Explicit Difference Schemes for Nonlinear Differential Functional Parabolic Equations with

Seria I: PRACE MATEMATYCZNE XLV (2) (2005), 205-217

Artur Poliński

Explicit Difference Schemes for Nonlinear Differential Functional Parabolic Equations with

Mixed Derivatives - Convergence Analysis

Abstract. We study the initial-value problem for parabolic equations with mixed partial derivatives and constant coefficients, and with nonlinear and nonlocal right- hand sides. Nonlocal terms appear in the unknown function and its gradient. We analyze convergence of explicit finite difference schemes by means of discrete funda- mental solutions.

2000 Mathematics Subject Classification: 35R10, 65M06, 65M12, 35K15.

Key words and phrases: Finite difference, stability, parabolic, nonlocal.

1. Introduction. The paper presents a convergence analysis for explicit finite difference methods (FDM’s) consistent with parabolic equations whose leading terms include mixed derivatives. The right-hand side contains nontrivial nonlocal operators (delays, integrals), acting on the unknown function and its derivatives. We show that discrete solutions and their spatial difference quotients converge uniformly to the exact solution and its gradient. Unlike [1], [3], [6], [7] the maximum principle is not applicable in that case, because the gradient essentially depends on functional arguments (delays, integrals). Applying a mixed combinatorial approach, together with recurrence inequalities, we generalize some results of [5], where an analogous convergence theorem was proved for nonlocal heat equations i.e., the leading term is just the Laplacean.

The paper is organized as follows: 1) formulation of the differential-functional problem and standard assumptions on coercivity and boundedness of the leading term, 2) formulation of the difference scheme and auxiliary lemmas on the positivity of its coefficients and properties of discrete fundamental solution, 3) the key role plays Lemma 1.6 on estimates of finite differences between two values of the fundamental

solution, 4) formulation of assumptions on the right-hand side function, natural in the consistency and stability theory of difference schemes, 5) concluding convergence theorem from consistency and stability lemmas, 6) the results are illustrated by numerical experiments performed in R².

1.1. Formulation of the Differential and Difference Problem. Suppose that we have b > 0, τ₀, τ₁, . . . , τ_n∈ R+and [−τ, τ ] = [−τ₁, τ₁] × · · · × [−τ_n, τ_n]. Let E = [0, b] × Rⁿ, E₀= [−τ₀, 0] × Rⁿ. Let C(B, R) denote the set of all continuous functions from B to R, where B = [−τ⁰, 0] × [−τ, τ ]. If u : E₀∪ E → R and (t, x) ∈ E, then we define the Hale-type functional u_(t,x): B → R by u(t,x)(s, y) = u(t + s, x + y) for (s, y) ∈ B. If U = (u₁, . . . , u_n) : E₀∪ E → Rⁿ, then U_(t,x) = (u₁)_(t,x), . . . , (u_n)_(t,x)
. Suppose that a_kl ∈ R, φ : E0 → R is continuous, Ω :=

E × C(B, R) × C(B, Rⁿ), f : Ω → R.

Consider the Cauchy problem (1) ∂_tu(t, x) =

n

X

k,l=1

a_kl∂_xkxlu(t, x) + f (t, x, u_(t,x), (∂_xu)_(t,x)) on E,

(2) u(t, x) = φ(t, x) on E0.

We will use the following assumptions:

Assumption 1 Suppose that akl = a_lk for k, l = 1, . . . , n and the matrix [a_kl]k,l=1,...,nis positive definite.

Assumption 2 There are positive real numbers C1, . . . , C_n such that 1 −

n

X

k=1

2C_k²a_kk+

n

X

k,l=1,k6=l

C_kC_l|a_kl| ≥ 0 on E

for k = 1, . . . , n.

