Numerical Solutions to Fractional Perturbed Volterra Equations

and Applied Analysis 3 where Θ is a weight function, is called the scalar product of functions f, g on the interval 0, t . Let us recall that two functions are orthonormal when ∀i,j 〈 φi t , φj t 〉 δij , 2.2 where δij is the Kronecker delta. We are looking for an approximate solution to 1.1 as an element of the subspaceHnφ , spanned on nφ first basic functions {φj : j 1, 2, . . . , nφ} unφ x, t nφ ∑ j 1 cj x φj t . 2.3 For simplicity of notations, let us consider 1.1 in one spatial dimension only. Inserting 2.3 into 1.1 , one obtains unφ x, t u x, 0 ∫ t 0 a t − s a ∗ k t − s Δunφ x, s ds ∫ t 0 b t − s unφ x, s ds nφ x, t , 2.4 where function nφ represents the approximation error function. From 2.3 and 2.4 , one gets nφ x, t nφ ∑ j 1 cj x φj t − ∫ t 0 a t − s a ∗ k t − s nφ ∑ j 1 d2 dx2 cj x φj s ds − ∫ t 0 b t − s nφ ∑ j 1 cj x φj s ds − u x, 0 . 2.5 Definition 2.2. The Galerkin approximation of 1.1 is the function unφ ∈ Hnφ , such that nφ ⊥ Hnφ , that is, ∀j 1,2,...,n 〈 nφ x, t , φj t 〉 0. 2.6 4 Abstract and Applied Analysis It follows from Definitions 2.2 and 2.1 and 2.5 that 0 ∫ t 0 ⎡ ⎣ nφ ∑ j 1 cj x φj τ ⎤ ⎦φi τ Θ τ dτ − ∫ t 0 u x, 0 φi τ Θ τ dτ − ∫ t 0 ⎡ ⎣ ∫ τ 0 a τ − s a ∗ k τ − s nφ ∑ j 1 d2 dx2 cj x φj s ds ⎤ ⎦φi τ Θ τ dτ − ∫ t 0 ⎡ ⎣ ∫ τ 0 b τ − s nφ ∑ j 1 cj x φj s ds ⎤ ⎦φi τ Θ τ dτ for i 1, 2, . . . , nφ. 2.7 Therefore ∫ t 0 u x, 0 φi τ Θ τ dτ ∫ t 0 ⎡ ⎣ nφ ∑ j 1 cj x φj τ ⎤ ⎦φi τ Θ τ dτ − ∫ t 0 ⎡ ⎣ ∫ τ 0 a τ − s a ∗ k τ − s nφ ∑ j 1 d2 dx2 cj x φj s ds ⎤ ⎦φi τ Θ τ dτ − ∫ t 0 ⎡ ⎣ ∫ τ 0 b τ − s nφ ∑ j 1 cj x φj s ds ⎤ ⎦φi τ Θ τ dτ, i 1, 2, . . . , nφ. 2.8 Using 2.2 , 2.8 can be written in an abbreviated form gi x ci x − nφ ∑ j 1 aij d2 dx2 cj x − nφ ∑ j 1 bijcj x , 2.9 where gi x u x, 0 ∫ t 0 φi τ Θ τ dτ, 2.10 aij ∫ t 0 [∫ τ 0 a τ − s a ∗ k τ − s φj s ds ] φi τ Θ τ dτ, 2.11 bij ∫ t 0 [∫ τ 0 b τ − s φj s ds ] φi τ Θ τ dτ. 2.12 In general aij / aji. The solution of the set of nφ coupled differential equations 2.9 for coefficients cj x , j 1, 2, . . . , nφ provides Galerkin approximation 2.3 to 1.1 . Abstract and Applied Analysis 5 3. Discretization Equations can be solved using discretization in a space variable. In one-dimesional case, let us introduce a grid of points x1, x2, . . . , xnh , where xl − xl−1 h. The grid approximation of a second derivative of a function f : R → R is given by f ′′ x ≈ f x − h − 2f x f x h h2 O ( h3 ) . 3.1and Applied Analysis 5 3. Discretization Equations can be solved using discretization in a space variable. In one-dimesional case, let us introduce a grid of points x1, x2, . . . , xnh , where xl − xl−1 h. The grid approximation of a second derivative of a function f : R → R is given by f ′′ x ≈ f x − h − 2f x f x h h2 O ( h3 ) . 3.1 Then the set of equations 2.9 takes the following form: gi xl ci xl 1 h2 nφ ∑ j 1 aij [−cj xl−1 2cj xl − cj xl 1 ] − nφ ∑ j 1 bijcj xl ci xl 1 h2 nφ ∑ j 1 [ −aijcj xl−1 ( 2aij − hbij ) cj xl − aijcj xl 1 ] , 3.2 where i 1, 2, . . . , nφ and l 1, 2, . . . , nh. In two-dimensional case, with the grid x1, x2, . . . , xnh × y1, y2, . . . , ynh , where xl − xl−1 ym − ym−1 h for l,m 2, 3, . . . , nh, the set of equations 2.9 takes the form gi ( xl, ym ) ci ( xl, ym ) 1 h2 nφ ∑ j 1 aij [−cj(xl−1, ym) − cj(xl, ym−1) 4cj ( xl, ym ) − cj(xl 1, ym) − cj(xl, ym 1)] − nφ ∑ j 1 bijcj ( xl, ym ) ci ( xl, ym ) 1 h2 nφ ∑ j 1 [ − aijcj ( xl−1, ym ) − aijcj(xl, ym−1) ( 4aij − hbij ) cj ( xl, ym ) − aijcj(xl 1, ym) − aijcj(xl, ym 1) ] . 3.3 Both sets of linear equations 3.2 and 3.3 can be written in a matrix form