In this paper, the existence and uniqueness of solution of singular Hammerstein-Volterra integral equation ( H-VIE) are considered. Toeplitz matrix ( TMM) and product Nystrom method ( PNM) to solve the H-VIE with singular logarithmic kernel are used. The absolute error is calculated.
The singular integral equations are considered to be of more interest than the others and a close form of solution is generally not available. Therefore, great attention must be considered for the numerical solution of these equations. Abdou in [
Consider:
μ ϕ ( x , t ) = f ( x , t ) + λ ∫ − a a K ( x , y ) γ ( y , t , ϕ ( y , t ) ) d y + λ ∫ 0 t F ( t , τ ) ϕ ( x , τ ) d τ (1)
This formula is measured in L 2 [ − a , a ] × C [ 0 , T ] , T < ∞ , where the FI term is measured with respect to position. While the VI term is considered in time, and f ( x , t ) is known function. λ is the parameter, while μ defines the kind of the integral Equation (1).
We assume:
1) K ( x , y ) ∈ C ( [ − a , a ] × [ − a , a ] ) , and satisfies:
[ ∫ − a a ∫ − a a | K ( x , y ) | 2 d y d x ] 1 2 = A 1 < ∞ , ( A 1 is a constant)
2) F ( t , τ ) ∈ C ( [ 0 , T ] × [ 0 , T ] ) , 0 ≤ τ ≤ t ≤ T ≤ ∞ , satisfies:
| F ( t , τ ) | ≤ A 2
3) f ( x , t ) is continuous in L 2 [ − a , a ] × C [ 0 , T ] where:
‖ f ( x , t ) ‖ = max 0 ≤ t ≤ T ∫ 0 t [ ∫ a b | f ( x , τ ) | 2 d x ] 1 2 d τ = A 3
4) γ ( x , t , ϕ ( x , t ) ) , satisfies for the constant B > B 1 , B > p , the following conditions:
a) ∫ 0 t ∫ a b ( | γ ( x , t , ϕ ( x , t ) ) | 2 d x d t ) 1 2 ≤ B 1 ‖ ϕ ( x , t ) ‖ L 2 [ a , b ] × C [ 0 , T ]
b) ‖ γ ( x , t , ϕ 1 ( x , t ) ) − γ ( x , t , ϕ 2 ( x , t ) ) ‖ ≤ N ( x , t ) | ϕ 1 ( x , t ) − ϕ 2 ( x , t ) |
where ‖ N ( x , t ) ‖ L 2 [ a , b ] × C [ 0 , T ] = p
In other words, we prove that the solution exists using the successive approximation method, also called the Picard method, that we pick up any real continuous function ϕ 0 ( x , t ) in L 2 [ − a , a ] × C [ 0 , T ] , we assume ϕ 0 ( x , t ) = f ( x , t ) , then construct a sequence ϕ n defined by
ϕ n ( x , t ) = f ( x , t ) + λ ∫ − a a K ( x , y ) γ ( y , t , ϕ n − 1 ( y , t ) ) d y + λ ∫ 0 t F ( t , τ ) ϕ n − 1 ( x , τ ) d τ , ( μ = 1 )
