (1.7) it was assumed that

𝑦𝑛+π‘˜ is known. However, for finding 𝑦𝑛+π‘˜ is to be known 𝑣𝑛+π‘˜

Note that in real calculations for finding the values𝑣𝑛+π‘˜ - it is used approximate values of 𝑦𝑛+π‘˜ calculated by the prediction method.

In this case, the proposed method for the solution of equation (1.5) is written in the

following form αΊŽπ‘›+π‘˜=Ξ£(π‘Žπ‘–π‘¦π‘›+π‘–π‘˜π‘–=0+β„Žπ›½π‘–π‘“π‘›+𝑖+β„Žπ›½π‘–π‘Žπ‘›+𝑖,𝑣𝑛+𝑖) 𝑣𝑛+π‘˜=Ξ£π‘Ž′𝑖𝑣𝑛+π‘–π‘˜−1𝑖=0+β„ŽΞ£π›½′𝑖𝑏𝑛+𝑖+π‘˜−1𝑖=0β„Žπ›½′𝑖𝑏(π‘₯𝑛+π‘˜,αΊŽπ‘›+π‘˜) 𝑦𝑛+π‘˜=Ξ£π‘Ž′𝑖𝑦𝑛+π‘–π‘˜−1𝑖=0+β„ŽΞ£π›½′𝑖(𝑓𝑛+𝑖+π‘˜−1𝑖=0π‘Žπ‘›+𝑖𝑣𝑛+𝑖)+β„Žπ›½′π‘˜(π‘₯𝑛+π‘˜,αΊŽπ‘›+π‘˜)+π‘Žπ‘›+π‘˜π‘£π‘›+π‘˜

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.

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