Assumption 3 Matrix [mkl]k,l=1,...,n = diag [1/C₁, . . . , 1/C_n][a_kl]diag [C₁, . . . , C_n] is diagonally dominant, i.e. m_kk > Pn

l=1,l6=k|m_kl| for all 1 ≤ k ≤ n, where diag [C₁, . . . , C_n] is the diagonal n × n matrix whose diagonal consists of C₁, . . . , C_n. Fix (¯h₀, ¯h⁰) ∈ (0, b) × (0, ∞)ⁿand C₁, . . . , C_n∈ (0, ∞). Define the set of admissible steps:

(3) I_C =

 h = (h₀, h⁰) : h₀∈ (0, b), h⁰= (h₁, . . . , h_n) ∈ Rⁿ+, h²_iC_i²= h₀, (i = 1, . . . , n)

 .

Here, for the sake of simplicity, we fix proportions of h₀to h²_i (in the literature it is assumed that h₀is sufficiently small compared with h²_i).

We introduce a regular mesh. Denote t^α = αh₀ and x^β = (β₁h₁, . . . , β_nh_n) for α ∈ Z and β ∈ Zⁿ. Let Z_h= {(t^α, x^β) : (α, β) ∈ Z¹⁺ⁿ}. Define

E⁰_h= E₀∩ Z_h, E_h= E ∩ Z_h, E˜_h⁰= E_h⁰∪ E_h, E⁺_h = {(t^α, x^β) ∈ E_h: (t^α+1, x^β) ∈ E_h}, B_h= B ∪ Z_h. We will use the difference operators δ⁺₀, δ_k and δ_kl(k, l = 1, . . . , n):

δ⁺₀u^(α,β)=u^(α+1,β)− u^(α,β)

h₀ ,

δ_ku^(α,β)= u^(α,β+ek)− u^(α,β−ek)

2h_k ,

δ_kku^(α,β)=u^(α,β+ek)− 2u^(α,β)+ u^(α,β−ek)

h²_k ,

δ_klu^(α,β)= (1

2δ_k⁺δ_l⁺u^(α,β)+ δ⁻_kδ_l⁻u^(α,β) , when akl≥ 0,

1

2δ_k⁻δ⁺_lu^(α,β)+ δ⁺_kδ_l⁻u^(α,β) , when a_kl< 0, where

δ_k⁺u^(α,β)= 1 h_k

h

u^(α,β+ek)− u^(α,β)i

, δ_k⁻u^(α,β)= 1 h_k

h

u^(α,β)− u^(α,β−ek)i , and e_k = (δ_1,k, . . . , δ_n,k), δ_j,k is the Kronecker symbol for j, k = 1, . . . , n. The difference operators δ⁺₀, δ_k, δ_kl approximate the respective partial derivatives ∂_t,

∂_xk, ∂_xkxl. Define

c₀= 1 −

n

X

k=1

2h₀ h²_k a_kk+

n

X

k,l=1,k6=l

|a_kl| h₀ h_kh_l,

c_{± ek} = h₀ h²_ka_kk−

n

X

l=1,l6=k

|akl| h₀ h_kh_l, c_{± ek,± el} = θ_kla_kl h₀

2h_kh_l, c_{± ek,∓ el}= −(1 − θ_kl)a_kl h₀

2h_kh_l, (k 6= l), where θ_kl∈0, 1 and θkl= 1 for a_kl≥ 0, θ_kl= 0 for a_kl< 0. Put c_s= 0 for the remaining multiindices s ∈ Zⁿ.

Lemma 1.1 Suppose that Assumptions 1-3 are satisfied and h ∈ IC, given by (3), then

X

s∈Zⁿ

c_s= 1, c_s≥ 0, for (t^α, x^β) ∈ R+× Rⁿ on E_h, and c±ek ≥  > 0 for k = 1, . . . , n.

Proof From the definition of coefficients we have that c± ek,± el, c_{± ek,∓ el} ≥ 0.