ϕ n − 1 ( x , t ) = f ( x , t ) + λ ∫ − a a K ( x , y ) γ ( y , t , ϕ n − 2 ( y , t ) ) d y + λ ∫ 0 t F ( t , τ ) ϕ n − 2 ( x , τ ) d τ , ( μ = 1 )
ψ n ( x , t ) = ϕ n ( x , t ) − ϕ n − 1 ( x , t ) = λ ∫ − a a K ( x , y ) [ γ ( y , t , ϕ n − 1 ( y , t ) ) − γ ( y , t , ϕ n − 2 ( y , t ) ) ] d y + λ ∫ 0 t F ( t , τ ) [ ϕ n − 1 ( x , τ ) − ϕ n − 2 ( x , τ ) ] d τ , n = 1 , 2 , ⋯
Then:
ϕ n ( x , t ) = ∑ i = 0 n ψ i ( x , t ) (2)
Hence
ψ n ( x , t ) = f ( x , t ) + λ ∫ − a a K ( x , y ) γ ( y , t , ψ n − 1 ( y , t ) ) d y + λ ∫ 0 t F ( t , τ ) ψ n − 1 ( x , τ ) d τ
Using the properties of the norm, we obtain:
‖ ψ n ( x , t ) ‖ ≤ | λ | ‖ ∫ − a a K ( x , y ) γ ( y , t , ψ n − 1 ( y , t ) ) d y ‖ + | λ | ‖ ∫ 0 t F ( t , τ ) ψ n − 1 ( x , τ ) d τ ‖
For n = 1 , we get
‖ ψ 1 ( x , t ) ‖ ≤ | λ | ‖ ∫ − a a K ( x , y ) γ ( y , t , ψ 0 ( y , t ) ) d y ‖ + | λ | ‖ ∫ 0 t F ( t , τ ) ψ 0 ( x , τ ) d τ ‖ ≤ | λ | ‖ ( ∫ − a a | K ( x , y ) | 2 d y ) 1 2 ( ∫ − a a | γ ( y , t , ψ 0 ( y , t ) ) | 2 d y ) 1 2 ‖ + | λ | ‖ ∫ 0 t | F ( t , τ ) | | ψ 0 ( x , τ ) | d τ ‖
Using Cauchy Schwarz inequality and from conditions (i)-(iv-a) with ψ 0 = f ( x , t ) and ‖ f ‖ = A 3 , we get
‖ ψ 1 ( x , t ) ‖ ≤ | λ | max ∫ 0 t [ ∫ − a a ( ∫ − a a | K ( x , y ) | 2 d y ∫ − a a | γ ( y , t , ψ 0 ( y , t ) ) | 2 d y ) d x ] 1 2 d τ 0 ≤ t ≤ T + | λ | A 2 ∫ 0 t ‖ ψ 0 ( x , τ ) ‖ d τ ≤ | λ | A 1 A 3 B 1 + | λ | A 2 A 3 ‖ t ‖
We have 0 ≤ τ ≤ t ≤ T ≤ ∞ , then m a x | t | = T = L , and then we have:
‖ ψ 1 ( x , t ) ‖ ≤ | λ | A 3 ( A 1 B 1 + A 2 L )
In general, we get:
‖ ψ 1 ( x , t ) ‖ ≤ | λ | n A 3 ( A 1 B 1 + A 2 L ) n = A 3 α n , α = | λ | ( A 1 B 1 + A 2 L ) (3)
This bound makes the sequence ψ n ( x , t ) converges if
α < 1 ⇒ | λ | < 1 A 1 B 1 + A 2 L (4)
The result (4), leads us to say that the formula (2) has a convergent solution. So let n → ∞ , we have:
ϕ ( x , t ) = ∑ i = 0 ∞ ψ i ( x , t ) = A 3 1 − α , ( α < 1 ) (5)
The infinite series of (5) is convergent, and ϕ ( x , t ) represents the convergent solution of Equation (1). Also each of ψ i is continuous, therefore ϕ ( x , t ) is also continuous.
