PREPRINT 2009:16. Existence and Uniqueness of the Solution of an Integro-Differential Equation with Weakly Singular Kernel FARDIN SAEDPANAH

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1 PREPRINT 29:16 Existence and Uniqueness of the Solution of an Integro-Differential Equation with Weakly Singular Kernel FARDIN SAEDPANAH Department of Mathematical Sciences Division of Mathematics CHALMERS UNIVERSITY OF TECHNOLOGY UNIVERSITY OF GOTHENBURG Göteborg Sweden 29

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3 Preprint 29:16 Existence and Uniqueness of the Solution of an Integro-Differential Equation with Weakly Singular Kernel Fardin Saedpanah Department of Mathematical Sciences Division of Mathematics Chalmers University of Technology and University of Gothenburg SE Göteborg, Sweden Göteborg, April 29

4 Preprint 29:16 ISSN Matematiska vetenskaper Göteborg 29

5 EXISTENCE AND UNIQUENESS OF THE SOLUTION OF AN INTEGRO-DIFFERENTIAL EQUATION WITH WEAKLY SINGULAR KERNEL FARDIN SAEDPANAH Abstract A hyperbolic type integro-differential equation with weakly singular kernel is considered together with mixed homogeneous Dirichlet and non-homogeneous Neumann bounadry conditions Existence and uniqueness of the solution is proved by means of Galerkin method Regularity estimates are proved and the limitations of the regularity is discussed 1 Introduction We study a model problem, which is a hyperbolic type integro-differential equation with weakly singular kernel This problem arises as a model for fractional order viscoelasticity The fractional order viscoelastic model, that is, the linear viscoelastic model with fractional order operators in the constitutive equations, is capable of describing the behavior of many viscoelastic materials by using only a few parameters There is an extensive literature regarding well-posedness and numerical treatment for integro-differential equations, see, eg, [1], [3], [9], [1], [11], [12], [13], [14], [15], [16], and [17] Existence, uniqueness and regularity of a reformed model has been studied in [12] by means of Fourier series One may also see [6], where the theory of analytic semigroups is used in terms of interpolation spaces An abstract Volterra equation, as an abstract model for equations of linear viscoelasticity, has been studied in [4] In a previous work [9], well-posedness and regularity of the model problem was studied in the framework of the semigroup of linear operators The drawback of the framework is that this does not admit non-homogeneous Neumann boundary condition While in practice mixed homogeneous Dirichlet and non-homogeneous Neumann boundary conditions are of special interest Here we investigate existence, uniqueness and regularity of the solution of the problem 25 by means of the Galerkin approximation method In the sequel, in 2 we describe the construction of the model and we define a weak generalized solution Then in 3 we study well-posedness, regularity and limitations for higher regularity Date: April 14, Mathematics Subject Classification 65M6, 45K5 Key words and phrases integro-differential equation, fractional order viscoelasticity, Galerkin approximation, weakly singular kernel, regularity, a priori estimate 1

6 2 F SAEDPANAH 2 The model problem and weak formulation Let σ ij, ɛ ij and u i denote, respectively, the usual stress tensor, strain tensor and displacement vector We recall that the linear strain tensor is defined by ɛ ij = 1 ui + u j 2 x j x i With the decompositions s ij = σ ij 1 3 σ kkδ ij, e ij = ɛ ij 1 3 ɛ kkδ ij, the constitutive equations are formulated in [2] as 21 s ij t + τ α1 σ kk t + τ α2 with initial conditions 1 Dα1 t 2 Dα2 t s ij t = 2G e ij t + 2Gτ α1 1 Dα1 t σ kk t = 3K ɛ kk t + 3Kτ α2 2 Dα2 t e ij t, s ij + = 2Ge ij +, σ kk + = 3Kɛ kk +, ɛ kk t, meaning that the initial response follows Hooke s elastic law Here G, K are the instantaneous unrelaxed moduli, and G, K are the long-time relaxed moduli Note that we have two relaxation times, τ 1, τ 2 >, and fractional orders of differentiation, α 1, α 2, 1, where the fractional order derivative is defined by D α t ft = D t D 1 α t ft = D t 1 Γ1 α t s α fs ds The contitutive equations 21 can be solved for σ by means of Laplace transformation, [7]: s ij t = 2G e ij t G G θ 1 t se ij s ds, G where σ kk t = 3K ɛ kk t K K K θ i t = d t dt E αi α i, E αi t = τ i θ 2 t sɛ kk s ds, n= t n Γ1 + nα i, and E αi is the Mittag-Leffler function of order α i We make the simplifying assumption synchronous viscoelasticity: α = α 1 = α 2, τ = τ 1 = τ 2, f = f 1 = f 2 Then we may define a parameter γ, a kernel β, and the Lamé constants µ, λ, γ = G G = K K G K, βt = γθt, µ = G, λ = K 2 3 G, and the constitutive equations become σ ij t = 2µɛ ij t + λɛ kk tδ ij βt s 2µɛ ij s + λɛ kk sδ ij ds, = σ ij t βt sσ ij s ds

