An Approximate Solution for Volterra Integral Equations of the Second Kind in Space with Weight Function

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1 International Journal of Mathematical Analysis Vol no HIKARI Ltd An Approximate Solution for Volterra Integral Equations of the Second Kind in Space with Weight Function M. E. Nasr Department of Mathematics Faculty of Science Benha University Benha Egypt M. F. Jabbar Department of Mathematics Faculty of Science Benha University Benha Egypt Copyright c 17 M. E. Nasr and M. F. Jabbar. This article is distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. Abstract The main aim of this study is to present a new and simple algorithm to prove that the function ϕ n (x) is a good approximation to the solution ϕ(x) for Volterra integral equations (VIEs) of the second kind in the space L p(x) [ with weight function p(x). This approximation is discussed in details with help of the Jackson s and Dirichlet s operators. Special attention is given to study the convergence analysis and estimation of an upper bound for the error of the approximated solution. Mathematics Subject Classification: 45D5 65R Keywords: Volterra integral equations Jackson s and Dirichlet s operators Convergence analysis 1 Introduction In this paper we present the approximate solution for Volterra integral equations of the second kind in the space L p(x) [ with weight function p(x) 1 p(x) is a summable function on [ ϕ(x) = f(x) + λ x k(x y)ϕ(y)dy x y (1)

2 85 M. E. Nasr and M. F. Jabbar the functions f(x) and k(x y) belong to L p(x) [ and are -periodic functions 1 is a regular value of the kernel k(x y) and the kernel k(x y) satisfies the following λ conditions a1: { x p(y) k(x y) dy} 1 = χ(x) L p(x) [ ; a: λ k(x y) L p < 1 k(x y) L p = k(x y) L p(x) [ = [ x p(x)p(y)[k(x y) dydx. The simplicity of finding a solution for Fredholm integral equations (FIEs) of the second kind with degenerate kernel naturally leads one to think of replacing the given equation (1) by FIE with degenerate kernel ( [1 [6 [1). The solution of the new equation is taken as an approximate solution of the original equation. The study employs Dzyadyk s method which is based on the linear polynomial operator ( [- [5). Eq.(1) can be written in the new form ϕ(x) = f(x) + λ k(x y)ϕ(y)dy () k(x y) = e(x y)k(x y) e(x y) = { 1 for y x for y > x. From (3) it is found that the kernel k(x y) in () satisfies the following conditions b1: { p(y) k(x y) dy} 1 = ρ(x) L p(x) [ ; b: λ k(x y) L p < 1 k(x y) L p [ = [ p(x)p(y)[ k(x y) dydx. Now instead of Eqs.(1) and () let us solve the following equations ϕ n (x) = U n (f; x) + λ U n [ k(. y); xϕ n (y)dy x y. (4) The notation U n [ k(. y); x will mean that the operator U n acts on k(x y) as a function of x and at the same time the variable y plays the role of a parameter. Now since the functions U n (f; x) and U n [ k(. y); x are both trigonometric polynomials of order n with respect to x the solution ϕ n (x) of Eq.(4) will also be trigonometric polynomial of order n in x. It is well-known that the problem of determination of the solution of Fredholm integral equation of the second kind with degenerate kernel is reduced (3)

