A piecewise approximation for linear two dimensional Volterra Integral Equation by Chebyshev polynomials S. Fazeli, G. Hojjati∗, H. Kheiri
Faculty of Mathematical Sciences, University of Tabriz, Tabriz - Iran Abstract In this paper we investigate piecewise approximate solution for linear two dimensional Volterra integral equation, based on the interval approximation of the true solution by truncated Chebyshev series. By discretization respect to spatial and time variables, the solution is approximated by using collocation method. Analysis of discretization error is discussed and efficiency of the method is shown by applying it on some problems. Keywords: Two dimensional Volterra integral equation, Chebyshev approximation, Collocation methods. Mathematics Subject Classification: 65R20
1
Introduction
Consider the following linear two dimensional Volterra integral equation Z tZ x u(t, x) = f (t, x) + G(t, z, x, y)u(z, y)dydz, (1) 0
0
where x ∈ J := [0, X] and t ∈ I := [0, T ]. We assume that the given realvalued function f := f (t, x) and G := G(t, x, z, y) are at least continuous on D := I × J and S = {(t, z, x, y); 0 ≤ z ≤ t ≤ T, 0 ≤ y ≤ x ≤ X}, respectively. Equation of type (1) often arises from the mathematical modeling of the mechanical problems. Existence and uniqueness of solution may be found in [5]. There are no more tasks for numerical solution of this type ∗
Corresponding author (email:
[email protected],
[email protected])
1
of problems. Recently some authors applied differential transform methods(DTM) for solving (1.1) [6]. Here, we propose an approximation by using Chebyshev polynomials. This article is organized as follows: In section 2 the method is described and in section 3 the order of convergence of the method is determined. In section 4 numerical examples are given to confirm the theoretical concepts.
2
Description of the method
Let ΠM : 0 = x0 < x1 < · · · < xM = X and ΠN : 0 = t0 < t1 < · · · < tN = T be partitions of [0, X] and [0, T ] with constant step size h = xi+1 − xi , i = 0, 1, · · · , M − 1 and τ = tj+1 − tj , j = 0, 1, · · · , N − 1. We approximate the solution u(t, x) in the subinterval (tl , tl+1 ] × (xi , xi+1 ] say Ui,l (t, x) that is in Sq0 (ΠN )×Sp0 (ΠM ) where Sp0 (ΠM ) := {v : v|(xm ,xm+1 ] ∈ πp , 0 ≤ m ≤ M − 1} is a real polynomial spline space of degree p with continuity property in nodes. Here the approximated solution is represented by means of Chebyshev polynomials. The approximate solution for (1) in every subinterval (tl , tl+1 ] × (xi , xi+1 ] is determined by using the collocation S method. The approximations in the subintervals ([0, T ] × [0, xi ]) ([0, tl ] × (xi , xi+1 ]) have been obtained from the previous stages, thus we have Z tZ x Ui,l (t, x) = Fi,l (t, x) + G(t, z, x, y)Ui,l (z, y)dydz, (2) tl
xi
where (t, x) ∈ (tl , tl+1 ] × (xi , xi+1 ] and Fi,l (t, x) = f (t, x) +
i−1 X l−1 Z X r=0 s=0
+
+
i−1 Z t Z X
s=0
2.1
Z
xr+1
G(t, z, x, y)Ur,s (z, y)dydz xr
ts
xr+1
G(t, z, x, y)Ur,l (z, y)dydz
r=0 tl xr l−1 Z ts+1 Z x X ts
ts+1
(3)
G(t, z, x, y)Ui,s (z, y)dydz.
