The existence of a solution and a numerical method for the Timoshenko nonlinear wave system

M2AN, Vol. 38, N 1, 2004, pp. 1–26 DOI: 10.1051/m2an:2004001

THE EXISTENCE OF A SOLUTION AND A NUMERICAL METHOD FOR THE TIMOSHENKO NONLINEAR WAVE SYSTEM

Jemal Peradze

Abstract. The initial boundary value problem for a beam is considered in the Timoshenko model.

Assuming the analyticity of the initial conditions, it is proved that the problem is solvable throughout the time interval. After that, a numerical algorithm, consisting of three steps, is constructed. The solution is approximated with respect to the spatial and time variables using the Galerkin method and a Crank–Nicholson type scheme. The system of equations obtained by discretization is solved by a version of the Picard iteration method. The accuracy of the proposed algorithm is investigated.

Mathematics Subject Classification. 35Q, 65M.

Received: July 24, 2003.

Part I. The problem and its solvability

1. Formulation of the problem

This paper deals with the problem of geometrically nonlinear vibration of a beam. To this end, we use the well-known Timoshenko model which is a theory of the second generation. As compared with the classical Kirchhoff–Love theory, the Timoshenko model allows us to take into account deformation produced by cross force and rotary inertia, which is important for a lot of problems.

Thus, let us consider the system of equations

∂²w

∂t²(x, t) = cd−a+b Z ₁

0

∂w

∂x(x, t) 2

dx

!

∂²w

∂x²(x, t)−cd∂ψ

∂x(x, t),

∂²ψ

∂t²(x, t) =c∂²ψ

∂x²(x, t)−c²d

ψ(x, t)−∂w

∂x(x, t)

, 0< x <1, 0< t≤T, (1.1) with the initial and boundary conditions

∂^lw

∂t^l (x,0) =w^l(x), ∂^lψ

∂t^l(x,0) =ψ^l(x), l= 0,1, (1.2) w(0, t) =w(1, t) = 0, ∂ψ

∂x(0, t) = ∂ψ

∂x (1, t) = 0, 0≤x≤1, 0≤t≤T. (1.3)

Keywords and phrases. Timoshenko nonlinear system, beam, Galerkin method, Crank–Nicholson scheme, Picard process.

1 Department of Applied Mathematics and Computer Sciences of Tbilisi State University, Tbilisi, 380043, R. of Georgia.

e-mail:j−peradze@yahoo.com

c EDP Sciences, SMAI 2004

System (1.1) describing the dynamic state of a beam was proposed in the work of Hirschhorn and Reiss [2]. The boundary conditions (1.3) correspond to the case of a beam with hinged ends. Linearizing (1.1), we obtain the system for the linear Timoshenko beam [3].

The functions w(x, t) and ψ(x, t) are respectively transverse deflection of the beam centerline and rotary displacement of the beam cross-section.

The constants

a, b, c, d >0 and cd−a >0. (1.4)

Here

a= AeS

M₁, b= Ae²

2M₁, c=Ae²

M₂, d=GM₂

EM₁, (1.5)

E is Young’s modulus, Gis the shear modulus, A is the cross-section area,e is the beam length, M₁ is the moment of inertia of the cross-section about the axis perpendicular to the beam centerline, M₂ is the polar moment of inertia of the cross-section, andS is the end shortening of the beam.

By virtue of (1.5) we come to a conclusion that the second relation in (1.4) is a natural requirement for moderately compressed slender beams since it is equivalent to the conditionS < ^eG_E·

The question of the existence of a solution of problem (1.1)–(1.3) was posed for the first time by Tucsnak in [3] who proved that the problem is solvable locally with respect to time if the functions w^l(x) and ψ^l(x), l= 0,1,are differentiable a finite number of times.

Here it is assumed that

w^l(x) andψ^l(x) are analytic functions of the types w^l(x) =

X∞ i=1

a^l_isiniπx, ψ^l(x) = b^l₀

√2+ X∞ j=1

b^l_jcosjπx, l= 0,1. (1.6)

2. Existence of a solution

To prove the existence of a global solution of problem (1.1)–(1.3), we will use the approach by means of which Bernstein obtained, in [1], the solvability of the equation

∂²u

∂t² =ϕ Z _l

0

∂u

∂x ₂

dx

!

