Existence of solution and solving the integro-differential equations system by the multi-wavelet Petrov-Galerkin method

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1 Int. J. Nonlinear Anal. Appl. 7 (6) No., 7-8 ISSN: 8-68 (electronic) Existence of solution and solving the integro-differential equations system by the multi-wavelet Petrov-Galerkin method Mohsen Rabbani Department of Mathematics, Sari Branch, Islamic Azad University, Sari, Iran (Communicated by M.B. Ghaemi) Abstract In this paper, we discuss about existence of solution for integro-differential system and then we solve it by using the Petrov-Galerkin method. In the Petrov-Galerkin method choosing the trial and test space is important, so we use Alpert multi-wavelet as basis functions for these spaces. Orthonormality is one of the properties of Alpert multi-wavelet which helps us to reduce computations in the process of discretizing and we drive a system of algebraic equations with small dimension which it leads to approximate solution with high accuracy. We compare the results with similar works and it guarantees the validity and applicability of this method. Keywords: System of Integro-differential equations; Multi-wavelet; Petrov-Galerkin; Regular pairs; Trial space; Test space. MSC: Primary 5J5, 65L6, 65T6; Secondary C5, 3K8.. Introduction and preliminaries Construction and applications of multi-wavelet have been explained in [ such that these bases are orthonormal and also in [5, 9 the Petrov-Galerkin method has been used for solving integro-differential equations and in [, 8 some integro-differential equations have been solved by using semiorthogonal spline wavelet. Some integro-differential system are solved in [, 5 such that in [5 convergence of the Petrov-Galerkin method has been discussed with some restrictions on degrees of polynomial basis, but these restrictions removed in [9 and convergence is held for every degree of polynomial basis. Existence of solution for integro-differential equations or system of integro-differential equations has address: mrabbani@iausari.ac.ir (Mohsen Rabbani) Received: December Revised: September 5

2 8 Rabbani not been discussed in the previous research articles, (see [,, 3). But we introduce a process for proof of existence of solution. In [ the Tau method has introduced an approximate solution of integro-differential equations. In [6, six order compact finite difference method is given for solving second order integro-differential equation with different boundary condition, also it solved a system of integro-differential equation. This paper by proper choosing of test and trial space, the purposed method leads to diagonal matrices, so we obtain a final system with less dimension and computation. For validity and applicability the above proposed method we compare our results with[, 3. Consider the integro-differential equations system as follows: [α q (t)u q (t) + β q (t)u q (t) q= [γ q (t)u q (t) + δ q (t)u q (t) q= i= i= u () = α, u () = η, u () = α, u () = η, k i (t, s)u i (s)ds = f (t), (.) k i+ (t, s)u i (s)ds = f (t), where q is differential order and f i (t) L [,, i =, and k i (t, s) L [,, i =,, 3,, are known functions and the u i (t), i =, are unknown functions which must be determined. For discussing about existence of solution of equations system (.) in conditions of the Petrov-Galerkin method we need operator form of equations system (.), so we suppose that X = L [,, and κ i : X X for i =,, 3,, then integral operators form are as follows where κ i κ i (u j ) = o k i (t, s)u j (s)ds, i =,, 3,, j =, (.) for i =,, 3,, are linear compact operators. Also we assume that D i : X X, i =,, 3,, are linear compact operators such that D (u ) = α q (t)u q (t), D (u ) = β q (t)u q (t), D 3 (u ) = q= γ q (t)u q (t), D (u ) = q= q= q= δ q (t)u q (t), without loos of generality, we can write equations system (.) in the following operator system: u (κ u + κ u D u D u ) = f, (.3) u (κ 3 u + κ u D 3 u D u ) = f.