xi
Discretization in spatial variable
We rewrite Ui,l (t, x) in terms of new variable α ∈ (−1, 1] instead of x ∈ (xi , xi+1 ] defined by h x = xi + (1 + α). 2 2
Thus we can write h ¯i,l (t, α). (1 + α)) = U 2
Ui,l (t, x) = Ui,l (t, xi +
The solution of integral equation (1.1) with respect to the second variable can be approximated by the following finite sum (see [4]) ¯i,l (t, α) ' U
p 00 X
ak (t)Tk (α),
(4)
k=0
where p 00
2X ak (t) = Ui,l (t, xi + ξw h)Tk (αw ), p
k = 0, 1, · · · , p,
1 + αj , 2
j = 0, 1, · · · , p,
(5)
w=0
π αj = cos(p − j) , p
ξj =
andPTk (α) is the standard Chebyshev polynomial of degree k. Double prime on indicates that both of the first and the last terms in the summation are halved. So by changing of variable y = xi + h2 (1 + α) in (2) and substituting (4) and (5) in it and collocating the both side of obtained equation in the points xi + ξj h, j = 1, · · · , p, it leads to a system of Volterra integral equations that contains p equations and p unknown functions Ui,l (t, xi + ξw h), w = 1, 2, · · · , p in the form of Ui,l (t, xi + ξj h) = Fi,l (t, xi + ξj h) +
h p
p 00 p 00 X X
(6) Z tZ
αj
G(t, z, xi + ξj h, xi
Tk (αw ) tl
k=0 w=0
−1
h + (1 + α))Tk (α)Ui,l (z, xi + ξw h)dαdz. 2 For simplifying, set [w]
= Ui,l (z, xi + ξw h),
w = 0, 1, · · · , p,
[j]
= Fi,l (t, xi + ξj h),
j = 1, 2, · · · , p.
Ui,l
Fi,l
As the approximated value of u(t, xi ) is obtained from the previous stage, for continuity of approximation solution, we set Ui−1,l (t, xp ) instead of Ui,l (t, x0 ). Thus Z p 0 h tX [j] [j] [w] Ui,l (t) = F i,l (t) + Kj,w (t, z)Ui,l (z)dz, j = 1, 2, · · · , p, (7) p tl w=1
3
where p 00
[j] F i,l
=
[j] Fi,l
hX Tk (α0 ) + 2p
Z tZ
k=0
αj
G(t, z, xi + ξj h, xi
(8)
−1
tl
h [p] + (1 + α))Tk (α)Ui−1,l (z)dαdz, 2
j = 1, · · · , p,
and Kj,w (t, z) =
p 00 X k=0
Z
αj
Tk (αw )
G(t, z, xi + ξj h, xi
(9)
−1
h + (1 + α))Tk (α)da, 2
j = 1, · · · , p, w = 1, · · · , p.
P Prime on indicates that the last term in summation is halved. Note that [p] when i = 0, we set f (z, 0) instead of Ui−1,l (z).
2.2
Discretization in time variable
In subsection 2.1, a system of linear Volterra integral equations with p un[w] known functions Ui,l , w = 1, · · · , p was obtained. Now we approximate the solution of this system for t ∈ (tl , tl+1 ] by Chebyshev polynomials. By [j] using the same manner, we rewrite Ui,l (t) in terms of new variable β that is defined by τ t = tl + (1 + β). 2 For solving (7), we seek an approximate solution in the form of following finite sum q 00 X [j] [j] ¯ Ui,l (t) = Ui,l (β) = br Tr (β), (10) r=0
where q 00
2 X [j] br = Ui,l (tl + τ ζs )Tr (β), r = 0, 1, · · · , q, q
(11)
s=0
1 + βj π βj = cos(q − j) , ζj = , j = 0, 1, · · · , q. q 2 By changing of variable z = tl + τ2 (1 + β), substituating (10) and (11) in (7) and collocating in the points tl + ζd τ , d = 1, 2, · · · , q, the following algebraic
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system is yield [j,d] Ui,l
=
F [j,d] i,l
q 0 p 0 hτ X X [w,s] + Gj,w,d,s Ui,l , j = 1, · · · , p, d = 1, · · · , q, (12) pq s=1 w=1
where F [j,d] i,l
=
[j,d] F i,l