To show that ϕ ( x , t ) is unique, we assume that ϕ ¯ ( x , t ) is also a continuous solution of (1) then, we write
ϕ ( x , t ) − ϕ ¯ ( x , t ) = λ ∫ − a a K ( x , y ) [ γ ( y , t , ϕ ( y , t ) ) − γ ( y , t , ϕ ¯ ( y , t ) ) ] d y + λ ∫ 0 t F ( t , τ ) [ ϕ ( x , τ ) − ϕ ¯ ( x , τ ) ] d τ , ( μ = 1 )
which leads us to the following:
‖ ϕ ( x , t ) − ϕ ¯ ( x , t ) ‖ ≤ | λ | ‖ ∫ − a a | K ( x , y ) | | γ ( y , t , ϕ ( y , t ) ) − γ ( y , t , ϕ ¯ ( y , t ) ) | d y ‖ + | λ | ‖ ∫ 0 t | F ( t , τ ) | | ϕ ( x , τ ) − ϕ ¯ ( x , τ ) | d τ ‖
Using conditions (iv-b), then we have:
‖ ϕ ( x , t ) − ϕ ¯ ( x , t ) ‖ ≤ | λ | max 0 ≤ t ≤ T ∫ 0 t [ ∫ − a a ∫ − a a ( | K ( x , y ) | d x d y ) 1 2 ( ∫ − a a N 2 ( x , t ) | ϕ ( x , t ) − ϕ ¯ ( x , t ) | 2 d y ) 1 2 ] d τ + | λ | ‖ ∫ 0 t | F ( t , τ ) | | ϕ ( x , t ) − ϕ ¯ ( x , t ) | d τ ‖
Finally, with the aid of conditions (i) and (ii):
‖ ϕ ( x , t ) − ϕ ¯ ( x , t ) ‖ ≤ α ‖ ϕ ( x , t ) − ϕ ¯ ( x , t ) ‖
Then:
( 1 − α ) ‖ ϕ ( x , t ) − ϕ ¯ ( x , t ) ‖ ≤ 0
Since ‖ ϕ ( x , t ) − ϕ ¯ ( x , t ) ‖ is necessarily non-negative, and α < 1 :
‖ ϕ ( x , t ) − ϕ ¯ ( x , t ) ‖ = 0 ⇒ ϕ ( x , t ) = ϕ ¯ ( x , t )
It follows that if (1) has a solution it must be unique.
Consider:
ϕ ( x , t ) = f ( x , t ) + λ ∫ − a a K ( x , y ) γ ( y , t , ϕ ( y , t ) ) d y + λ ∫ 0 t F ( t , τ ) ϕ ( x , τ ) d τ (6)
when t = 0 Equation (13) becomes:
ϕ 0 ( x ) = f 0 ( x ) + λ ∫ − a a K ( x , y ) γ ( y , ϕ 0 ( y ) ) d y (7)
where ϕ 0 ( x ) = ϕ ( x , 0 ) , f 0 ( x ) = f ( x , 0 ) .
The formula (7) represents HIE of the second kind at t = 0 . Divide the interval [ 0 , T ] , 0 ≤ t ≤ T < ∞ as 0 = t 0 ≤ t 1 < ⋯ < t k < ⋯ < t N = T , then using the quadrature formula, the Volterra integral term in (6) becomes:
∫ 0 t k F ( t , τ ) ϕ ( x , τ ) d τ = ∑ j = 0 k u j F ( t k , t j ) ϕ ( x , t j ) + o ( ℏ i p ˜ + 1 ) , ( ℏ k → 0 , p ˜ > 0 ) (8)
where ℏ k = max 0 ≤ j ≤ k h j , h j = t j + 1 − t j
Using (8) in (6), we have:
ϕ k ( x ) = f k ( x ) + λ ∫ − a a K ( x , y ) γ ( y , t k , ϕ k ( y ) ) d y + λ ∑ j = 0 k u j F k j ϕ j ( x ) (9)
where ϕ k ( x ) = ϕ ( x , t k ) , f k ( x ) = f ( x , t k ) , F k j = F ( t k , t j ) .
μ n ϕ n ( x ) = G n ( x ) + λ ∫ − a a K ( x , y ) ϕ n ( y ) d y (10)
where μ n = 1 − λ F n n u n , G n ( x ) = f n ( x ) + λ ∑ j = 0 n − 1 u j F n j γ ( x , t j , ϕ j ( x ) ) , n = 0 , 1 , ⋯ , N .