∂²u

∂x²·

2.1. The infinite system

Let us consider the series w(x, t) =

X∞ i=1

w_i(t) siniπx, ψ(x, t) =ψ₀(t)

√2 + X∞ j=1

ψ_j(t) cosjπx, (2.1)

where the coefficientsw_i(t) andψ_j(t) are defined by the system of ordinary differential equations d²w_i

dt² (t) + cd−a+b 2π²

X∞ k=1

k²w²_k(t)

!

π²i²w_i(t)−cdπiψ_i(t) = 0, d²ψ_j

dt² (t) +cπ²j²ψ_j(t) +c²d(ψ_j(t)−πjw_j(t)) = 0, 0< t≤T, (w₀(t) = 0), (2.2) with the initial condition

d^lw_i

dt^l (0) =a^l_i, d^lψ_j

dt^l (0) =b^l_j, l= 0,1, i= 1,2, . . . , j= 0,1, . . . (2.3)

2.2. The finite system

Let us write a finite-dimensional analogue of problem (2.2), (2.3). Assume that for n≥1, w_n(x, t) =

Xn i=1

w_ni(t) siniπx, ψ_n(x, t) =ψ_n0(t)

√2 + Xn J=1

ψ_nj(t) cosjπx, (2.4) where for 0< t≤T

d²w_ni

dt² (t) + cd−a+ b 2π²

Xn k=1

k²w_nk² (t)

!

π²i²w_ni(t)−cdπiψ_ni(t) = 0, d²ψ_nj

dt² (t) +cπ²j²ψ_nj(t) +c²d(ψ_nj(t)−πjw_nj(t)) = 0, (w_n0(t) = 0), (2.5) and

d^lw_ni

dt^l (0) =a^l_i, d^lψ_nj

dt^l (0) =b^l_j, l= 0,1, i= 1,2, . . . , n, j= 0,1, . . . , n. (2.6)

2.3. Solvability of the finite system

Lemma 1. System(2.5, 2.6) has a solution for 0< t≤T.

Proof. Multiply the first equation of system (2.5) by 2^dw_dtⁿⁱ(t) and the second by 2¹_c^dψ_dt^nj(t). After that sum each of them overi= 1,2, . . . , nandj= 0,1, . . . , n,respectively, and add the obtained relations. This results in

dE_n

dt = 0, (2.7)

where E_n(t) =

Xn i=1

"

dw_ni dt (t)

₂

+π²i²ψ_ni² (t)

# +

Xn j=0

"

1 c

dψ_nj dt (t)

₂

+cd(ψ_nj(t)−πjw_nj(t))²

#

+1

b −a+b 2π²

Xn i=1

i²w_ni² (t)

!₂

, (w_n0(t) = 0). (2.8) By (2.7, 2.8) and also by (1.6, 2.6), we conclude thatE_n(t) =E_n(0), lim

n→∞E_n(0) =E₀andE_n(0)≤E₀,where

E₀= 2 Z ₁

0

"

w¹(x)₂ +cd

ψ⁰(x)−dw⁰ dx (x)

₂ + 1

2b −a+b Z ₁

0

dw⁰ dx (x)

₂ dx

!₂

+1

c ψ¹(x)₂ +

dψ⁰ dx (x)

₂# dx.

Therefore

E_n(t)≤E₀, (2.9)

i.e. E_n(t) is bounded by the constant that does not depend onnandt.

From the Cauchy theorem follows the existence of twice differentiable functions w_ni(t) and ψ_nj(t) which satisfy the system of equations (2.5) for 0< t≤T and the initial condition (2.6).

The latter fact and (2.5) imply the following property, which we will use below.

Corollary 1. For 0 < t < T the functions w_ni(t)and ψ_nj(t) have continuous derivatives up to fourth order inclusive,i= 1,2, . . . , n, j= 0,1, . . . , n.

2.4. Solvability of the infinite system

d^lwni

dt^l (t) _∞

n=1and d^lψnj

dt^l (t) _∞

n=0, l= 0,1,are bounded for each fixed i and j. Hence, when passing, if required, to subsequences, we see that, on [0, T], the sequencesw_ni(t) and ψ_nj(t) uniformly tend to some continuous functions as n → ∞, which in order to avoid the introduc- tion of new notations, are denoted by w_i(t) and ψ_j(t), respectively. Let us show that these functions satisfy system (2.2, 2.3).