Z ζd p 0 q 00 τ hτ X X [w,0] Kj,w (tl + ζd τ, tl + (1 + β))Tr (β)dβ. Tr (β0 )Ui,l + 2pq 2 −1 w=1 r=0
Here, Gj,w,d,s for j, w = 1, 2, · · · , p and d, s = 1, 2, · · · , q is in the form Gj,w,d,s =
q 00 X r=0
Z
βd
Tr (βs ) −1
τ Kj,w (tl + ζd τ, tl + (1 + β))Tr (β)dβ. 2
(13)
Similar to discretization in spatial, for continuity of the solution, the value [j,0] of Ui,l is substituted by approximated value from the previous stage i.e [j,q]
[w,0]
Ui,l−1 and for l=0, the value of Ui,0 is substituted by f (0, xi + ξw h). Thus we have the approximated solution Ui,l (t, x) in the form of p 00 p 00 q 00 q 00 4X X X X 2 2 [w,s] Ui,l (t, x) = Tk (αw )Tr (βs )Tr ( (t−tl )−1)Tk ( (x−xi )−1)Ui,l . pq τ h k=0 w=0 r=0 s=0
3
Convergence analysis
The collocation methods for one dimensional Volterra integral equations have order p for any choice of collocation parameters, 0 ≤ ξ1 < ξ2 < · · · < ξp ≤ 1, and can be achieved local superconvergence in the mesh points by opportunely choosing the collocation parameters, as stated by the following theorem[2]. Theorem 3.1 Suppose that the given functions in the following VIE Z t u(t) = g(t) + k(t, τ, u(τ ))dτ , t ∈ I := [0, T ], 0
satisfy k ∈ C (p) (D × R), g ∈ C (p) (I), where D := {(t, τ ) : 0 ≤ τ ≤ t ≤ T }. Then, for any choice of the collocation parameters 0 ≤ ξ1 < ξ2 < · · · < ξp ≤ 1, the error (t) = u(t) − up (t), which up (t) is the approximated solution by the collocation method, satisfies kk∞ = O(hp ). 5
Suppose moreover that k ∈ C (2p−ν) (D × R), g ∈ C (2p−ν) (I), for some ν ∈ {0, 1, 2}, then If the collocation parameters are the Radau II points for (0, 1] we have, for ν = 1 max |ε(tn )| = O(h2p−1 ).
n=0,...,N
If the collocation parameters are p Lobatto points for [0, 1] or p − 1 GaussLegedre points for [0, 1] with ξp = 1, we have, for ν = 2, max |ε(tn )| = O(h2p−2 ).
n=0,...,N
Theorem 3.2 Assume that f (t, x) and G(t, z, x, y) are sufficiently differ˜ (t, x) is the approximated solution entiable functions of their variables and U 0 0 of (1) in Sq (ΠN ) × Sp (ΠM ) that is of degree p and q with respect to spatial and temporal variables, respectively. Then the total error E(t, x) satisfies kEk∞ = O(τ q+1 ) + O(hp+1 ),
(t, x) ∈ D,
(14)
˜i,l (t, x) − u(t, x). where Ei,l (t, x) := E(t, x)|(tl ,tl+1 ]×(xi ,xi+1 ] = U Proof : By approximating the solution of (2) with respect to the second variable by collocation methods, we have sup |u(t, xi + α∈[−1,1]
h h (1 + α)) − Ui,l (t, xi + (1 + α))| = O(hp+1 ). 2 2
˜ (tl + τ (1+β), xi +ξw h) as approximate solution of VIEs Also, by assuming U 2 (6), we have ˜i,l (tl + τ (1 + β), xi + ξw h) − Ui,l (tl + τ (1 + β), xi + ξw h)| = O(τ q+1 ). sup |U 2 2 β∈[−1,1] Hence ˜k ≤ ku − U
h h (1 + α)) − Ui,l (t, xi + (1 + α))| + 2 2 α∈[−1,1] τ ˜i,l (tl + (1 + β), xi + ξw h) − Ui,l (tl + τ (1 + β), xi + ξw h)| sup |U 2 2 β∈[−1,1] sup |u(t, xi +
= O(hp+1 ) + O(τ q+1 ). 2 6
4
Numerical examples
In this section, we present the numerical results obtained using our new scheme on some problems. The aim of this investigation is to demonstrate numerically the generally superior performance of the new method.