3 Existence of solution and solving the integro-differential... 7 (6) No., Existence of solution By considering the above mentioned operator system, we introduce operators κ and D in the following form( ) ( ) κ κ κ = D D, D = κ 3 κ D 3 D with suppose ˆκ i = κ i D i, we can write ˆκ = κ D, such that ˆκ operator can be introduced by ( ˆκ : L [, ) ( L [, ),( so from ) (.3) and the above explanations we have, u u f ˆκ =. u u ( ) f ( ) u f By choosing U = and F = the operator form of Eq. (.3) is given by Eq. (.), u f (I ˆκ) U = F. (.) In the case of one dimensional in Eq. (.), if ˆκ is an compact operator and it does not have as an eigenvalue, so has a unique solution u X = L [,, (see [3, 5). But for Eq. (.) in the case of two-dimensional { we} must prove ˆκ is a compact operator and (I ˆκ) is invertible. So we suppose u,n {U n } n =, is a bounded sequence in L u [, with a suitable norm such as,n n ( ) U n = max ( u,n, u,n ), x(t) = x(t) dt, (.) so, {u i,n } n for i =, are bounded sequences, on the other hand because κ i and D i for i =,, 3, are assumed linear compact operator in L [, then ˆκ i s for i=,,3, are linear compact operator. So, every one of the following sequences has a convergence subsequence in L [,, { ˆκ u,n } n, { ˆκ u,n } n, { ˆκ 3 u,n } n, { ˆκ u,n } n, { ˆκ u,n + ˆκ u,n } and { ˆκ 3 u,n + ˆκ u,n }, then { } ˆκ u {ˆκU n }=,n + ˆκ u,n ˆκ 3 u,n + ˆκ u,n, n has a convergence subsequence in L [,, so ˆκ is a linear compact operator. Also I ˆκ operator introduce as, I ˆκ : L [, ( L [,, ) I ˆκ where I ˆκ= ˆκ and I ˆκ 3 I ˆκ is an identity two-dimensional operator and I is an identity one-dimensional operator, so similar to one dimensional in [5 and concept of eigenvalues for I ˆκ we can rewrite (I ˆκ) U=λU, so ( )( ) λ λ λ u =, λ 3 λ λ u where λ is an eigenvalue of I ˆκ and λ i s are eigenvalues of ˆκ i s for i =,, 3,. So we have λ ( λ λ ) λ + ( λ ) ( λ ) λ λ 3 = thus for obtaining λ for (I ˆκ) and invertibility of the I ˆκ operator,we assume there is a relation among eigenvalues of κ i s operators for i =,, 3, as follows: ( λ ) ( λ ) λ λ 3, (.3) thus (I ˆκ) is invertible, so Eq. (.) and equations system (.) have a unique solution as u L [,.

4 Rabbani 3. Multi-wavelet Bases We have introduced the multi-wavelet bases constructed by Alpert in [ for L [,. The interval [, is divided to m subinterval where f(x) has degree less than k in all subintervals. The set Sm k is introduced as: { { } P olynomial with degree < k, Sm k n < t < n+, n = f f = m m m., Otherwise It is obvious that dim(s k m) = m k and S k S k. Assume that R k m is the complement of S k m in S k m+, then S k m+ = S k m R k m, S k m R k m. If we assume that h, h,..., h k : R R are the orthogonal basis functions for R k, then R k R k. In [ by orthonormalizing h, h,..., h k, a basis for space Rm k is obtained as: { Rm k = linear span h n j,m h n j,m(t) = m hj ( m t n), } j =,,..., k, n =,,..., m. On the other hand, S k m = S k m R k m = S k m p= R k p, for example if {p, p,..., p k } are the Legendre orthonormal polynomials on [,, then Sm k = span B { k = span {p,..., p k } } {h n j,p p =,,..., m, n =,,..., ( p ), j =,,..., k} = span{b j } m k j=. By choosing S k = m=sm k then S k = L [, and B k is known as the multi-wavelet basis of order k for L [, (see [, 9). Now we choose trial and test spaces by the multi-wavelet bases and introduce B matrices: If X n = S = span{, { { 3(t )}, }, < t < Y n = S = span,, < t <, then If B = ( 3 X n = S = span{, 3(t ), 5(6t 6t + ), { 7(t 3 3t + t )}, Y n = S = span, 3(t ), ). { 3( t + ), < t < 3(t 3), < t <, { 6t, < t < 6t 5, < t < },

5 Existence of solution and solving the integro-differential... 7 (6) No., 7-8 then If X n = S 6 and Y n = S 3 then B = B 6 6 = { X n = S = span, { 3( t + ), < t < 3(t ), 3(t 3), { Y n = S = span B = < t <, { {, < t <,, < t <, < t <,, < t <, , { 6t, < t < 6t 5, < t < {, < t < 3, 3 < t < }, }, where B T.B = D, which D is diagonal matrix with the positive elements. Thus (B T ) = B D exists. To know how to prove this part which the matrices are n n diagonal see [9. Also if X n = S and Y n = S, then the matrix B 8 8 is the following form: B 8 8 = Conditions of the Petrov-Galerkin Method In the case of one dimensional integro-differential equations, conditions of the Petrov-Galerkin and their theorems have been proved in [9, we try to expand them for a system of integro-differential 3