From (1.6) it follows that there exist constantsQ >0 andR >1 such that [1]

i^5−2l a^l_i2

< Q

Rⁱ, j^5−2l b^l_j−12

< Q

R^j, l= 0,1, i, j= 1,2, . . . (2.10) Next, note that if, for somet,the finite series

r_n(t, R) = Xn i=1

iRⁱ

"

π²i²w²_ni(t) + ∆⁻¹_n (t) dw_ni

dt (t) ₂

+π²i²ψ_ni²(t) +1 c

dψ_ni dt (t)

₂

+cdψ_ni² (t)

#

, (2.11) where

∆_n(t) =cd−a+ b 2π²

Xn j=1

j²w_nj² (t)

has the boundary not depending onn, then the same value is the boundary for the series r(t, R) =

X∞ i=1

iRⁱ

"

π²i²w_i²(t) + ∆⁻¹(t) dw_i

dt (t) ₂

+π²i²ψ²_i(t) +1 c

dψ_i dt (t)

₂

+cdψ_i²(t)

#

, (2.12)

where

∆(t) =cd−a+b 2π²

X∞ j=1

j²w_j²(t), d^lw_i

dt^l (t) = lim

n→∞

d^lw_ni

dt^l (t), d^lψ_i

dt^l (t) = lim

n→∞

d^lψ_ni

dt^l (t), l= 0,1.

Hence, for brevity, we will consider the seriesr(t, R) directly.

Lemma 2. If the series r(t, R) converges for some t = t₀, then the series r(t,1) converges uniformly for

|t−t₀| ≤ _µ¹lnR, where µis some constant not depending on t.

Proof. Consider the function

F_i(t) =π²i²w_i²(t) + dw_i

dt (t) ₂

+π²i²ψ²_i(t) +1 c

dψ_i dt (t)

₂

+cdψ_i²(t).

From (2.2) it follows that dF_i

dt (t) = 2π²i²dw_i

dt (t)w_i(t)(1−∆(t)) + 2cdπid

dt(w_i(t)ψ_i(t)).

Based on (2.8, 2.9), we conclude that ∆(t) is uniformly bounded. Hence ^dFi

dt (t)≤µiF_i(t), where µ is some constant not depending ontand therefore if exp(µ|t−t₀|)≤R, then

F_i(t)≤F_i(t₀) exp(µi|t−t₀|)≤F_i(t₀)Rⁱ.

Now, taking (2.12) into account, we obtain the validity of the lemma.

From Lemma 2 and the fact that r(0, R) converges by virtue of (2.10) it follows in particular that the function r(t,1) is finite and continuous fort≤ _µ¹lnR.

Lemma 3. There exists no finite value oft for which the series r(t, R)is not convergent.

Proof. After differentiating (2.12) and applying (2.2), we find dr

dt(t, R) = X∞ i=1

iRⁱ



−bπ²∆⁻²(t) dw_i

dt (t) ₂X_∞

j=1

j²dw_j

dt (t)w_j(t) + 2cdπi

∆⁻¹(t)dw_i

dt (t)ψ_i(t) +w_i(t)dψ_i dt (t)

.

This and (2.12) imply

dr dt(t, R)

≤r(t, R)(a₁r(t,1) +a₂), (2.13)

where

a₁=πb

2(cd−a)⁻¹², a₂=cdmax¹² (cd−a)⁻¹, c . Assume now thatr(t, R) is finite for somet=t₁. Thenr(t₁,1) is finite, too.

Denote

g(t) = 1 +a₂

a₁r⁻¹(t,1), p(t) = [1−g(t₁) exp(a₂(t₁−t))]⁻¹. ForR= 1,from (2.13) we obtain|dlng(t)|< a₂dtand therefore if

0< t−t₁< 1

a₂lng(t₁), (2.14)

then r(t,1) <−^a_a²₁p(t). The latter inequality together with (2.13) gives ^{d lnr}

dt (t, R)< a₂(1−p(t)). Therefore ln_r(t^r(t,R)_1,R)≤ln[(1−g(t₁))p(t)].

Hence we conclude that when (2.14) is fulfilled, the estimate

r(t, R)≤r(t₁, R) exp(a₂(t−t₁)) 1 1 +a₁

a₂r(t₁,1)[1−exp(a₂(t−t₁))]

(2.15)

is true.