7 WELL-POSEDNESS OF THE SOLUTION OF AN INTEGRO-DIFFERENTIAL EQ 3 Note that the viscoelastic part of the model contains only three parameters: < γ < 1, < α < 1, τ > The kernel is weakly singular: βt = γ d t α dt E α = γ α t 1+αE t α τ τ τ α 22 τ Ct 1+α, t, and we note the properties βt, β L1R + = βt dt = γ E α E α = γ < 1 The equations of motion now become, we denote time derivatives with : ρü i σ ij,j = f i in Ω, u i = on Γ D, σ ij n j = g i on Γ N, where ρ is the constant mass density, and f and g represent, respectively, the volume and the surface loads We let Ω R d, d = 2, 3, be a bounded polygonal domain with boundary Γ D Γ N = Ω, Γ D Γ N = and measγ D We set Au i = 2µɛ ij u + λɛ kk uδ ij,j, that is, Au = σ u, and we write the equations of motion 24 in the form 25 ρüx, t + Aux, t βt saux, s ds = fx, t in Ω, T, ux, t = on Γ D, T, σu; x, t n = gx, t on Γ N, T, ux, = u x in Ω, ux, = v x in Ω We introduce the function spaces H = L 2 Ω d, H ΓN = L 2 Γ N d, and V = {v H 1 Ω d : v ΓD = } We define the bilinear form with the usual summation convention au, v = 2µɛij uɛ ij v + λɛ ii uɛ jj v dx, u, v V, Ω which is coercive on V We denote the norms in H and H ΓN by and ΓN, respectively, and we equip V with the inner product a, and norm v 2 V = av, v Now we define a weak solution to be a function u = ux, t that satisfies u L 2, T ; V, u L 2, T ; H, ü L 2, T ; V, ρ üt, v + aut, v u = u, u = v βt saus, v ds = ft, v + gt, v ΓN, v V, ae t, T,

8 4 F SAEDPANAH Here gt, v ΓN = Γ N gt v ds, and, denotes the pairing of V and V We note that 26 implies, by a classical result for Sobolev spaces, that u C[, T ]; H, u C[, T ]; V so that the initial conditions 28 make sense for u H, v V 3 Existence, uniqueness and regularity In this section we prove existence and uniqueness as well as regularity of a weak solution of 25 using Galerkin method, in a similar way as for hyperbolic PDE s in [8], [5] To this end, we first introduce the Galerkin approximation of a weak solution of 25 in a classical way, and we obtain a priori estimates for approximate solutions These will be used to construct a weak solution and then we will verify uniqueness as well as regularity We recall 22, 23 and we define the function 31 ξt = γ and it is easy to see that 32 βs ds = t βs ds = γe α t, D t ξt = βt <, ξ = γ, lim t ξt =, < ξt γ Besides, ξ is a completely monotone function, that is, 1 j D j t ξt, t,, j N, since the Mittag-Leffler function E α, α [, 1] is completely monotone Consequently, an important property of ξ, is that it is a positive type kernel, that is, it is continuous and, for any T, satisfies 33 ξt sφtφs ds dt, φ C[, T ] 31 Galerkin approximations Let {λ j, ϕ j } j=1 be the eigenpairs of the weak eigenvalue problem 34 aϕ, v = λϕ, v, v V It is known that {ϕ j } j=1 can be chosen to be an ON-basis in H and an orthogonal basis for V Now, for a fixed positive integer m N, we seek a function of the form m 35 u m t = d j tϕ j to satisfy 36 ρü m t, ϕ k + au m t, ϕ k j=1 βt sau m s, ϕ k ds = ft, ϕ k + gt, ϕ k ΓN, k = 1,, m, t, T, with initial conditions m m 37 u m = u, ϕ j ϕ j, u m = v, ϕ j ϕ j j=1 j=1