6 Rabbani equations. So for X = L [, follows In this way, three conditions of the Petrov-Galerkin method are: we consider an operator form of integro-differential equations as (I ˆκ) u = f. (.). X is Banach space and ˆκ is a compact linear operator it doesn t have as an eigenvalue.. For n N, X n X, Y n X, dimx n = dimy n <, i.e. X n, Y n are finite dimensional subsets of X, X sequentially. 3. H-Condition: For each x X and y X, there exists x n X n and y n Y n such that x n x and y n y as n +. The Petrov-Galerkin method for equation (.) is a numerical method for finding u n X n such that: < u n ˆκ n, y n >=< f, y n >, y n Y n. (.) Definition.. For linear operator P n : X X n, given x X, an element P n x X n is called a generalized best approximation from X n to x with respect to Y n if it satisfies the equation < x P n x, y n >=, y n Y n, similarly, given y X, an element P ny Y n is called a generalized best approximation from Y n to y with respect to X n if it satisfies the equation < x n, y P ny >=, x n X n. Above three conditions with Definition. illustrate that in the Petrov-Galerkin method, we are looking for the generalized best approximation x X with respect to Y n in [5. Also Eq. (.) is equivalent to u n P nˆκu n = f. (.3) This equation indicates that the Petrov-Galerkin method is a projection method. By assuming X = L [, in the case of two-dimensional and Eq. (.), and by considering definition of operator ˆκ, we show three conditions of the Petrov-Galerkin are held. Condition is held because we proved ˆκ is linear and compact operator. For invertibility we construct a condition on eigenvalues of ˆκ as follows, ( λ p ) ( λ q ) λ r λ f 3, (.) where λ i s are eigenvalues of ˆκ i operators for i =,, 3, and p, q, r and f are number of eigenvalue of ˆκ, ˆκ, ˆκ 3 and ˆκ respectively. ( ) S For condition we choose: X = L [,, Xn k = m Sm k, and also X = L [, (see ( ) S [7), and Yn k m = X n, where Sm, k Sm k have been introduced in [9. Of course X n in [9 is S k m one dimensional but in this paper Xn is two dimensional, also according to Sm k in [9 dim Xn = ( m k) =dim Yn, so condition is satisfied. By considering X = L [, and X = L [, condition 3 holds.

7 Existence of solution and solving the integro-differential... 7 (6) No., Solution of the Integro-Differential Equations System Again we consider to the equations system (.): we suppose u (t), u (t) are linear combination of a basis in X = L [,, so U n = combination of a basis in Xn L [,. Regarding to Definition () we can write, m=s k ( ) P n : X = L [, m = S Xn k = m, m=s k m S k m ( u,n u,n ) is a linear ( n i= P n (u(t)) = U n (t) = c ) ib i (t) n j= c, u(t) X. jb j (t) In other words {b i (t), b j (t)} is a base for subspace Xn from L [,. P n is an operator which explained completely in [9. By substituting u (t), u (t) in system (.) we have m n q= i= α q(t)c i b q i (t) + β q(t)c i b q i (t) [ n i,j= k ijb i (t)b j (s) [ n i,j= k ijb i (t)b j (s) [ n k= c kb k (s) ds = n i= f i(t)b i (t), m n q= i= γ q(t)c i b q i (t) + δ q(t)c i b q i (t) [ n i,j= k 3ijb i (t)b j (s) [ n i,j= k ijb i (t)b j (s) [ n k= c kb k (s) ds = n i= f i(t)b i (t). We assume that g (t) = α q (t)b q i (t), g (t) = g 3 (t) = q= γ q (t)b q i (t), g (t) = q= q= [ n k= c kb k (s) ds [ n k= c kb k (s) ds β q (t)b q i (t), (5.) δ q (t)b q i (t), q= with substituting (5.) in last system we have [ [c i g i (t) + c i g i (t) k ij b i (t)c j + i= i,j= [ [c i g 3i (t) + c i g i (t) k 3ij b i (t)c j + i= i,j= k ij b i (t)c j = i,j= k ij b i (t)c j = i,j= Now we project g ij (t), i =,, 3, on X n, so [ [c i g ij b i (t) + c i g ij b i (t) (k ij c j + k ij c j ) b i (t) = i,j= i,j= [ [c i g 3ij b i (t) + c i g ij b i (t) (k 3ij c j + k ij c j ) b i (t) = i,j= i,j= f i b i (t), i= f i b i (t). i= f i b i (t), i= f i b i (t). (5.) i=