Further, if the seriesr(t, R) converges for all positivet < t₀, then by Lemma 2 the series r(t,1) converges uniformly whent₀−¹_µlnR ≤t≤t₀. Therefore there exists a finite valueH >0 such thatr(t,1)< H in this interval. In view of inequalities (2.14), (2.15) we conclude that ift₁< t₀is chosen so close tot₀thatt₀−t₁< ε, where ε= _a¹

2ln(1 +^a_a²

1H⁻¹),then r(t₀, R) is also finite. Thus there exists not₀ such that the seriesr(t, R) is convergent for all t < t₀ and is not convergent fort=t₀+¹₂ε.

This lemma and the arguments given in its proof imply the following propositions that we will use later on.

Corollary 2. The seriesP_∞

i=1i^l|w_i(t)| andP_∞

j=1j^l|ψ_j(t)| converge,l=−2,−1, . . . ,2, 0≤t≤T.

Corollary 3. The series r_n(t, R) andr(t, R) are uniformly bounded, the former with respect to n and t,and the latter with respect to t, n= 1,2, . . . , 0< t≤T.

Each pair of equations (2.5) with the boundary conditions (2.6) is equivalent to the equations w_ni(t) =a⁰_i+a¹_it+

Z _t

0

(τ−t)

"

cd−a+ b 2π²

X∞ k=1

k²w²_nk(τ)

!

π²i²w_ni(τ)−cdπiψ_ni(τ)

# dτ, ψ_nj(t) =b⁰_j+b¹_jt+

Z _t

0

(τ−t)

cπ²j²ψ_nj(τ) +c²d(ψ_nj(τ)−πjw_nj(τ))

dτ, (w_nk(t) = 0 fork= 0 andk > n).

(2.16) Since all functions w_ni(t) and ψ_nj(t) satisfy these equations for i = 1,2, . . . , n, j = 0,1, . . . , n, and for any 0≤t≤T and, moreover, for giveniandj the functionsw_ni(t) andψ_nj(t) tend uniformly, asn→ ∞,tow_i(t) andψ_j(t),for which (2.3) is fulfilled, while, as follows from (2.9) and (2.15),P_n

k=1k²w²_nk(t) tends uniformly to P_∞

k=1k²w_k²(t),we see that w_i(t) andψ_j(t) satisfy the same equations (2.16) for anyiand j.

Thus for 0 < t ≤ T the functions w_i(t) and ψ_j(t) are twice differentiable and satisfy the system of equa- tions (2.2) and the initial conditions (2.3).

2.5. Solvability of the initial problem

W^(l)(x, t) =− 1 π^l−1

X∞ i=1

1

i^l+1w_i(t)d^l−1siniπx dx^l−1 , Ψ^(l)(x, t) =− 1

π^l−1 X∞ j=1

1

j^l+1ψ_j(t)d^l−1cosjπx

dx^l−1 , l= 1,2. (2.17)

By virtue of Corollary 2, the series in equalities (2.1) and (2.17) converge absolutely. Hence, after multiplying, fori, j = 1,2, . . . ,the equations of system (2.2) by _i¹₂siniπx and _j¹₂cosjπx, respectively, and summing them, we obtain the absolute convergence of the series

−X^∞

i=1

1 i²

d²w_i

dt² (t) siniπx= ∂²W⁽¹⁾

∂t² (x, t), −X^∞

j=1

1 j²

d²ψ_j

dt² (t) cosjπx= ∂²Ψ⁽¹⁾

∂t² (x, t).

Thus the system of equations

∂²W⁽¹⁾

∂t² (x, t) = cd−a+b Z ₁

0

∂w

∂x(x, t) ₂

dx

!

π²w(x, t)−cdπΨ⁽²⁾(x, t),

∂²Ψ⁽¹⁾

∂t² (x, t) =cπ²

ψ(x, t)−ψ₀(t)

√2

−c²d

Ψ⁽¹⁾(x, t)−πW⁽²⁾(x, t)

(2.18) is fulfilled for 0< x <1 and 0< t≤T.

Let us twice differentiate the equations of system (2.18) with respect to x. That this operation is possible follows from Corollary 2. In particular, this gives

∂²W⁽¹⁾

∂x² (x, t) =π²w(x, t), ∂²Ψ⁽¹⁾

∂x² (x, t) =π²ψ(x, t).

We also take into account the equation ^d_dt^2ψ₂⁰(t) +c²dψ₀(t) = 0 from system (2.2). This leads us to system (1.1).