Assumption 2 gives that c₀ ≥ 0, while the Assumption 3 gives that there exists

 > 0 such that c_±ek≥  > 0 for k = 1, . . . , n. The resultP

s∈Zⁿc_s= 1 follows from

direct summation of all coefficients c_s. _

The finite difference approximation of problem (1)-(2) takes the form (4) δ₀⁺u^(α,β)=

n

X

k,l=1

a_klδ_klu^(α,β)+ f [u]^(α,β) on E_h⁺,

(5) u^(α,β)= ¯φ^(α,β) on E_h⁰,

where ¯φ is a discrete perturbed counterpart of the function φ,

(6) f [u]^(α,β)= f_h(t^α, x^β, u_[α,β], (δu)_[α,β]), δu = (δ₁u, . . . , δ_nu), and

u_[α,β](t^α˜, x^β˜) = u(t^α+α˜ , x^β+β˜ ) for (t^α˜, x^β˜) ∈ B_h, (δu)_[α,β]= (δ₁u)_[α,β], . . . , (δ_nu)_[α,β]
. Equation (4) can be rewritten in the explicit form

(7) u^(α+1,β)= X

s∈Zⁿ

c_su^(α,β+s)+ h₀f [u]^(α,β) on E_h⁺.

Formula (7) is crucial in further theoretical considerations. First we investigate basic properties of solutions of such equations.

Lemma 1.2 Suppose that Assumptions 1-3 are satisfied. If h ∈ IC, g : E_h⁺ → R, and u : ˜E_h→ R satisfies the equation

(8) u^(α+1,β)= X

s∈Zⁿ

c_su^(α,β+s)+ h₀g^(α,β) on E_h⁺,

then

(9) ku^(α)k∞≤ ku⁽⁰⁾k∞+ h₀

α−1

X

µ=0

kg^(µ)k∞≤ ku⁽⁰⁾k∞+ t^αkg^(α−1)k∞,

where kv^(α)k∞= sup_{α≤α,β∈Z˜} n|v^{( ˜α,β)}| for any discrete function (α, β) 7→ v^(α,β), and there is the unique representation of the solution

(10) u^(α,β)= X

η∈Zⁿ

Γ^(α,β,0,η)u^(0,η)+ h₀

α

X

ζ=1

X

η∈Zⁿ

Γ^{(α,β,ζ,η)}g^(ζ−1,η) on E_h,

where Γ^{(α,β,ζ,η)} is the discrete fundamental solution, determined by the relations Γ^{(α,β,α,η)} = δ_0,|β−η|,

(11)

Γ(α+1,β,ζ,η) = X

s∈Zⁿ

c_sΓ(α,β+s,ζ,η) 0 ≤ ζ ≤ α, (12)

where δ_0,|β−η| is the Kronecker symbol.

Proof Formula (9) follows from Lemma 1.1. Formulas (10) and (11)-(12) are

proved by induction on α. _

Remark 1.3 It follows from Lemma 1.2 that Γ^{(α,β,ζ,η)}= X

s1∈Zⁿ

· · · X

sα−ζ∈Zⁿ α−ζ

Y

i=1

c^(α−i)_s

i δ_{0,|η−β−}Pα−ζ i=1 si|.

We give further properties of the discrete fundamental solution. The following lemma is a simple consequence of Lemma 1.1 and the recurrence relations (11) and (12).

Lemma 1.4 Under the assumptions of Lemma 1.2 we have

(13) Γ^{(α,β,ζ,η)}≥ 0, X

η∈Zⁿ

Γ^{(α,β,ζ,η)}= 1 for α, ζ = 0, 1, . . . , β ∈ Zⁿ, α ≥ ζ.

To obtain apriori estimates of the difference operators for the fundamental solu- tion we will use the following auxiliary lemma. The symbol [r] stands for the integer part of r ∈ R, i.e., the integer k such that k ≤ r < k + 1.

Lemma 1.5 If 0 < b ≤ 1/2, then

i

X

k=0

 i k



(1 − 2b)^i−kb^k

 k [k/2]



≤ 1

p2b(i + 1). Proof The proof is based on the estimate

 i [i/2]



≤ 2ⁱ

√i + 1.