9 WELL-POSEDNESS OF THE SOLUTION OF AN INTEGRO-DIFFERENTIAL EQ 5 Theorem 1 For each m N, there exists a unique function u m of the form 35 satisfying Moreover, if u V, v H, f L 2, T ; H, g W1 1, T ; H ΓN, there is a constant C = CΩ, γ, ρ, T such that, 38 u m L,T ;V + u m L,T ;H + ü m L2,T ;V C { u V + v + g W 1 1,T ;H Γ N + f L2,T ;H} Proof Using 35 and the fact that {ϕ j } j=1 is an ON-basis for H and a solution of the eigenvalue problem 34, we obtain from 36 that, 39 ρ d k t + λ k d k t λ k β d k t = f k t + g k t, k = 1,, m, t, T, where denotes the convolution, and f k t = ft, ϕ k, g k t = gt, ϕ k ΓN This is a system of second order ODE s with the initial conditions 31 d k = u, ϕ k, d k = v, ϕ k, k = 1,, m The Laplace transform can be used, for example, to find the unique solution of the system Indeed, the Laplace transform of the Mittag-Leffler function is, LE α ax α = Hence for the kernel β defined in 22 we have, sα 1 s α a, s > a 1/α 311 Lβt = γsle α τ α t α + γe α = γs s α + τ α + γ = γ γ s α + τ α = γ τs α + 1 s α 1 s α Then, taking the Laplace transform of 39 we get, 312 ρs 2 + λ k λ k Lβs Ld k s = Lf k s + Lg k s + ρd k s + ρd k, where the inverse Laplace transform is computable Therefore, there is a unique solution for the system 39 with the initial conditions 31 Now we prove the a priori estimate 38 Since βt s = D s ξt s, by 32, we can write 36, after partial integration, as ρü m t, ϕ k + γau m t, ϕ k + = ft, ϕ k + gt, ϕ k ΓN ξt sa u m s, ϕ k ds ξtau m, ϕ k, k = 1,, m, t, T, where γ = 1 γ Then multiplying by d k t and summing k = 1,, m, we have, ρü m t, u m t + γau m t, u m t + = ft, u m t + gt, u m t ΓN ξtau m, u m t, t, T ξt sa u m s, u m t ds

10 6 F SAEDPANAH Then integrating with respect to t, we have, ρ u m t 2 + γ u m t 2 V + 2 r = ρ u m 2 + γ u m 2 V fr, u m r dr + 2 ξr sa u m s, u m r ds dr ξrau m, u m r dr gr, u m r ΓN dr Since ξ is a positive type kernel, by 33, the third term of the left hand side is non-negative Then integration by parts in the last two terms at the right side yields ρ u m t 2 + γ u m t 2 V ρ u m 2 + γ u m 2 V fr, u m r dr ġr, u m r ΓN dr + 2gt, u m t ΓN 2g, u m ΓN βrau m, u m r dr 2ξtau m, u m t + 2ξau m, u m, that using the Cauchy-Schwarz inequality, the trace theorem, β L1R + = γ, and ξ = γ, we obtain ρ u m t 2 + γ u m t 2 V ρ u m 2 + γ u m 2 V 2 + 2/C 1 max u mr 2 + C 1 fr dr r t + 2C T race /C 2 max u mr 2 V + 2C T race C 2 r t + 2C T race /C 3 u m t 2 V + 2C T race C 3 gt 2 Γ N + 2C T race /C 4 u m 2 V + 2C T race C 4 g 2 Γ N + 2/C 5 u m 2 V + 2γ 2 C 5 max u mr 2 V r t + 2/C 6 u m 2 V + 2C 6 ξ 2 t u m t 2 V + 2γ u m 2 V 2 ġr ΓN dr Hence, considering the facts that C T race = CΩ, u m v, and u m V u V, in a standard way we get, u m 2 L,T ;H + u m 2 L,T ;V C { v 2 + u 2 V + g 2 L,T ;H Γ N + f 2 L 1,T ;H + ġ 2 L 1,T ;H Γ N },