8 Rabbani According to the petrov-galerkin method we multiply both sides of Eq. (5.) in bases elements of Y n (Y n is a dual space of X n ) and from orthonormality of test and trial spaces X n, Y n we have c i g ij < b j (t), b p(t) > + c i g ij < b j (t), b p(t) > i,j= i,j= [ (k ij c j + k ij c j ) < b j (t), b p(t) > = i,j= c i g 3ij < b j (t), b p(t) > + i,j= f i < b j (t), b p(t) >, i= c i g ij < b j (t), b p(t) > i,j= [ (k 3ij c j + k ij c j ) < b j (t), b p(t) > = i,j= f i < b j (t), b p(t) >, i= where < b j (t), b p(t) >, are elements of the B matrix, so the inner product equal to when j = p and it s when j p. So, we can write, [ c i g ip + c i g ip (k pj c j + k pj c j ) = f p, i,j= c i g 3ip + i,j= i,j= i,j= [ c i g ip (k 3pj c j + k pj c j ) = f p. i,j= i,j= By using boundary conditions we have following system and P =,,..., n, instead of two other equations we apply boundary conditions, so c i (g pi k ip ) + c i (g pi k ip ) = f p, (5.3) i= i= b i ()c i = α, i= c i (g 3pi k 3ip ) + i= b i ()c i = η, i= c i (g pi k ip ) = f p, i= b i ()c i = α, i= where in the system of (5.3) we have k ij = k 3ij = f p = b i ()c i = η. i= k (t, s)b i (t)b j (s), k ij = k 3 (t, s)b i (t)b j (s), k ij = f (t)b p (t)dt, f p = Matric form of system (5.3) is the following form ( ) g t B k g t k n n g3 t k 3 g t C k n = F. k (t, s)b i (t)b j (s), k (t, s)b i (t)b j (s), f (t)b p (t)dt.

9 Existence of solution and solving the integro-differential... 7 (6) No., Application of proposed method Here before solving systems we are going to show that condition (6) is held. To this end we have chosen some kernels. By solving them it s clear that condition (6) is held. (Other kernels are the same.) Example 6.. In this example we solve u u u (t) u (t) e ts u (s)ds e t+s u (s)ds (t + s)u (s)ds = + e t e t+ ( 3 + t ), (t + s) u (s)ds = + et + 3et t t u () =, u () =, u () =, u () =, ( + t 3 + t ), with exact solution of u (t) = t, u (t) = t. At first we show that condition (6) is held for these kernels. k (u(s)) = k 3 (u(s)) = e t+s u(s)ds, k (u(s)) = e ts u(s)ds, k (u(s)) = (t + s)u(s)ds, (t + s) u(s)ds, eigenvalue for k i th kernel is k i u = λ i u, i, so for k k u = λ u, also kernel we have e t+s u(s)ds = λ u(t), e t e s u(s)ds = λ u(t), by choosing a = es u(s)ds, we have (6.) u(t) = a λ e t, (6.) and by substituting (6.) in (6.) eigenvalue for k write k u = λ u, so is like this λ = e. Also for kernel k we can (t + s)u(s)ds = λ u(t), we suppose u(s)ds = a, su(s)ds = a, so we have and( by substituting (6.) in (6.3), we obtain λ ( t + )a 3 + ( ) t + ) a = ta + a, (6.3) a t + a = λ u(t), (6.)

10 6 Rabbani by reordering following relation and by knowing that {, t} are independent and a i s are not zero we will ( ( reach the following system λ ) ( ) ( ) ) ( a λ =, 3 ) a so eigenvalues for kernel κ are { λ = 3, 3 λ = Also for kernel k 3 we can write k 3 u(s) = λ 3 u(t). So, and according to following expansion e ts u(s)ds = λ 3 u(t) (6.5) and by choosing a i = ti i- by substituting (6.6) in (6.5) we have e st = + (st) + (st) +..., si u(s)ds, i=,, 3, in Eq. (6.5) we obtain u(t) = a + ta + t a 3 λ 3 (6.6) ( a + a + a 3 3 a λ 3 ) + t ( a + a 3 + a 3 λ 3 a ) + t ( a 6 + a 8 + a 3 λ 3a 3 ) =. Because {, t, t } are linearly independent so the last system will be ( λ 3 ) ( a 3 λ 3 3) ( λ ) a = a 3 So eigenvalues for k 3 will be produced as: λ 3 = i.83 7, λ 3 = i , λ 3 3 =.89 i Also for kernel k we can write k u(s)=λ u(t). So, (t + s) u(s)ds=λ u(t), u(s)ds + t su(s)ds + t s u(s)ds = λ u(t), by choosing a i = ti i- si u(s)ds, i=,, 3, we can write u(t)= a 3+ta +t a t ( a 3 + a ) ( + a 3 λ a + t a + a ) ( 3 + a 3 λ a + a 3 + a + a ) 3 3 λ a 3 =. According to previous, eigenvalues for k are λ = i , λ = i , λ 3 =.6753 i λ, so