Example 4.1 Consider the following linear integral equation 1 u(t, x) = x cos(t) − t + xt2 − x sin(x) sin(t) + x2 cos(x) sin(t) 2 1 2 1 − xt cos(x) − x2 t sin2 (t) + xt sin(t) − xt2 cos(t) 2 4 Z tZ x + (x sin(y) + t sin(z))u(z, y)dydz, 0
0
where t ∈ [0, 1], x ∈ [0, 1] and u(t, x) = x cos(t) − t is the exact solution of it. We use the method by p = q = 3 and m = M = 4, 8, 16, 32. Absolute error of the method in some points is reported in Table 1. Figure 1 shows the numerical solution of problem that is continues. Figure 2 represents the error function where it is seen that the error growths when t and x increase. Table 1: Absolute error of example 4.1, p=q=3 (t,x) (0.1,0.2) (0.3,0.5) (0.5,0.5) (0.6,0.3) (0.8,0.6) (0.9,0.1) (1.0,1.0)
M=N=4 3.29×10−6 6.76×10−5 1.27×10−5 1.56×10−4 2.03×10−4 1.24×10−3 4.06×10−3
M=N=8
M=N=16
M=N=32
5.03×10−6 5.84×10−6 2.55×10−5 1.44×10−5 3.23×10−5 5.83×10−4 7.00×10−4
2.33×10−7 5.68×10−6 1.05×10−5 4.94×10−5 8.14×10−5 1.11×10−5 2.77×10−4
1.66×10−7 1.15×10−6 2.40×10−6 1.85×10−6 4.30×10−6 3.44×10−5 7.27×10−5
7
1.0
0.5
0.0
−0.5
−1.0 1.0
0.75
0.5
0.25
0.0 0.0
0.25
0.5
0.75
1.0
t
x
Figure 1: The plot of the numerical solution of example 4.1, for p=q=3 and M=N=4
Example 4.2 Consider the problem Z tZ u(t, x) = 0
x
xt(xy + tz)u(z, y)dydz + sin(xt) + xt
0
− x3 t −
t5 x3 t3 x5 − t3 x + x2 sin(xt) − + t2 sin(xt) 6 6
where t, x ∈ [0, 1]. The exact solution of this problem is u(t, x) = sin(xt) + xt. In Table 2. the absolute error of the method in some points is reported. In Figures 3 and 4, the plot of numerical solution and the function of error are shown, respectively.