Thus we have proved.

Theorem 1. If the initial functionsw^l(x)andψ^l(x),l= 0,1,satisfy condition (1.6), then problem (1.1)–(1.3) has a solution, namely, analytic with respect to 0≤x≤1for 0≤t≤T functionsw(x, t)andψ(x, t)which are representable by series (2.1).

Part II. Numerical method

3. Approximation with respect to a spatial variable 3.1. Galerkin method

In this stage of construction of the numerical algorithm for problem (1.1)–(1.3) we make use of the Galerkin method. This method has already been used above. Thus we will approach the sought functions w(x, t) andψ(x, t) by means of sums (2.4). The functionsw_ni(t) andψ_nj(t), i= 1,2, . . . , n, j= 0,1, . . . , n,in (2.4) are solutions of the system of ordinary differential equations (2.5), provided that the values

d^lw_ni

dt^l (0), d^lψ_nj

dt^l (0) are given, l= 0,1. (3.1)

A representation of the initial conditions forw_ni(t) andψ_nj(t) in form (3.1) and not by equalities (2.6) means that we may come across the case where the coefficients a^l_i andb^l_j in expansions (1.6), which as a rule have to be calculated, are given inexactly.

Let us assume that the analogues of inequalities (2.10)

i^5−2l

d^lw_ni dt^l (0)

2

< Q

Rⁱ, j^5−2l

d^lψ_n,j−1 dt^l (0)

2

< Q

R^j, l= 0,1, i= 1,2, . . . , n, j= 1,2, . . . , n+ 1, (3.2) hold, where, for simplicity, we use the same constantQ >0 andR >1.

3.2. Matrix notation of the system

wn(t) = (w_ni(t))ⁿ_i=1, ψ_n(t) = (ψ_nj(t))ⁿ_j=0 (3.3) and alsow^l_n= (a^l_i)ⁿ_i=1,ψ^l_n= (b^l_j)ⁿ_j=0, l= 0,1.

Let us introduce the scalar product and norms inR^n+l, l= 0,1.Ifαandβare vectors of the same dimension and their kth components are equal to α_k and β_k, then the scalar product (α, β)_n = P

kα_kβ_k, where the summation involves all components of α and β and the norm ||α||n = (α, α)_n¹². If D is a diagonal matrix with nonnegative elements whose order coincides with the dimension of the vector α, then the energy norm

||α||_D= (Dα, α)_n¹².

In addition to this, let us define two diagonal matrices and one (n+ 1)×nmatrix

K_n=πdiag(1,2, . . . , n), L_n=πdiag(0,1, . . . , n), U_n=







0 0 . . . 0 1 0 . . . 0

· · · · 0 0 . . . 1





. (3.4)

The above notation allows us to rewrite system (2.5) as d²wn

dt² (t) +

cd−a+ b

2||wn(t)||²_K2 n

K_n²wn(t)−cdU_n⁰L_nψ_n(t) = 0, d²ψ_n

dt² (t) +cL²_nψ_n(t) +c²d(ψ_n(t)−L_nU_nw_n(t)) = 0, 0< t≤T. (3.5) Condition (3.1) means that the vectors

d^lwn

dt^l (0), d^lψ_n

dt^l (0) are given, l= 0,1. (3.6)

3.3. Truncation error

Let us consider the Fourier expansion of the exact solution of problem (1.1)–(1.3) w(x, t) =

X∞ i=1

w_i(t) siniπx, ψ(x, t) = ψ₀(t)

√2 + X∞ j=1

ψ_j(t) cosjπx

and construct the vectorsp_nw(t) = (w_i(t))ⁿ_i=1, p_nψ(t) = (ψ_j(t))ⁿ_j=0. By (2.2) and (2.3) we obtain the system of equations

d²p_nw dt² (t) +

cd−a+b

2||p_nw(t)||²_K2 n

K_n²p_nw(t)−cdU_n⁰L_np_nψ(t) +ξ_n(t) = 0, d²p_nψ

dt² (t) +cL²_np_nψ(t) +c²d(p_nψ(t)−L_nU_np_nw(t)) = 0 (3.7) with the initial condition

d^lp_nw

dt^l (0) =w^l_n, d^lp_nψ

dt^l (0) =ψ^l_n, l= 0,1. (3.8)

In (3.7)ξ_n(t) is a truncation error for which the following equality holds ξ_n(t) = b

2π² X∞ i=n+1

i²w²_i(t)

!