11 Existence of solution and solving the integro-differential... 7 (6) No., Table. Numerical results for Example 6. error of u (t) error of u (t) X n = S Y n = S X n = S Y n = S condition (6) is held for kernels k i, i. By using system (5.3) for solving Example 6., the absolute errors in some different points are shown in Table. Example 6.. In this example we solve and compare our method with system of integro-differential equations which was solved in [, 3. Results are shown in Table. (also this example is solved by the Tau method in [ and Yousufouglu solved the same example with another method in [3. Differences are in Table 3. Of course condition (6) is held as before. u (t) + u (t) u (t) + u (t) 3 tsu (s)ds ( t s )u (s)ds 6 u () =, u () =, u () =, u () =, 6tsu (s)ds = 3t + 3t + 8, (t + s )u (s)ds = t + 5, exact solutions are u (t) = 3t +, u (t) = t 3 + t. The absolute errors in some different points are shown in Table 3. Table. Numerical results for Example 6. by proposed method error of u (t) error of u (t) X n = S, Y n = S X n = S, Y n = S N = N =

12 8 Rabbani Table 3. Numerical results for Example 6. by [, 3 Eu (t) Eu (t) Eu (t) Eu (t) in [ in [ in [3 in [3 N = N = Conclusion In this paper, we proved existence of solution for integro-differential system and also we used Alpert multi-wavelet bases to solve system of integro-differential equations. Suitable choices of test and trial space is very important. These orthogonal bases reduce the computations. So final systems have small dimension and enough accuracy. Of course there is not restriction on degrees of chosen polynomial basis. References [ A. Ayad, Spline approximation for first-order Fredholm integro-differential equation, Studia universitatis babes- Bolyai. Math. (996) 8. [ B.K. Alpert, A class of a bases in L for the sparse representaion of integral operators, SIAM J. Math. Anal. (993) 6 6. [3 K.E. Atkinson, The numerical solution of integral equations of the second kind, Cambridge University Press, 997. [ S.H. Behiry and H. Hashish, Wavelet methods for the numerical solution of Fredholm integro-differential equations, Int. J. Appl. Math. () [5 Z. Chen and Y. Xu, The Petrov-Galerkin and iterated Petrov-Galerkin methods for second kind integral equations, SIAM J. Num. Anal. 35 (998) 6 3. [6 Jichao zhao, Robert M.Corless, Compact finit difference method for integro-differential equations, Appl. Math. Comput. 77 (6) [7 E. Kreyszing, Introductory functional analysis with applications, John Wiley & Sons Inc., 978. [8 M. Lakestani, M. Razzaghi and M. Dehghan, Semiorthogonal spline wavelets approximation for Fredholm integrodifferential equations, Mathematical Problems in Engineering Vol.6, (6), Article ID 968,. [9 K. Maleknejad, M. Rabbani, M. Aghazade and M. karami, A Wavelet Petrov-Galerkin for solving integrodifferential equations, Int. J. comput. Math., vol., No., Month (8) 9. [ K. Maleknejad and M. Karami, Using the WPG Method for Solving Integral Equations of the Second Kind, Appl. Math. Comput. 66 (5) 3 3. [ M. Rabbani and R. Jamali, Solving Nonlinear System of Mixed Volterra-Fredholm Integral Equations by Using Variational Iteration Method, J. Math. Comput. Sci. 5 () [ J. pourmahmoud, M. Y. Rahimi-Ardabili, S. Shahmorad,Numerical solution of the system of fredholm integrodifferential equations by the Tau methods, Appl. Math. Comput. 68 (5) [3 Yusufouglu(Agadjanov), Numerical solving initial value problem for Fredholm type linear integro-differential equation system, Appl. Math. Comput. 36 (9)