3 Volterra integral equations 851 to the solution of a corresponding system of linear algebraic equations ( [14 [15). In this study it will be proved that the function ϕ n (x) is a good approximation to the solution ϕ(x) of Eq.(1) on the space L p(x). This approximation is discussed in details with the help of the Jackson s and Dirichlet s operators. Mathematical modeling of real-life problems usually results in functional equations like ordinary or partial differential equations integral and integro-differential equations stochastic equations. Many mathematical formulations of physical phenomena contain integral equations these equations arise in many fields like physics astronomy potential theory fluid dynamics biological models and chemical kinetics. Most of integral equations are usually difficult to solve analytically so it is required to obtain an efficient approximate solution [8. Recently several numerical methods to solve such integral equations have been given such as spline functions expansion [11 and collocation method ( [7 [16). Preliminaries and basic assumptions Starting from the known linear polynomial operators U n (g; x) which are good approximation to the function g(x) in the space L p(x) and have the form: λ (n) k U n (g; x) = 1 g(t)u n (x t)dt = 1 U n (x) = 1 + n k=1 are constants which define the method of approximation. g(x t)u n (t)dt (5) λ (n) k cos(kx) (6) Theorem.1. [9 For k(x y) belongs to L p(x) [ such that λ k(x y) L p f(x) belongs to L p(x) then the integral equation < 1 and has an unique solution ϕ(x) in L p(x) [. ϕ(x) = f(x) + λ k(x y)ϕ(y)dy Now with the help of the following theorem we will find the condition by which Eq.(4) han an unique solution. Theorem.. [9 If A and B are two bounded linear operators in Banach space E while A has an inverse and B E A 1 E < 1 then the operator (A + B) has also an inverse and ) 1. (A + B) 1 E A 1 E (1 B E A 1 E

4 85 M. E. Nasr and M. F. Jabbar To find this condition we rewrite both of Eqs.() and (4) in the operator form (I λ K)ϕ = f (I λu n ( K))ϕ n = f n Kϕ = k(x y)ϕ(y)dy U n ( K)ϕ n = U n [ k(. y); xϕ n (y)dy. It is obvious that I λ K = A λ( K U n ( K)) = B are two bounded linear operators in the space L p(x). It is well-known that the operator I λ K has an inverse for each λ such that 1 is a regular value of K λ [9. So Eq.() has an unique solution and we can write ϕ = (I + λr)f = f + λrf (I λ K) 1 = I + λr and R is the resolvent of the operator K. From theorem if λ (I λ K) 1 E K U n ( K) E < 1 then (I λu n ( K)) has also an inverse thereby Eq.(4) has an unique solution and can be written in the form ϕ n = (I + λr n )f n = f n + λr n f n (I λu n ( K)) 1 = I + λr n and R n is the resolvent of the operator U n ( K). Now we return to the functional representation of resolvents R(x y; λ) R n (x y; λ) and equations () and (4). Knowing the resolvent R(x y; λ) we at once obtain the solution of the original equation () with an arbitrary right hand side f(x) in the following form ϕ(x) = f(x) + λ R(x y; λ)f(y)dy. Also the solution of Eq.(4) can be represented through the resolvent as follows ϕ n (x) = f n (x) + λ R n (x y; λ)f n (y)dy. Theorem.3. [3 For any kernel k(x y) L p[ if the linear polynomial operator U n of order n is defined in L p(x) and if the function f(x) L p(x) then U n [ k(. y)f(y)dy; x = U n [k(. y); xf(y)dy.

5 Volterra integral equations Auxiliary definitions and theorems Definition 3.1. The averaged-modulus of continuity of the kernel k(x y) L p[ is defined as follows w L p (k; t) = w L p (t) = 1 [ sup s t x s Lemma 3.. The function w L p (t) has the following properties i1: w L p (t) for t ; i: w L p (t) is positive and monotonic increasing; i3: w L p (t 1 + t ) w L p (t 1 ) + w L p (t ); i4: w L p (t) is continuous; p(x)p(y)[k(x s y) k(x y) dxdy. (7) i5: for any positive real number η the following inequality holds w L p (ηt) (1 + η)w L p (t). Also by the averaged-modulus of continuity with respect to x and y of a function k(x y) = e(x y)k(x y) defined in [ we mean the following function Ω L p (t) Ω L p ( k; t) = Ω L p (t) = 1 [ sup p(x)p(y)[k(x y)[e(x s y) e(x y) dxdy. s t (8) It is evident that the function Ω L p (t) satisfies the above properties of the modulus of continuity (i1-i5). Definition 3.3. The value of the following norm δ n ( k) = δ( k; U n ) = U n ( k(. y); x) k(x y) L p = [ p(x)p(y)[u n ( k(. y); x) k(x y) dxdy (9) will play an important role in estimating the error arising from replacement of Eq.(1) by Eq.(4). The following theorem provides an estimate of δ( k; U n ). Theorem 3.4. For any kernel k(x y) L p[ and for any linear polynomial operator U n (g; x) we always have the following inequality δ n ( k) [w Lp ( 1n ) + Ω ( 1n ) Lp [n t + 1 U n (t) dt. (1)