K_n²p_nw(t). (3.9)

3.4. Equations for the error

Under the error of the method we understand the vectors ∆wn(t) = wn(t)−p_nw(t) and ∆ψ_n(t) = ψ_n(t)−p_nψ(t).To write equations for these vectors, we subtract (3.7) from (3.5), and obtain

d²∆w_n dt² (t) +

cd−a+b

2||w_n(t)||²_K2 n

K_n²w_n(t)

−

cd−a+b

2||p_nw(t)||²_K2 n

K_n²p_nw(t)−cdU_n⁰L_n∆ψ_n(t) =ξ_n(t), d²∆ψ_n

dt² (t) +cL²_n∆ψ_n(t) +c²d(∆ψ_n(t)−L_nU_n∆wn(t)) = 0. (3.10) As for the initial condition, according to (3.6) and (3.8) they can be written as

d^l∆w_n

dt^l (0) = d^lw_n

dt^l (0)−w^l_n, d^l∆ψ_n

dt^l (0) = d^lψ_n

dt^l (0)−ψ^l_n, l= 0,1.

Our aim is to estimate the norms of ∆wn(t) and ∆ψ_n(t).For this, we need to derive some auxiliary inequalities.

3.5. A priori estimates

Here we obtain the required norm estimates of the vectors which form nonlinearities in systems (3.5) and (3.7).

The boundaries of these norms are calculable.

Lemma 4. The inequality

||p_nw(t)||²_K2

n≤s (3.11)

is fulfilled, wheresis defined by means of the functions from the initial condition(1.2)and the parameters from the first relation of (1.4)

s= 2a b +

( 8 b

Z ₁

0

"

(w¹(x))²+cd

ψ⁰(x)−dw⁰ dx (x)

₂

+1

2b −a+b Z ₁

0

dw⁰ dx (x)

2

dx

!₂ 1

c ψ¹(x)2

+ dψ⁰

dx(x) 2

dx





12

· (3.12)

Proof. Multiply the equations of system (1.1) by 2^∂w_∂t(x, t) and 2¹_c^∂ψ_∂t(x, t), respectively, integrate them with respect toxfrom 0 to 1 and sum the obtained inequalities. Also taking into account (1.2, 1.3) and equality (1.6), we come to the formulas

d dt

Z ₁

0

"

∂w

∂t(x, t) ₂

+cd

ψ(x, t)−∂w

∂x(x, t) ₂

+ 1

2b −a+b Z ₁

0

∂w

∂x(x, t) ₂

dx

!₂

+1 c

∂ψ

∂t(x, t) ₂

+ ∂ψ

∂x(x, t) ₂#

dx= 0.

This, (1.2) and also the inequality||pnw(t)||²_K2 n ≤2R₁

0 ∂w

∂t(x, t)2

dxgive (3.11).

Further we will need the value s_n defined by the vectors from the initial condition (3.6) and the prescribed parameters

s_n= 2a b +

( 4 b

"

dwn

dt (0) ²

n

+cd||ψ_n(0)−U_nK_nwn(0)||²_n

+1 b

−a+b

2||wn(0)||²_K2 n

₂ +1

c dψ_n

dt (0) ²

n

+||ψ_n(0)||²_L2 n

#)¹₂

· (3.13)

As follows from (3.2), s_n is bounded uniformly with respect to n. After comparing (3.12) and (3.13), we see that in the particular case, where the initial condition (3.6) has the form

d^lwn

dt^l (0) =w^l_n, d^lψ_n

dt^l (0) =ψ^l_n, l= 0,1, we haves_n→sasn→ ∞and, moreover,s_n ≤s.

Lemma 5. There exists a solution of problem (3.5, 3.6) and the estimate

||wn(t)||²_K2

n≤s_n (3.14)

is true.

Proof. Multiplying the equations of system (3.5) scalarly by 2^dw_dtⁿ(t) and 2¹_c^dψ_dtⁿ(t),respectively, and summing the resulting equations, we obtain the equality

d dt

"

dw_n dt (t)

²

n

+cd||ψ_n(t)−U_nK_nw_n(t)||²_n+1 b

−a+b

2||w_n(t)||²_K2 n

₂ +1

c dψ_n

dt (t) ²

n

+||ψ_n(t)||²_L2 n

#

= 0,

which ensures the fulfillment of (3.14).