Then, after simple calculations, using Schwarz’s inequality and formula

 i k

 1

k + 1 = i + 1 k + 1

 1

i + 1,

we get the assertion of the Lemma 1.5. _

Lemma 1.6 Suppose that Assumptions 1 - 3 are satisfied and h ∈ IC. Then

α

X

ζ=1

X

η∈Zⁿ

|Γ^{(α,β+ej,ζ,η)}− Γ^{(α,β−ej,ζ,η)}| ≤ 2√

√2α

d +4(n − 1)√

√ 2α

 ,

where

d = c_+ej+

n

X

i=1,i6=j

(c_+ej±ei+ c_±ei+ej), 0 <  = min

j=1,...,n,c_+ej.

Proof Without loss of generality we assume that j = 1. Define

∆Γ^(α,ζ)= X

η∈Zⁿ

|Γ^{(α,β+e1,ζ,η)}− Γ^{(α,β−e1,ζ,η)}|.

Thus from Remark 1.3 we have

∆Γ^(α,ζ)= X

η∈Zⁿ

X

s1∈Zⁿ

· · · X

sα−ζ∈Zⁿ α−ζ

Y

i=1

c_siδ_{0,|η−β−e}

1−Pα−ζ i=1 si|

− X

s1∈Zⁿ

· · · X

sα−ζ∈Zⁿ α−ζ

Y

i=1

c_siδ_0,|η−β+e

1−Pα−ζ i=1 si|

  .

Because each multiindex in Zⁿ can be decomposed to me₁+ η, where η₁ = 0 and m ∈ Z, we rearrange the above formula as follows

(14) ∆Γ^(α,ζ)= X

m∈Z

X

η∈Zⁿ,η1=0

X

P

isi=(m+1)e1+η α−ζ

Y

i=1

c_si− X

P

isi=(m−1)e1+η α−ζ

Y

i=1

c_si       .

Since c_+e1= c_−e1, a suitable replacement +e₁by −e₁(or −e₁by +e₁) in the set of multiindices (s₁, . . . , s_α−ζ) allows us to rewrite (14) in the form

∆Γ^(α,ζ)= X

η∈Zⁿ,η1=0

X

P

isi=+e1+η α−ζ

Y

i=1

c_si− X

P

isi=−e1+η α−ζ

Y

i=1

c_si      

+2X

m∈N

X

η∈Zⁿ,η1=0

X

P

isi=(m+1)e1+η α−ζ

Y

i=1

c_si− X

P

isi=(m−1)e1+η α−ζ

Y

i=1

c_si       . Hence

(15) ∆Γ^(α,ζ)= 2 X

η∈Zⁿ,η1=0



 X

P isi=0+η

α−ζ

Y

i=1

c_si+ X

P

isi=+e1+η α−ζ

Y

i=1

c_si



+ 2X

m∈N

A_m,

where A_m consists of products of coefficients c_si whose sum of indices is equal to me₁+ η and which are not cancelled by any product of coefficients c_si whose sum of indices is equal to (m − 2)e₁+ η.

Now we consider the first term in (15). It follows from (15) that we have to consider only sequences (s₁, . . . , s_α−ζ) such that

(16) X

i

s_i= 0 + η, η₁= 0

!

or X

i

s_i= +e₁+ η, η₁= 0

! .

These sequences will be classified according to the appearance of three categories of coefficients:

• the first one for which s_i= +e₁or s_i= +e₁±e_kor s_i= ±e_k+ e₁, k = 2, . . . , n,

• the second one for which s_i = −e₁ or s_i = −e₁± e_k or s_i = ±e_k − e₁, k = 2, . . . , n,

• the third one - all remaining indices.