11 WELL-POSEDNESS OF THE SOLUTION OF AN INTEGRO-DIFFERENTIAL EQ 7 for some constant C = CΩ, γ, ρ, T This, and the facts that f L1,T ;H C f L2,T ;H, and by Sobolev s inequality implies 313 g L,T ;H Γ N C g W 1 1,T ;H Γ N, u m 2 L,T ;H + u m 2 L,T ;V C { v 2 + u 2 V + g 2 W 1 1,T ;HΓ N + f 2 L 2,T ;H} Now we need to find a bound for ü m For any fixed v V with v V 1, we write v = v 1 + v 2, where v 1 span{ϕ j } m j=1, v2 span{ϕ j } m j=1 Then from 36 we obtain, ρ ü m t, v = ρü m t, v 1 = ft, v 1 + gt, v 1 ΓN au m t, v 1 + βt sau m s, v 1 ds, that, using the Cuachy-Schwarz inequality and the trace theorem, implies { ρ ü m t 2 V dt ft v 1 + C T race gt ΓN v 1 V + u m t V v 1 V + This, using v 1 V 1 and 313, concludes } βt s u m s V v 1 V ds dt ρ ü m 2 L 2,T ;V C Ω{ f 2 L2,T ;H + g 2 L 2,T ;H Γ N + C γ,t u m 2 L,T ;V Therefore, for some constant C = CΩ, γ, ρ, T, ü m 2 L 2,T ;V C { f 2 L 2,T ;H + g 2 L 2,T ;H Γ N + v 2 + u 2 V + g 2 W 1 1,T ;HΓ N } C { v 2 + u 2 V + g 2 W 1 1,T ;HΓ N + f 2 L 2,T ;H} This and 313 imply the estimate 38, and the proof is complete 32 Existence and uniqueness of the weak solution Now, we use Theorem 1 to prove existence and uniqueness of the weak solution of 25, that is, a solution of Theorem 2 If u V, v H, g W 1 1, T ; H ΓN, f L 2, T ; H, there exists a unique weak solution of 25 Proof 1 We note that the estimate 38 does not depend on m, so we have } That is, 314 u m L,T ;V + u m L,T ;H + ü m L2,T ;V K = KΩ, γ, T, u, v, f, g {u m } m=1 is bounded in L, T ; V L 2, T ; V, { u m } m=1 is bounded in L, T ; H L 2, T ; H, {ü m } m=1 is bounded in L 2, T ; V

12 8 F SAEDPANAH 2 First we prove existence From 314 and a classical result in functional analysis, we conclude that the sequences {u m } m=1, { u m } m=1, {ü m } m=1 are weakly precompact That is there are subsequences of {u m } m=1, { u m } m=1, {ü m } m=1, such that, 315 u l u in L 2, T ; V, u l u in L 2, T ; H, ü l ü in L 2, T ; V, where the index l is a replacement of the label of the subsequences and denotes weak convergence Consequently, 26 holds true and we need to verify 27 and 28 To show 27 we fix a positive integer N and we choose v C[, T ]; V of the form 316 vt = N h j tϕ j j=1 Then we take l N and from 36 we have 317 ρ ü l, v + au l, v βt sau l s, v ds dt = f, v + g, vγn dt This, by 315, implies in the limit, 318 ρ ü, v + au, v βt saus, v ds dt = f, v + g, vγn dt Since functions of the form 316 are dense in L 2, T ; V, this equality then holds for all functions v L 2, T ; V, and further it implies 27 Now, we need to show that u satisfies the initial conditions 28 Let v C 2 [, T ]; V be any function which is zero in a neighborhood of T or simply vt = vt = Then by partial integration in 317 we have ρ u l, v + au l, v = βt sau l s, v ds dt f, v + g, vγn dt ρul, v + ρ u l, v, so that, recalling 315 and 37, in the limit we conclude, ρ u, v + au, v = βt saus, v ds dt f, v + g, vγn dt ρu, v + ρv, v

13 WELL-POSEDNESS OF THE SOLUTION OF AN INTEGRO-DIFFERENTIAL EQ 9 On the other hand integration by parts in 318 gives, ρ u, v + au, v βt saus, v ds dt = f, v + g, vγn dt ρu, v + ρv, v Compairing the last two identities we conclude 28, since v, v are arbitrary Hence u is a weak solution of 25 3 It remains to prove uniqueness To this end, we show that u = is the only solution of for u = v = f = g = Let us fix r [, T ] and define r uω dω t r, vt = t r t T We note that 319 vt V, vr =, vt = ut Then inserting v in 27 and integrating with respect to t, we have 32 r ρ ü, v dt + For the last term we obtain r r au, v dt βt saus, vt ds dt = r = r r s r = γ + r βt saus, vt ds dt = D t ξt saus, vt dt ds ξr saus, vr ds r r ξaus, vs ds s r r ξt saus, vt dt ds aus, vs ds ξt saus, ut ds dt, where we changed the order of integrals and we used integration by parts, ξ = γ from 32 and vr = from 319 Therefore integration by parts in the first term of 32, recalling γ = 1 γ, yields ρ r u, v dt + γ r au, v dt + This, using v = u from 319, implies r ρ ur 2 ρ u 2 γ vr 2 V + γ v 2 V + 2 r ξt saus, ut ds dt = ξt saus, ut ds dt = Consequently, recalling 33, vr =, and u =, we have u = ae, and this completes the proof