6 854 M. E. Nasr and M. F. Jabbar Proof. Using Minkowski inequality and equalities (5) and (7) we obtain δ n ( k) = U n ( k(. y); x) k(x y) L p = = 1 [ k(x t y) k(x y)u n (t)dt = = 1 [ + U n (t) L p [ p(x)p(y) U n (t)[ k(x t y) y)dt k(x dydx [ [ U n (t) [ [[ U n (t) [ [[ U n (t) p(x)p(y)[ k(x t y) k(x y) dydx p(x)p(y)[e(x t y)k(x t y) e(x y)k(x y) dydx dt p(x)p(y)[e(x t y)[k(x t y) k(x y) dydx p(x)p(y)[k(x y)[e(x t y) e(x y) dydx dt dt p(x)p(y)[k(x t y) k(x y) dydx + p(x)p(y)[k(x y)[e(x t y) e(x y) dydx U n (t) [w L p ( t ) + Ω L p ( t )dt [w Lp ( 1n ) + Ω ( 1n ) Lp [n t + 1 U n (t) dt. dt Definition 3.5. We define the error of approximation of k(x y) as follows Enm( k) L p = k(x y) Tnm(x y) L p [ = inf T nm(xy) p(x)p(y)[ k(x y) T nm (x y) dxdy En ( k) L p = k(x y) Tn (x y) L p [ = inf T n (xy) p(x)p(y)[ k(x y) T n (x y) dxdy E m( k) L p = k(x y) T m(x y) L p [ = inf T m(xy) p(x)p(y)[ k(x y) T m (x y) dxdy

7 Volterra integral equations 855 T nm(x y) denotes the trigonometric polynomial in x of order n and in y of order m of best approximation of k(x y) in the metric L p[ T n (x y) denotes the trigonometric polynomial in x of order n of best approximation of k(x y) in the metric L p[ and T m(x y) denotes the trigonometric polynomial in y of order m of best approximation of k(x y) in the metric L p[. The estimates of how rapidly the quantities E nm( k) L p E n ( k) L p and E m( k) L p tend to zero as n m are given in [14 E nm( k) L p as n m and then E nm( k) L p E n ( k) L p and E nm( k) L p E m( k) L p E n ( k) L p as n (11) E m( k) L p as m. (1) Now we will mention the bounds of the norm (9) for various linear polynomial operators U n as the following cases: Case 1: Jackson s method [5: U n = J n we have From (13) and (1) we get From definition 3 we have δ( k; J n ) 1 E n ( k) L p k(x y) J n [ k(. y); x L p 1 Case : Dirichlet s method [5: U n = D n we have 1 From Eq.(16) and definition 3 we get [n t + 1 J n (t) dt < 6. (13) [w Lp ( 1n ) + Ω Lp ( 1n ). (14) [w Lp ( 1n ) + Ω Lp ( 1n ). (15) D n (t) dt < + ln n n. (16) δ( k; D n ) = k(x y) D n ( k(. y); x) L p = k(x y) Tn (x y) D n [ k(. y) Tn (. y); x L p En ( k) L p + 1 [ D n (t) p(x)p(y)[ k(x t y) T n (x t y) dydx [ D n (t) dt E n ( k) L p (3 + ln n)e n ( k) L p (17) dt