If, in addition, we use (3.2), then we have that the norms of the vectors ^d_d^lw_ltlⁿ(t) and ^dl_dt^ψ_lⁿ(t), l = 0,1,are bounded uniformly with respect tonandt.This fact implies the solvability of problem (3.5, 3.6).

3.6. Method error

Theorem 2. For the error of Galerkin method for0< t≤T there holds an estimate

z_n(t)≤C₀z_n(0) +C₁qⁿ, (3.15)

where

z_n(t) = d∆w_n

dt (t) n

+ d∆ψ_n

dt (t) n

+||∆w_n(t)||_Kn² +||∆ψ_n(t)||_L²_n, (3.16) C₀, C₁ andq <1 are some positive constants not depending onnandt.

Proof. Multiplying scalarly the equations of system (3.10) by 2^d∆w_dtⁿ(t) and 2¹_c^d∆ψ_dtⁿ(t),respectively, and sum- ming the resulting relations, we come to the equality

dΦ_n dt (t) = b

2||∆wn(t)||²_K2 n

d

dt||wn(t)||²_K2 n+b

||p_nw(t)||²_K2

n− ||wn(t)||²_K2 n

×

K_n²p_nw(t),d∆w_n dt (t)

n

+ 2cdd

dt(U_n⁰L_n∆ψ_n(t),∆wn(t))_n+ 2

ξ_n(t),d∆w_n dt (t)

n

(3.17) with the notation

Φ_n(t) = d∆w_n

dt (t) ²

n

+1 c

d∆ψ_n dt (t)

²

n

+

cd−a+b

2||wn(t)||²_K2 n

||∆wn(t)||²_K2

n+||∆ψ_n(t)||²_L2

n+cd||∆ψ_n(t)||²_n. (3.18) In our further proof we will use some constantsc_i>0, i= 0,1, . . . ,6,which are expressed in terms of the initial data of the problem and the known values. This dependence, which is not quite obvious only for the first two parameters, is not given here for the sake of simplicity. A concrete definition of the parametersc_i turns out useful when it is necessary to define the constants C₀ andC₁from (3.15).

Let us derive some estimates. For this, we use the first formula from (3.4) and definitions (2.11) and (2.12).

Note, that the uniform boundedness of the seriesr_n(t, R),which was said in Corollary 3, takes place also in the

case where (3.1, 3.2) are fulfilled instead of (2.6, 2.10). We also apply (3.14). As a result, we have d

dt||wn(t)||²_K2 n≤2π²

Xn i=1

i²|w_ni(t)| dw_ni

dt (t)

≤πmax

1, cd−a+b 2s_n

r_n(t,1)≤c₀. (3.19) Again using Corollary 3, this time its part that concerns the seriesr(t, R),we see that

K_n²p_nw(t)²

n =π⁴ Xn i=1

i⁴w_i²≤π²i₀r(t, R)≤c₁, (3.20) wherei₀is a smallest natural number such thati < Rⁱ holds fori=i₀+ 1, i₀+ 2, . . .

By (3.20), as well as by (3.11) and (3.14), we can write

||p_nw(t)||²_K2

n− ||wn(t)||²_K2

n K_n²p_nw(t),d∆wn

dt (t)

n

≤ ||p_nw(t)||_K²_n +||w_n(t)||_K²_n

||∆w_n(t)||_Kn²K_n²p_nw(t)

n

d∆w_n dt (t)

n

≤c₂||∆w_n(t)||_K²_n d∆w_n

dt (t) n

. (3.21) Furthermore, we obtain an estimate for the norm ofξ_n(t).Since by virtue of Theorem 1 the functionw(x, t) is analytic with respect to the variable x,there exist values 0< q <1 and Ω>0 not depending on tsuch that i²w²_i(t)<Ωqⁱ holds for the coefficientsw_i(t) from (2.1). This fact, (3.9) and (3.20) imply

||ξ_n(t)||_n≤c₃qⁿ. (3.22)

After integrating (3.17) with respect totand using estimates (3.19)–(3.22), we obtain Φ_n(t)≤Φ_n(0) +

Z _t

0

b

2c₀||∆w_n(τ)||²_K2

n+bc₂||∆w_n(τ)||_Kn² d∆w_n

dτ (τ) n

+2cd

d∆ψ_n dτ (τ)

n||∆wn(τ)||_Kn² +||∆ψ_n(τ)||_L²_n d∆wn

dτ (τ) n

+ 2c₃qⁿ d∆wn

dτ (τ) n

dτ.