The formula (10) represents SHIEs of the second kind, and we have N unknown ϕ n ( x ) .
In this section, we present the TMM to obtain numerical solution for HIE of the second kind with singular kernel. Consider:
ϕ ( x ) = f ( x ) + λ ∫ − a a K ( | x − y | ) γ ( y , ϕ ( y ) ) d y (11)
Write the integral term in the form:
∫ − a a K ( | x − y | ) γ ( y , ϕ ( y ) ) d y = ∑ n = − N N − 1 ∫ n h n h + h K ( | x − y | ) γ ( y , ϕ ( y ) ) d y , ( h = 2 a N ) (12)
Approximate the integral in the right hand side of Equation (12) by:
∫ n h n h + h K ( | x − y | ) γ ( y , ϕ ( y ) ) d y = A n ( x ) γ ( n h , ϕ ( n h ) ) + B n ( x ) γ ( n h + h , ϕ ( n h + h ) ) + R (13)
where A n ( x ) and B n ( x ) are two arbitrary functions. Putting ϕ ( x ) = 1 , x in Equation (13), where in this case we choose R = 0 . By solving the result, then we take:
A n ( x ) = 1 h [ γ ( n h + h , n h + h ) I ( x ) − γ ( n h + h , 1 ) J ( x ) ] (14)
And
B n ( x ) = 1 h [ γ ( n h + h , 1 ) J ( x ) − γ ( n h , n h ) I ( x ) ] (15)
where:
I ( x ) = ∫ n h n h + h K ( | x − y | ) γ ( y , 1 ) d y (16)
J ( x ) = ∫ n h n h + h K ( | x − y | ) γ ( y , y ) d y (17)
The relation (12), becomes:
∫ − a a K ( | x − y | ) γ ( y , ϕ ( y ) ) d y = ∑ n = − N N D n ( x ) γ ( n h , ϕ ( n h ) )
where
D n ( x ) = { A − N ( x ) , n = − N A n ( x ) + B n ( x ) , − N < n < N B N − 1 ( x ) , n = N (18)
The IE (11) becomes:
ϕ ( x ) − λ ∑ n = − N N D n ( x ) γ ( n h , ϕ ( n h ) ) = f ( x ) (19)
Putting x = m h , we have:
ϕ m − λ ∑ n = − N N D n , m γ n ( ϕ n ) = f m , − N ≤ m ≤ N (20)
where ϕ m = ϕ ( m h ) , D n , m = D n ( m h ) , f m = f ( m h ) .
The matrix D n , m may be written as D n , m = G n , m + E n , m , where:
G n , m = A n ( m h ) + B n − 1 ( m h ) , − N ≤ n , m ≤ N (21)
Is a Toeplitz matrix of order 2 N + 1 and:
E n , m ( x ) = { B − N − 1 ( x ) , n = − N , m = − N + i 0 , − N < n < N A N ( x ) , n = N , m = − N + i (22)
where 0 ≤ i ≤ 2 n . The solution of the formula (20):
ϕ m = [ I − λ ( G n , m + E n , m ) ] − 1 f m , | I − λ ( G n , m + E n , m ) | ≠ 0 (23)
Also
R = | ∫ − a a K ( | x − y | ) γ ( y , ϕ ( y ) ) d y − ∑ n = − N N D n m γ ( n h , ϕ ( n h ) ) | (24)
Consider:
ϕ ( x ) − λ ∫ − a a p ( x , y ) K ¯ ( x , y ) γ ( y , ϕ ( y ) ) d y = f ( x ) (25)
where p and K ¯ are badly behaved and well-behaved functions of their arguments, respectively. Then, we get:
ϕ ( x i ) − λ ∑ j = 0 N w i j K ¯ ( x i , y j ) γ ( y j , ϕ ( y j ) ) = f ( x i ) (26)