8 856 M. E. Nasr and M. F. Jabbar and from (15) we get δ( k; D n ) 1(3 + ln n)[w L p ( 1 n ) + Ω L ( 1 ). (18) p n Now from (14) (17) and (18) it is clear that δ n ( k) as n for Jackson s method for every periodic function k(x y) L p[ while in the case of Dirichlet s method δ( k; D n ) as n if w L p ( 1 ) = o(1/ln n) Ω n L ( 1 ) = o(1/ln n) or p n En ( k) L p = o(1/ln n). The following quantities will play an important role in estimating the error of our approximation ξ n = k(x y)[ϕ(y) U n (ϕ; y)dy (19) L p γ m = γ m (U n ; ϕ) = m i=1 E n (ϕ) L p = inf T n ϕ(x) T n (x) L p T n (x) is a trigonometric polynomial of order n in x m n. 1 λ (n) i E i 1 (ϕ) L p () Theorem 3.6. For any kernel k(x y) L p[ and for linear polynomial operator U n (g; x) the following inequality holds ξ n ( k) = ξ( k; U n ; ϕ) = k(x y)[ϕ(y) U n (ϕ; y)dy E m( k) L p ϕ(y) U n (ϕ; y) L p + for any positive integer m n. L p [ γ m(u n ; ϕ) p(x)dx [ k(x y) Lp + E m( k) Lp (1) Proof. For any function ϕ(x) L p with Fourier coefficients c i and d i in view of Bunyakovskii inequality and p(x) 1 we obtain 1 c i cos ix + d i sin ix = inf T i 1 (t) [ϕ(t) T i 1 (t)cos(i(x t))dt 1 [ [ inf p(t)[ϕ(t) T i 1 (t) [cos(i(x t)) 1 dt. dt T i 1 (t) p(t) [ inf p(t) ϕ(t) T i 1 (t) dt T i 1 (t) E i 1(ϕ) L p

9 Volterra integral equations 857 therefore Letting c i cos ix + d i sin ix L p T m (x y) = ( ) 1 E i 1(ϕ) L p p(x)dx. m a i (x)cos iy + b i (x)sin iy i= E m( k) L p = inf y) a i b i k(x m a i (x)cos iy + b i (x)sin iy L p i= and taking into consideration () and using Bunyakovskii inequality we obtain ξ n = ξ( k; U n ; ϕ) = k(x y)[ϕ(y) U n (ϕ; y)dy = [ [ p(x) + L p [ p(x) k(x y)[ϕ(y) U n (ϕ; y)dy dx [ [ p(x) inf T m(xy) k(x y) T m (x y) ϕ(y) U n (ϕ; y) dy ( k(x y) + T m (x y) k(x dx y))(ϕ(y) U n (ϕ; y))dy inf T m(xy) [ k(x y) T m (x y) ϕ(y) U n (ϕ; y) dy dx [ [ + p(x) inf ( k(x y) + T m (x y) k(x y)). T m(xy) ( m ) dx (1 λ (n) 1 i )(c i cos iy + d i sin iy) dy i=1 E m( k) L p ϕ(y) U n (ϕ; y) L p + [ γ m(u n ; ϕ) p(x)dx [ k(x y) Lp + E m( k) Lp. 4 The approximate solution and its error bounds The following theorem shows that for sufficiently good linear methods U n (g; x) the difference between the polynomials ϕ n (x) and the original solution ϕ(x) is sufficiently small. Theorem 4.1. If the kernel k(x y) of () satisfies the assumptions (b1 b) all functions appearing in () are periodic in x and y then for any linear polynomial operator U n (g; x) if λ Rδ( k; U n ) < 1 and if Eq.(1) is replaced by Eq.(4) the following inequality holds ϕ(x) ϕ n (x) L p (1 + α n ( k)) ϕ(x) U n (ϕ; x) L p ()