We transform this inequality and, using (3.16) and (3.18), replace Φ_n(t) byz_n(t). As a result, z_n²(t)≤c₄z_n²(0) +c₅q²ⁿ+c₆

Z _t

0

z²_n(τ)dτ.

From this we conclude by virtue of Gronwall’s lemma that estimate (3.15) is valid.

4. Approximation with respect to the time variable 4.1. Transformation to a system of first order equations

Let us write the collection of vectors

un(t), vn(t), f_n(t), ϕ_n(t), ψ_n(t), (4.1) of which the first four are the new ones defined by the formulas

u_n(t) =dw_n

dt (t), v_n(t) =K_nw_n(t), f_n(t) = dψ_n

dt (t), ϕ_n(t) =L_nψ_n(t). (4.2)

Applying (3.5) and (4.2), we obtain a system of equations for vectors (4.1) dun

dt (t) +

cd−a+b

2||v_n(t)||²_n

K_nv_n(t)−cdU_n⁰ϕ_n(t) = 0, dvn

dt (t) =K_nun(t), df_n

dt (t) +cL_nϕ_n(t) +c²d(ψ_n(t)−U_nvn(t)) = 0, dϕ_n

dt (t) =L_nf_n(t), dψ_n

dt (t) =f_n(t). (4.3)

At the initial moment the values of vectors (4.1)

un(0), vn(0), f_n(0), ϕ_n(0), ψ_n(0) are given (4.4) by virtue of (3.6).

Thus problem (3.5, 3.6) is replaced by the equivalent problem (4.3, 4.4). After solving the latter problem, the vectorw_n(t),which is missing in the collection (4.1), is constructed by the formulaw_n(t) =K_n⁻¹v_n(t).

4.2. Scheme of Crank–Nicholson type

Let us solve problem (4.3, 4.4) by the difference method. On the interval [0, T], we introduce the grid {t_m|0 =t₀< t₁<· · ·< t_M =T}with variable pitchτ_m=t_m−t_m−1, m= 1,2, . . . , M.Denote the approximate values of vectors (4.1) on the mth time layer,i.e. fort=t_m, m= 0,1, , . . . , M,by

u^m_n, v^m_n, f^m_n, ϕ^m_n, ψ^m_n. (4.5) We use the modified Crank–Nicholson scheme form= 1,2, . . . , M

u^m_n −u^m−1_n

τ_m +

cd−a+b 2

||v^m_n||²_n+||v^m−1_n ||²_n 2

K_nv^m_n +v^m−1_n

2 −cdU_n⁰ϕ^m_n +ϕ^m−1_n

2 = 0,

v^m_n −v^m−1_n

τ_m =K_nu^m_n +u^m−1_n

2 ,

f^m_n −f^m−1_n

τ_m +cL_nϕ^m_n +ϕ^m−1_n 2 +c²d

ψ^m_n +ψ^m−1_n

2 −U_nv^m_n +v^m−1_n 2

= 0, ϕ^m_n −ϕ^m−1_n

τ_m =L_nf^m_n +f^m−1_n

2 , ψ^m_n −ψ^m−1_n

τ_m = f^m_n +f^m−1_n

2 , (4.6)

assuming that the vectors

u⁰_n, v⁰_n, f_n⁰, ϕ⁰_n, ψ⁰_n are given. (4.7) (4.7) written in a more detailed form means that the vectors

u⁰_n=un(0) + ∆u⁰_n, v⁰_n=vn(0) + ∆v⁰_n, f⁰_n=f_n(0) + ∆f⁰_n,

ϕ⁰_n =ϕ_n(0) + ∆ϕ⁰_n, ψ⁰_n=ψ_n(0) + ∆ψ⁰_n (4.8) are known, where

∆u⁰_n, ∆v⁰_n, ∆f⁰_n, ∆ϕ⁰_n, ∆ψ⁰_n (4.9) are the vectors of possible errors of the given initial values for the difference scheme.

We assume that

the norms|| ||n of the vectors (4.9) and ∆ϕ⁰_n−L_n∆ψ⁰_n are bounded uniformly with respect ton. (4.10)