8
10−3
4
3
2
1
0 1.0
1.0 0.75 0.75
0.5 0.5 x
0.25
0.25 0.0
t
0.0
Figure 2: The plot of the error function of example 4.1, for p=q=3 and M=N=4
Table 2: Absolute error of the method for example 4.2. (t,x) (0.1,0.2) (0.3,0.5) (0.5,0.5) (0.6,0.3) (0.8,0.6) (0.9,0.1) (1.0,1.0)
M=N=4
M=N=8
7.08×10−6 6.86×10−5 4.21×10−4 7.61×10−5 4.57×10−4 1.15×10−5 7.68×10−4
5.69×10−7 3.24×10−6 1.25×10−5 2.57×10−5 1.39×10−5 2.44×10−5 3.21×10−4
M=N=16 3.26×10−7 2.43×10−6 9.29×10−5 4.71×10−5 2.22×10−5 5.81×10−5 4.70×10−4
M=N=32 9.71×10−8 2.29×10−6 1.97×10−5 6.12×10−6 1.38×10−5 7.43×10−6 9.91×10−5
Example 4.3 Consider the following linear integral equation u(t, x) = xet + tex − x2 t + xt2 e−x − xt2 + x2 tt2 e−(x+t) + x2 te−t Z tZ x + xte−(z+y) u(z, y)dydz, 0
0
where t ∈ [0, 1], x ∈ [0, 1]. The exact solution of this problem is u(t, x) = xet + tex . We use the method by p = q = 3 and m = M = 4, 8, 16, 32. The absolute error of the method in some points is reported in Table 3. Figures 5 and 5 show the the numerical solution of the problem and its function error, respectively. 9
1.5
1.0
0.5
0.0 1.0
0.75
0.5
0.25
0.01.0
0.75
0.5
0.25
0.0
x
t
Figure 3: The plot of the numerical solution of example 4.2, for p=q=3 and M=N=4
10−3 7.5 5.0 2.5 0.0 −2.5 −5.0 1.0
1.0
0.75
0.75
0.5
0.5 x
0.25
0.25 0.0 0.0
t
Figure 4: The plot of the error function of example 4.2, for p=q=3 and M=N=4
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Table 3: Absolute error of the method for example 4.3. (t,x) (0.1,0.2) (0.3,0.5) (0.5,0.5) (0.6,0.3) (0.8,0.6) (0.9,0.1) (1.0,1.0)
5
M=N=4
M=N=8
9.46×10−5 3.90×10−5 8.30×10−4 2.60×10−4 3.78×10−4 1.37×10−4 4.14×10−3
5.11×10−6 7.36×10−5 2.20×10−4 1.81×10−6 3.71×10−4 1.54×10−5 9.19×10−4
M=N=16
M=N=32
9.08×10−7 2.86×10−6 6.82×10−5 2.72×10−5 5.78×10−4 9.56×10−5 2.95×10−4
7.00×10−7 8.88×10−6 6.51×10−5 7.87×10−6 4.11×10−6 5.14×10−5 1.29×10−5
Conclusion
The presented method can be apply on the two dimensional Volterra integral equations to obtain a solution in the form of piecewise polynomials that is continues in discretization nodes with good accuracy properties. Although, theoretically for getting higher accuracy we can set the method with larger values of M and N and also larger of the degree of approximation, p and q, but it leads to solving M N linear systems of size pq × pq, that can be have its difficulties.
11
5
4
3
2
1
0 1.0
0.75
0.5
0.25
t
0.75 0.5 0.0 1.0 x
0.25
0.0
Figure 5: The plot of the numerical solution of example 4.3, for p=q=3 and M=N=4
10−3 4
2
0
−2
1.0
0.75
0.5 x
1.0 0.75 0.5 t 0.25 0.25
0.0
0.0
Figure 6: The plot of the error function of example 4.3, for p=q=3 and M=N=4
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References [1] K. E. Atkinson, The Numerical Solution of Integral Equation of the second Kind., Cambridge Univ Press, Cambridge, 1997. [2] H. Brunner, P. J. Van Der Houwen, The numerical solution of nonlinear Volterra equations, CWI Monographs, Vol. 3, North-Holland, Amesterdam, 1986. [3] H. Brunner, Collocation method for Volterra Integral and related functional Differential Equation, Cambridge University Press, 2004. [4] L. Fox, I. B. Parker, Chebyshev Polynomials in Numerical Analysis, Oxford University Press, London, 1968. [5] A. Tari, Numerical solution of two dimensional integral and integrodifferential equations with expansion and operetional matrices method, Ph.D. thesis, University of Tabriz, Tabriz, 2008. [6] A. Tari, M. Y. Rahimi, S. Shahmorad, F. Talati, Solving a class of two-dimentional linear and nonlinear Volterra integral equations by the differential transform method, J. Comput. Appl. Math. 228 (2009) 70–76.
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