10 858 M. E. Nasr and M. F. Jabbar in which [ ξ( k; U n ; ϕ) / α n ( k) = λ R δ( k; U n ) + [1 λ Rδ( k; U n ) (3) ϕ(x) U n (ϕ; x) L p δ( k; U n ) and ξ( k; U n ; ϕ) are defined in (9) and (19) respectively and R = 1 + λ R(x y) L p R(x y) denotes the resolvent of the kernel k(x y). Proof. Using theorem 3 and Eq.() we represent the solution ϕ n (x) of Eq.(4) in the form ϕ n (x) = U n (f; x) + λu n [ k(. y)ϕ n (y)dy; x [ = U n (f; x) + λu n k(. y)[ϕ n (y) ϕ(y)dy + = λ = λ it follows that g n (x) =λ U n [ k(. y); x[ϕ n (y) ϕ(y)dy + U n [f(.) + λ U n [ k(. y); x[ϕ n (y) ϕ(y)dy + U n (ϕ; x) ϕ n (x) U n (ϕ; x) = λ k(. y)ϕ n (y)dy; x k(. y)ϕ(y)dy; x (4) k(x y)[ϕ n (y) U n (ϕ; y)dy + g n (x) (5) [U n ( k(. y); x) k(x y)[ϕ n (y) ϕ(y)dy + λ k(x y)[u n (ϕ; y) ϕ(y)dy. Thus by Eqs.(9) (1) and (19) we get the estimate g n (x) L p λ [U n ( k(. y); x) k(x y)[ϕ n (y) ϕ(y)dy L p + λ k(x y)[u n (ϕ; y) ϕ(y)dy L p λ δ( k; U n ) [ ϕ n (x) U n (ϕ; x) L + U p n (ϕ; x) ϕ(x) L + λ ξ( k; U p n ; ϕ). In view of λ k(x y) L p < 1 Eq.(5) has an unique solution given by ϕ n (x) U n (ϕ; x) = g n (x) + λ R(x y)g n (y)dy. (6)

11 Volterra integral equations 859 Therefore [ ϕ n (x) U n (ϕ; x) L p g n (x) L p 1 + λ R(x y) L p = R g n (x) L p [ R λ δ( k; U n ) [ ϕ n (x) U n (ϕ; x) L + U p n (ϕ; x) ϕ(x) L p Taking into consideration λ Rδ( k; U n ) < 1 we obtain Therefore + ξ( k; U n ; ϕ). ϕ n (x) U n (ϕ; x) L p λ R[δ( k; U n ) U n (ϕ; x) ϕ(x) L p + ξ( k; U n ; ϕ). 1 λ Rδ( k; U n ) ϕ(x) ϕ n (x) L p ϕ(x) U n (ϕ; x) L p + ϕ n (x) U n (ϕ; x) L p ϕ(x) U n (ϕ; x) L p + λ R[δ( k; U n ) U n (ϕ; x) ϕ(x) L p + ξ( k; U n ; ϕ) 1 λ Rδ( k; U n ) (1 + α n ( k)) ϕ(x) U n (ϕ; x) L p (7) α n is given by (3). Thus the inequality () is proved. 5 The results It is well-known that in [1 one cannot achieve an error less than the corresponding to the best approximation. The error estimate in () with rate of convergence α n ( k) means that the rate of convergence of ϕ n (x) to ϕ(x) is comparable with the rate of convergence of the best approximation which means that the error estimate () is optimal. Applying theorem 6 and also the corresponding results from section 3 we obtain the following results: In the case of the application of Jackson s method: The quantity α n ( k) in the relation () will not tend to zero for any solution ϕ(x) but will tend to zero only under the condition that the solution ϕ(x) belongs to some subclasses of integrable functions. Restricting ourselves to the Hölder classes W (r) H β (L p) r is a non-negative integer and < β 1 we obtain the following case: In order that α n ( k) as n considering (14) and [14 it is sufficient that the following conditioned be satisfied ϕ(x) W (r) H β (L p(x)) r + β < < β 1. In the case of the application of Dirichlet s method: From [14 and () we have ϕ(x) ϕ n (x) L p (1 + α n ( k))(3 + ln n)e n(ϕ) L p