where x i = y i = a + i h , i = 0 , 1 , ⋯ , N with h = 2 a N , N even and w i j are the weights. When x = x i , we write:
∫ − a a p ( x i , y ) K ¯ ( x i , y ) γ ( y , ϕ ( y ) ) d y = ∑ j = 0 N − 2 2 ∫ y 2 j y 2 j + 2 p ( x i , y ) K ¯ ( x i , y ) γ ( y , ϕ ( y ) ) d y (27)
Form relation (25) through (27) we find:
∑ j = 0 N w i j K ¯ ( x i , y j ) γ ( y j , ϕ ( y j ) ) = ∑ j = 0 N − 2 2 ∫ y 2 j y 2 j + 2 p ( x i , y ) K ¯ ( x i , y ) γ ( y , ϕ ( y ) ) d y (28)
Then, we obtain:
∫ − a a p ( x i , y ) K ¯ ( x i , y ) γ ( y , ϕ ( y ) ) d y = ∑ j = 0 N − 2 2 ∫ y 2 j y 2 j + 2 p ( x i , y ) { ( y 2 j + 1 − y ) ( y 2 j + 2 − y ) 2 h 2 γ ( y 2 j , ϕ ( y 2 j ) ) + ( y − y 2 j ) ( y 2 j + 2 − y ) h 2 γ ( y 2 j + 1 , ϕ ( y 2 j + 1 ) ) + ( y 2 j + 1 − y ) ( y 2 j − y ) 2 h 2 γ ( y 2 j + 2 , ϕ ( y 2 j + 2 ) ) } d y
Therefore:
w i , 0 = β 1 ( y i ) w i , 2 j + 1 = 2 γ j + 1 ( y i ) w i , 2 j = α i ( y i ) + β j + 1 ( y i ) w i , N ( y i ) = α N 2 ( y i ) (29)
where:
α j ( y i ) = 1 2 h 2 ∫ y 2 j − 2 y 2 j p ( y i , y ) ( y − y 2 j − 2 ) ( y − y 2 j − 1 ) d y β j ( y i ) = 1 2 h 2 ∫ y 2 j − 2 y 2 j p ( y i , y ) ( y − y 2 j − 1 ) ( y − y 2 j ) d y γ j ( y i ) = 1 2 h 2 ∫ y 2 j − 2 y 2 j p ( y i , y ) ( y − y 2 j − 2 ) ( y 2 j − y ) d y (30)
We now introduce the change of variable y = y 2 j − 2 + ζ h , 0 ≤ ζ ≤ 2 thus the system (30) becomes:
α j ( y i ) = h 2 ∫ 0 2 ζ ( ζ − 1 ) p ( y 2 j − 2 + ζ h , y i ) d ζ
β j ( y i ) = h 2 ∫ 0 2 ( ζ − 1 ) ( ζ − 2 ) p ( y 2 j − 2 + ζ h , y i ) d ζ
γ j ( y i ) = h 2 ∫ 0 2 ζ ( 2 − ζ ) p ( y 2 j − 2 + ζ h , y i ) d ζ
If we define:
ψ i = ∫ 0 2 ζ i p ( y 2 j − 2 + ζ h , y i ) d ζ
For p ( x , y ) = p ( x − y ) , we have:
ψ i = ∫ 0 2 ζ i p ( y i , y 2 j − 2 + ζ h ) d ζ , i = 0 , 1 , 2 (31)
When y i − y 2 j − 2 = ( i − 2 j + 2 ) h . If we assume z = i − 2 j + 2 , then:
α j ( y i ) = h 2 ∫ 0 2 ζ ( ζ − 1 ) p ( z − ζ ) d ζ β j ( y i ) = h 2 ∫ 0 2 ( ζ − 1 ) ( ζ − 2 ) p ( z − ζ ) d ζ γ j ( y i ) = h 2 ∫ 0 2 ζ ( 2 − ζ ) p ( z − ζ ) d ζ (32)
Hence, the system (29) becomes:
w i , 0 = h 2 [ 2 ψ 0 ( z ) − 3 ψ 1 ( z ) + ψ 2 ( z ) ] , z = i w i , 2 j + 1 = h [ 2 ψ 1 ( z ) − ψ 2 ( z ) ] , z = i − 2 j
w i , 2 j = h 2 [ ψ 2 ( z ) − ψ 1 ( z ) + 2 ψ 0 ( z − 2 ) − 3 ψ 1 ( z − 2 ) + ψ 2 ( z − 2 ) ] , z = i − 2 j + 2 w i , N = h 2 [ ψ 2 ( z ) − ψ 1 ( z ) ] , z = i − N + 2 (33)
Therefore, the integral Equation (25) is reduced to SLAEs as in (26) or: ( I − λ W ) ϕ = F
Which has the solution:
ϕ = ( I − λ W ) − 1 F , | I − λ W | ≠ 0 (34)
The PNM is said to be convergent of order r in [ − a , a ] . If for N sufficiently large, there exists a constant C > 0 independent of N such that:
‖ ϕ ( x ) − ϕ N ( x ) ‖ ≤ C N − r
We using TMM and PNM at N = 20 , 40 , T = 0.03 , 0.7 , λ = 1 , and μ = 1 . In Tables 1-4:
ϕ E x a c t ® Exact solution, ϕ T ® appro. sol. of TMM, E T ® the absolute error of TMM, ϕ N ® appro. sol. of PNM, E N ® the absolute error of PNM.
Example 1
Consider:
ϕ ( x , t ) = f ( x , t ) + λ ∫ − 1 1 ln | x − y | ( y t ) 2 d y + λ ∫ 0 t τ 2 ϕ ( x , τ ) d τ
| T | x | ϕ E x a c t | ϕ T | E T | ϕ N | E N |
|---|---|---|---|---|---|---|
| 0.03 | −1.0 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1.0 | −0.03000000 −0.02400000 −0.01800000 −0.01200000 −0.00600000 0 0.006000000 0.012000000 0.018000000 0.024000000 0.030000000 | −0.030701591 −0.023838516 −0.017855743 −0.011903314 −0.005955438 0.000000452 0.0059710686 0.0011959300 0.0179646910 0.0239827313 0.029999603 | 7.0159E−4 1.6148E−4 1.4425E−4 9.6685E−5 4.4561E−5 4.5279E−7 2.8931E−5 4.6995E−5 3.5308E−5 1.7268E−5 3.9617E−7 | −0.03002738 −0.02403073 −0.01803683 −0.01203176 −0.00632818 −0.00002599 0.005976067 0.011983505 0.017987883 0.023995269 0.030003419 | 2.7386E−5 3.0733E−5 3.6835E−5 3.1761E−5 3.2818E−5 2.5995E−5 2.3932E−5 1.6494E−5 1.2116E−5 4.7302E−6 3.4192E−6 |
| 0.7 | −1.0 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1.0 | −0.70000000 −0.56000000 −0.42000000 −0.28000000 −0.14000000 0 0.140000000 0.280000000 0.420000000 0.560000000 0.700000000 | −0.716358582 −0.556088118 −0.416502750 −0.277643540 −0.138874201 0.0001142305 0.1394831440 0.2792953877 0.4195246008 0.5600309299 0.7003734045 | 1.6358E−2 3.9118E−3 3.4972E−3 2.3564E−3 1.1257E−3 1.1423E−4 5.1685E−4 7.0461E−4 4.7539E−4 3.0929E−5 3.7340E−4 | −0.70031893 −0.56033293 −0.42054763 −0.28051997 −0.14061900 −0.00050225 0.139540294 0.279740200 0.419885056 0.560081782 0.700159944 | 3.1893E−4 3.3293E-4 5.4763E−4 5.1997E−4 6.1900E−4 5.0225E−4 4.5979E−4 2.5979E−4 1.1494E−4 8.1782E−4 1.5994E−4 |
| T | x | ϕ E x a c t | ϕ T | E T | ϕ N | E N |
|---|---|---|---|---|---|---|