Let J = {1, . . . , α − ζ}. Denote by A and B all disjoint subsets of J such that their cardinal numbers #A and #B either are equal to each other or satisfy the relation

#B = #A − 1. The set A is related to the indices s = +e₁, s = +e₁± e_k and s = ±e_k+ e₁, k = 2, . . . , n, while the set B to the indices s = −e₁, s = −e₁± e_kand s = ±e_k− e₁, k = 2, . . . , n, so condition (16) is met. Since A ∩ B = ∅ and A ∪ B ⊂ J , it is obvious that 2#A ≤ #J + 1. We have c_+e1 = c−e1 and A ∩ B = ∅, so it will cause no confusion if we use the notation c_±e1 instead of c_+e1 and c_−e1 to simplify some of derived formulas. Fixed in (15) any k ∈ J and any (s_i)_i6=k, the sum over s_k ∈ Zⁿ, s_k 6= ±e₁, s_k 6= ±e₁± e_l and s_k 6= ±e₁∓ e_lyields (1 − 2d)Q

i∈J,i6=kc_si. Thus we can represent the first term in formula (15) as follows

(17) 2 X

A,B⊂J A∩B=∅

#A−#B∈{0,1}

Y

k∈A

dY

k∈B

d Y

k∈J \(A∪B)

(1 − 2d),

where d = c_+e1 +Pn

i=2(c_+e1±ei + c_±ei+e1). Let i = #(A ∪ B). Observe that 0 ≤ i ≤ α − ζ. Thus, by basic combinatorics, we have the estimate of the first term in (15) by

2

α−ζ

X

i=0

α − ζ i

  i [i/2]



(1 − 2d)^α−ζ−idⁱ. Now we estimate the term 2P

m∈NA_m in (15). Without loss of generality we assume that c_+e1−ek = 0 for k = 2, . . . , n. Consider all sequences (s₁, . . . , s_α−ζ) such that η₁> 0, where η = (η₁, . . . , η_n) =P

is_i. These sequences can be divided into n categories:

(a) the first one for which #{i : s_i= +e₁} > #{i : s_i= −e₁},

(b) the categories j = 2, . . . , n for which #{i : s_i= +e₁+e_j} > #{i : s_i= −e₁−e_j},

not belonging to the category (a). _

The sequences from the category (a) vanish, because there is a replacement of indices s = +e₁by s = −e₁like in the case without mixed derivatives in (1). The sequences from the category (b) for j = 2 vanish if there is a replacement of (+e₁+ e₂, −e₂) by (−e₁− e₂, +e₂), so it depends on indices s = ±e₂, as we have the inequality

#{i : s_i = +e₁+ e₂} > #{i : s_i = −e₁− e₂}. Hence the remaining products of coefficients, whose sequences of indices belong to the category (b) for j = 2 can be estimated similarly as in the case without mixed derivatives in (1). In that case the indices can be divided into three categories like it the estimation of the first term in (15). The first category is related to the indices s = −e₂(the set A), the second category to the indices s = +e₂(the set B) and the third category to the remaining

indices. So we get the estimate of the products of the coefficients whose sequences of indices belong to the category (b) for j = 2

4

α−ζ

X

i=0

α − ζ i

  i [i/2]



(1 − 2c_±e2)^α−ζ−icⁱ_±e

2.

In the same way we treat sequences from that category for j = 3, . . . , n. Hence we get the estimate of the products of the coefficients whose sequences of indices belong to the category (b) for any j = 2, . . . , n

4

α−ζ

X

i=0

α − ζ i

  i [i/2]



(1 − 2c_±ej)^α−ζ−icⁱ_±e

j. Applying Lemma 1.5 we get

(18) ∆Γ^(α,ζ)≤ 2

p2d(α − ζ + 1)+

n

X

k=2

4

p2c_±ek(α − ζ + 1)

Observe that, if c_+e1±ek = 0 for some 2 ≤ k ≤ n, then k-th member of the above sum is equal to 0. Summing over ζ in (18) and using Assumption 3 yields

α

X

ζ=1

X

η∈Zⁿ

|Γ^{(α,β+e1,ζ,η)}− Γ^{(α,β−e1,ζ,η)}| ≤2√

√2α

d +4(n − 1)√

√ 2α

 .

Remark 1.7 Suppose that Assumption 3 is not satisfied, and c±ek = 0 for all k.

Then we have

X

η∈Zⁿ

|Γ^{(α,β+ej,ζ,η)}− Γ^{(α,β−ej,ζ,η)}| = 2.

This means that the solution of scheme (4) will not converge to the solution of (1).