14 1 F SAEDPANAH 33 Regularity Here we study the regularity of the unique weak solution of 25, that is, a solution of Corollary 1 If u V, v H, g W 1 1, T ; H ΓN, and f L 2, T ; H, then for the unique solution u of we have 321 u L, T ; V, u L, T ; H, ü L 2, T ; V Moreover we have the estimate 322 u L,T ;V + u L,T ;H + ü L2,T ;V C { u V + v + g W 1 1,T ;H Γ N + f L2,T ;H} Proof It is known that if u m u, then 323 u lim m inf u m Then, recalling 315 and the a priori estimates 38, we conclude 321 and 322 It is known from the theory of the elliptic operators, that global higher spatial regularity can not be obtained with mixed boundary conditions Therefore we specialize to the homogeneous Dirichlet boundary condition, that is Γ N =, and assume that the polygonal domain Ω is convex We recall the usual Sobolev space H 2 = H 2 Ω and we note that here V = H 1 Ω We then use the extension of the operator Au = σ u to an abstract operator A with DA = H 2 Ω d V such that au, v = Au, v for sufficiently smooth u, v We note that, the elliptic regularity holds, that is, 324 u H 2 C Au, u H 2 Ω d V Theorem 3 We assume that Γ N = If u H 2, v V, and f L 2, T ; H, then for the unique solution u of we have 325 u L, T ; H 2, u L, T ; V, ü L, T ; H, u L 2, T ; V Moreover we have the estimate 326 Proof Writing u L,T ;H 2 + u L,T ;V + ü L,T ;H + u L2,T ;V C { u H 2 + v V + f H 1,T ;H} βt sau m s, ϕ k ds = βsau m t s, ϕ k ds, differentiating 36 with respect to time, and writing v = v, we have 327 ρü m t, ϕ k + au m t, ϕ k βt sau m s, ϕ k ds = ft, ϕ k + βtau m, ϕ k, k = 1,, m, t, T,

15 WELL-POSEDNESS OF THE SOLUTION OF AN INTEGRO-DIFFERENTIAL EQ 11 with the initial conditions 328 u m = u m = u m = ü m = m v, ϕ j ϕ j, j=1 m f Aum, ϕ j ϕj j=1 Then, using βt s = D s ξt s from 32 and partial integration, we have ρü m t, ϕ k + γau m t, ϕ k + = ft, ϕ k + βtau m, ϕ k ξt sa u m s, ϕ k ds ξtau m, ϕ k, k = 1,, m, t, T Now, multiplying by d k t and summing k = 1,, m, we have ρü m t, u m t + γau m t, u m t + = ft, u m t + βtau m, u m t ξtau m, u m t, t, T ξt sa u m s, u m t ds Integrating over [, t] and partial integration in the last term, we have ρ u m t 2 + γ u m t 2 V ρ u m 2 + γ u m 2 V fr, u m r dr + 2 βrau m, u m r dr βrau m, u m r dr 2ξtau m, u m t + 2ξau m, u m, that using the Cauchy-Schwarz inequality, the trace theorem, β L1R + = γ, and ξ = γ, we obtain ρ u m t 2 + γ u m t 2 V This implies, for some constant C, u m 2 L,T ;H + u m 2 L,T ;V ρ u m γ u m 2 V + 2/C 1 max u mr 2 + C 1 fr dr r t + 2/C 2 u m 2 H 2 + 2γ2 C 2 max r t u mr 2 + 2/C 3 u m 2 V + 2γ 2 C 3 max r t u m r 2 V + 2/C 4 u m 2 V + 2γ 2 C 4 u m t 2 V C { u m 2 + u m 2 V + u m 2 H 2 + f 2 L 1,T ;H} 2