| 0.03 | −1.0 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1.0 | −0.03000000 −0.02400000 −0.01800000 −0.01200000 −0.00600000 0 0.006000000 0.012000000 0.018000000 0.024000000 0.030000000 | −0.031354257 −0.023848333 −0.017860806 −0.011906018 −0.005956673 0.0000002134 0.005971517 0.0119602026 0.0179658449 0.0239839397 0.0300003812 | 1.3542E−3 1.5156E−4 1.3919E−4 9.3981E−5 4.3326 E−5 2.1349E−7 2.8482E−5 3.9797E−5 3.4155E−5 1.6060E−5 3.8126E−7 | −0.03002817 −0.02403524 −0.01803612 −0.01203478 −0.00603193 −0.00002798 0.059768287 0.011982306 0.017988279 0.0239946215 0.030002272 | 2.8170E−5 3.5249E−5 3.6128E−5 3.4780E−5 3.1939E−5 2.7985E−5 2.3171E−5 1.7693E−5 1.1720E−5 5.3784E−6 2.2728E-6 |
| 0.7 | −1.0 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1.0 | −0.70000000 −0.56000000 −0.42000000 −0.28000000 −0.14000000 0 0.140000000 0.280000000 0.420000000 0.560000000 0.700000000 | −0.731583858 −0.556312874 −0.416615893 −0.277703301 −0.138901367 0.0001086431 0.1394919213 0.2793130480 0.4195464227 0.5600523036 0.7003829896 | 3.1585E−2 3.6871E−3 3.3841E−3 2.2966E−3 1.0986E−3 1.0864E−4 5.0807E−4 6.8695E−4 4.5357E−4 5.2303E−5 3.8298E−4 | −0.700337200 −0.560438272 −0.420531161 −0.280590379 −0.140598518 −0.000548653 0.1395580310 0.2791722263 0.4198942815 0.560066655 0.7001331775 | 3.3720E−4 4.3827E−4 5.3116E−4 5.9037E−4 5.9851E−4 5.4865E−4 4.4196E−4 2.8777E−4 1.0571E−4 6.6665E−5 1.3317E−4 |
| T | x | ϕ E x a c t | ϕ T | E T | ϕ N | E N |
|---|---|---|---|---|---|---|
| 0.03 | −1.0 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1.0 | −0.015000000 −0.012000000 −0.009000000 −0.006000000 −0.003000000 0 0.0030000000 0.0060000000 0.0090000000 0.0120000000 0.0150000000 | −0.015350899 −0.119193964 −0.008927989 −0.005951798 −0.002977777 −0.176673371 0.0029854757 0.0059795673 0.0089822306 0.0119912274 0.0149997058 | 3.5089E−4 8.0603E−5 7.2013E−5 4.8260E−5 2.2222E−5 1.7667E−7 1.4524E−5 2.0432E−5 1.7769E−5 8.7725E−6 2.9414E−7 | −0.0150137693 −0.0120155049 −0.0090185327 −0.0060159635 −0.0030164679 −0.0001304779 0.00298797512 0.00599166981 0.00899382675 0.01199749658 0.01500161353 | 1.3789E−5 1.5504E−5 1.8532E−5 1.5963E−5 1.6467E−5 1.3047E−5 1.2024E−5 8.3301E-6 6.1732E−6 2.5034E−6 1.6135E−6 |
| 0.7 | −1.0 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1.0 | −0.350000000 −0.280000000 −0.210000000 −0.140000000 −0.700000000 0 0.7000000000 0.1400000000 0.2100000000 0.2800000000 0.3500000000 | −0.358222414 −0.278111233 −0.208307768 −0.138862748 −0.069466949 0.0000300394 0.069707680 0.1395986121 0.209637503 0.279932202 0.3501244044 | 8.2224E−3 1.8887E−3 1.6922E−3 1.1372E−3 5.3305E−4 3.0039E−5 2.9231E−4 4.0138E−4 3.0262E−4 6.7797E−5 1.2440E−4 | −0.3502016929 −0.2802337016 −0.2103304274 −0.1403011727 −0.0703395348 −0.000278348 0.0697361145 0.1398209052 0.209873904 0.2799576210 0.3500176757 | 2.0169E−4 2.3370E−4 3.3042E−4 3.0117E−4 3.3953E−4 2.7834E−4 2.6388E−4 1.7909E−4 1.2609E−4 4.2379E−4 1.7675E−5 |