12 86 M. E. Nasr and M. F. Jabbar E n(ϕ) L p is the best approximation of ϕ(x) by trigonometric polynomial and depends on the smoothness of ϕ(x) which in turn depends on the smoothness of f(x) and k(x y). As a consequence of (17) (18) (1) (3) and [14 we obtain the following estimate for α n ( k) α n ( k) λ R (3 + ln n)e n ( k) L p + E m( k) L p 1(3 + ln n)w λ R (1/n) + L E p m( k) L p 1 λr(3 + ln n)en ( k) L p 1 1λR(3 + ln n)w L p (1/n) α n ( k) as n for ϕ(x) L p(x) k(x y) L p(x) [ whenever w L p (1/n) = o(1/ln n) and Ω L p (1/n) = o(1/ln n) or E n ( k) L p = o(1/ln n). 6 Conclusion and remarks In this article we presented the approximate solutions of the Volterra integral equations of the second kind in the space L p(x) [ with weight function p(x) with the help of the Jackson s and Dirichlet s operators. In the same time we proved that the function ϕ n (x) is a good approximation to the exact solution ϕ(x) for the Volterra integral equations. References [1 S. Ashirov and N. Kurbanmamedov Investegation of the solution of a class of integral equations of Volterra type (in Russian) Izv. Vyssh. Uohebn. Zaved. Mat. 81 (1987). [ V. K. Dzyadyk V. T. Gavrilyuk and O. I. Stepanets On the best approximations of Hölder class functions by Rogozinski polynomials Dokl. Akad. Nauk Ukr. SSR. Ser. A 3 (1969). [3 V. K. Dzyadyk On the applications of linear methods to the approximation by polynomials of functions which are solution of Fredholm integral equation of the second kind I Urain. Matem. Zh. (197) [4 V. K. Dzyadyk V. T. Gavrilyuk and O. I. Stepanets On a precise upper bound of approximations on classes of differentiable periodic functions by Rogozinski polynomials Urainski. Matem. Zh. (197). [5 V. K. Dzyadyk Approximation Methods for Solutions of Differential and Integral Equations The Netherlands VSP [6 M. I. Hessein On the approximation of solutions of Fredholm integral equations of the second kind in the space L p Urain. Matem. Zh. 6 (1974) [7 M. M. Khader On the numerical solutions for the fractional diffusion equation Communications in Nonlinear Science and Numerical Simulations 16 (11)

13 Volterra integral equations 861 [8 M. M. Khader and N. H. Sweilam On the approximate solutions for system of fractional integro-differential equations using Chebyshev pseudo-spectral method Applied Mathematical Modelling 37 (13) [9 A. N. Kolmogorov and S. V. Fomin Elements of the Theory of Functions and Functional Analysis Dover Pubns [1 P. P. Korovkin Linear Operators and Approximation Theory Hindustan (India) (Translated from the Russian) 196. [11 S. T. Mohamed and M. M. Khader Numerical solutions to the second order Fredholm integro-differential equations using the spline functions expansion Global Journal of Pure and Applied Mathematics 8 (1) [1 R. V. Poljukov The Approximation Solution of Systems of Linear Integral Equations by means of Splines Nauk Moscow [13 I. G. Sokolov The remainder term of Fourier series of differentiable functions Dokl. Akad. Nauk SSSR 13 (1955). [14 A. F. Timan Theory of Approximations of Functions of a Real Variable Dover New York [15 F. G. Tricomi Integral Equations Dover New York [16 S. Yousefi and M. Razzaghi Legendre wavelet method for the nonlinear Volterra-Fredholm integral equations Math. Comp. Simul. 7 (5) Received: July 9 17; Published: September 17 17