16 12 F SAEDPANAH Then recalling u = u, the initial data from 328, and using we have u m H 2 u H 2, u m V v V, 329 ü m 2 L,T ;H + u m 2 L,T ;V C { u 2 H 2 + v 2 V + f 2 + f 2 L 1,T ;H} We now find a bound for u m t H 2 We recall the eigenvalue problem 34 with eigenpairs {λ j, ϕ j } j=1, au, v = Au, v Then we multiply 36 by λ kd k t and add for k = 1,, m to obtain This implies au m, Au m = f ρü m, Au m + Au m t 2 C ɛ βt sau m s, Au m t ds ft 2 + ρ ü m t 2 + ɛ Au m t 2 + γ max r t Au ms 2, that gives us, by elliptic regularity 324, u m 2 L,T ;H 2 C f 2 L,T ;H + ü m 2 L,T ;H From this and 329 we conclude ü m 2 L,T ;H + u m 2 L,T ;V + u m 2 L,T ;H 2 C { u 2 H 2 + v 2 V + f 2 L,T ;H + f 2 L 1,T ;H}, that using f L,T ;H C f W 1 1,T ;H, by Sobolev inequality, we have ü m 2 L,T ;H + u m 2 L,T ;V + u m 2 L,T ;H 2 C { u 2 H 2 + v 2 V + f 2 W 1 1,T ;H } C { u 2 H 2 + v 2 V + f 2 H 1,T ;H} Finally from 327, similar to the proof of Theorem 1, we obtain u m 2 L 2,T ;V C{ u 2 H 2 + v 2 V + f 2 H 1,T ;H} The last two estimates then, in the limit, imply the desired estimate 326, and the proof is now complete Remark 1 If we continue differentiation in time to investigate more regularity, we obtain from 327 ρ u m t, ϕ k + a u m t, ϕ k = ft, ϕ k + βtau m, ϕ k Further the term βtau m, ϕ k leads to βt sa u m s, ϕ k ds + βtau m, ϕ k, k = 1,, m, t, T, βtau m, ü m t

17 WELL-POSEDNESS OF THE SOLUTION OF AN INTEGRO-DIFFERENTIAL EQ 13 But β is not integrable Besides, after integration in time, we can not use partial integration to transfer one time derivative from β to ü m t, since there is not enough regularity to handle u m This means that we can not get more regularity with weakly singular kernel β This also indicates that with smoother kernel we can get higher regularity in case of homogeneous Dirichlet boundary condition under the appropriate assumption on the data, that is, more regularity and compatibility References 1 K Adolfsson, M Enelund, and S Larsson, Space-time discretization of an integro-differential equation modeling quasi-static fractional-order viscoelasticity, J Vib Control 14 28, R L Bagley and P J Torvik, Fractional calculus a different approach to the analysis of viscoelastically damped structures, AIAA J , H Brunner and P J van der Houwen, The Numerical Solution of Volterra Equations, CWI Monographs, vol 3, North-Holland Publishing Co, C M Dafermos, An abstract Volterra equation with applications to linear viscoelasticity, J Differential Equations 7 197, R Dautray and J L Lions, Mathematical Analysis and Numerical Methods for Science and Technology Vol 5, Springer-Verlag, W Desch and E Fašanga, Stress obtained by interpolation methods for a boundary value problem in linear viscoelasticity, J Differential Equations , M Enelund and P Olsson, Damping described by fading memory-analysis and application to fractional derivative models, International J Solids Structures , L C Evans, Partial Differential Equations, American Mathematical Society, Providence, Rhode Island, S Larsson and F Saedpanah, The continuous Galerkin method for an integro-differential equation modeling dynamic fractional order viscoelasticity, IMA J Numer Anal, to appear 1 Ch Lubich, I H Sloan, and V Thomée, Nonsmooth data error estimates for approximations of an evolution equation with a positive-type memory term, Math Comp , W McLean, I H Sloan, and V Thomée, Time discretization via Laplace transformation of an integro-differential equation of parabolic type, Numer Math 12 26, W McLean and V Thomée, Numerical solution of an evolution equation with a positive-type memory term, J Austral Math Soc Ser B , A K Pani, V Thomée, and L B Wahlbin, Numerical methods for hyperbolic and parabolic integro-differential equations, J Integral Equations Appl , M Renardy, W J Hrusa, and J A Nohel, Mathematical Problems in Viscoelasticity, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol 35, Longman Scientific & Technical, B Riviére, S Shaw, and J R Whiteman, Discontinuous Galerkin finite element methods for dynamic linear solid viscoelasticity problems, Numer Methods Partial Differential Equations 23 27, S Shaw and J R Whiteman, A posteriori error estimates for space-time finite element approximation of quasistatic hereditary linear viscoelasticity problems, Comput Methods Appl Mech Engrg , O J Staffans, Well-posedness and stabilizability of a viscoelastic equation in energy space, Trans Amer Math Soc , Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, SE Göteborg, Sweden address: fardin@chalmersse