| T | x | ϕ E x a c t | ϕ T | E T | ϕ N | E N |
|---|---|---|---|---|---|---|
| 0.03 | −1.0 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1.0 | −0.015000000 −0.012000000 −0.009000000 −0.006000000 −0.003000000 0 0.0030000000 0.0060000000 0.0090000000 0.0120000000 0.0150000000 | −0.015677228 −0.011924354 −0.008930517 −0.005953091 −0.002978395 0.0000000570 0.0029857000 0.0059800185 0.0089828075 0.0119918316 0.0150009457 | 6.7722E−4 7.5645E−5 6.9482E−5 4.6908E−5 2.1604E−5 5.7023E−8 1.4299E−5 1.9981E−5 1.7192E−5 8.1683E−6 9.457E−8 | −0.015014180 −0.120177632 −0.009018179 −0.006017473 −0.003016028 −0.000014042 0.002988355 0.0059910700 0.008994024 0.0119971724 0.015001040 | 1.4180E−5 1.7763E−5 1.8179E−5 1.7473E−5 1.6028E−5 1.4042E−5 1.1644E−5 8.9299E−6 5.9752E−6 2.8275E−6 1.0403E−6 |
| 0.7 | −1.0 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1.0 | −0.350000000 −0.280000000 −0.210000000 −0.140000000 −0.700000000 0 0.7000000000 0.1400000000 0.2100000000 0.2800000000 0.3500000000 | −0.36583575 −0.27822360 −0.20836433 −0.13889262 −0.06948053 0.000027245 0.697120697 0.139607442 0.209704661 0.279942888 0.350129197 | 1.5835E−2 1.7763E−3 1.6356E−3 1.1073E−3 5.1946E−4 2.7245E−5 2.8793E−4 3.9255E−4 2.9533E−4 5.7111E−4 1.2919E−4 | −0.350210888 −0.280286378 −0.210322188 −0.14033638 −0.070329283 −0.000301555 0.6974498644 0.139806915 0.209878520 0.279950061 0.3500042976 | 2.1082E−4 2.8637E−4 3.2218E−4 3.3638E−4 3.2928E−4 3.0155E−4 2.5501E−4 1.9308E−4 1.2147E−4 4.9938E−5 4.2976E−6 |
Exact solution: ϕ ( x , t ) = x t
Example 2
Consider:
ϕ ( x , t ) = f ( x , t ) + λ ∫ − 1 1 ln | x − y | ( y t 2 ) 2 d y + λ ∫ 0 t t τ ϕ ( x , τ ) d τ
Exact solution: ϕ ( x , t ) = x t 2
The goal of this work is to study the H-VIE with singular kernel of the second kind. TMM and PNM are successive to solve this equation numerically. As N is increasing, the errors are decreasing. As t is increasing, the errors are increasing.
The author declares no conflicts of interest regarding the publication of this paper.
Al-Bugami, A.M. (2021) Singular Hammerstein-Volterra Integral Equation and Its Numerical Processing. Journal of Applied Mathematics and Physics, 9, 379-390. https://doi.org/10.4236/jamp.2021.92026