Remark 1.8 Changing definition of category (b) for all j = 2, . . . , n, such that there are disjoint, it allows to obtain the estimate

α

X

ζ=1

X

η∈Zⁿ

|Γ^{(α,β+e1,ζ,η)}− Γ^{(α,β−e1,ζ,η)}| ≤ 2√

√2α d +2√

√2α

 . However, the proof is more complicated in that case.

Corollary 1.9 It follows form Lemmas 1.2 and 1.6 that

|δ_ju^(α,β)| ≤ kδ_ju⁽⁰⁾k∞+ C_j

√2t^α+1

√

d +2(n − 1)√ 2t^α+1

√

!

kg^(α)k∞.

2. Stability and convergence. The defect of scheme (4)-(5) will be defined by

Θ[u, h]^(α,β)= δ⁺₀u^(α,β)−

n

X

k,l=1

a_klδ_klu^(α,β)− f [u]^(α,β).

If Θ[u, h]^(α,β) ≡ 0, then {u^(α,β)} is an exact solution of (4)-(5). We formulate sufficient consistency and stability conditions. All supremum norms will be denoted by k · k∞.

Assumption 4 Suppose that there are constants L1, L₂∈ R+such that

|f_h(t, x, p, q) − f_h(t, x, ˜p, ˜q)| ≤ L₁kp − ˜pk∞+ L₂kq − ˜qk∞

for all (t, x, p, q), (t, x, ˜p, ˜q) ∈ E_h⁺×^BhR × (^BhR)ⁿ.

Assumption 5 Let the discrete function fh= f [u]^(α,β)(see (6)) satisfies

|f_h(t^α, x^β, u_[α,β], (δu)_[α,β]) − f (t^α, x^β, u_(tα,x^β), (∂_xu)_(tα,x^β))| ≤ Ckhk∞, with a constant C dependent on u, ∂_x1u, . . . , ∂_xnu and Lipschitz constant of ∂_xiu with respect to x.

It can be shown (see [2]) that there exists an interpolation operator T_h such that T_hw ∈ C(B, R) for arbitrary w : Bh→ R, and there exist constants C, ¯C > 0 such that

|Th(v_|Bh)(x) − v(x)| ≤ Ckhk_∞ for v ∈ C¹(B) and

kT_hw − T_hwk¯ ∞≤ ¯Ckw − ¯wk∞ for w, ¯w : B_h→ R.

If we define

f_h(t^α, x^β, w, δw) = f (t^α, x^β, T_hw, T_h(δw))

for w : B_h→ R, then Assumption 5 is satisfied, provided that f fulfills the Lipschitz condition with respect to p, q and u ∈ C²(E, R).

Assumption 6 Suppose that there exists the unique solution u ∈ C(E0∪ E, R) ∩ C^1,3(E, R) of the Cauchy problem (1), (2).

Lemma 2.1 (consistency) Suppose that Assumptions 5-6 are satisfied. Then scheme (4), (5) is consistent with the Cauchy problem (1), (2), on its solution u ∈ C(E₀∪ E, R) ∩ C^0,2(E, R).

Proof The consistency is obtained by using Taylor expansions at nodal points. 

Lemma 2.2 (stability) If u, v : ˜E_h→ R and Assumption 4 is satisfied, and

|u^(α,β)− v^(α,β)| ≤ ˇCkhk_∞ on E_h⁰,

|δ_j(u − v)^(α,β)| ≤ ˜Ckhk∞ on E_h⁰, j = 1, . . . , n, Θ[u, h]^(α,β)= 0, |Θ[v, h]^(α,β)| ≤ ¯Ckhk∞ on E_h⁺,

L₁b + ˆC r2b

d +2(n − 1)√

√ 2b



! L₂< 1, where

C =ˆ max

j∈{1,...,n}C_j, 0 <  = min

j=1,...,nc_+ej, d = max

j=1,...,n



c_+ej+

n

X

i=1,i6=j

(c_+ej±ei+ c±ei+ej)



, and there is ˜α > 0 such that for all α ≤ ˜α, t^α≤ b, then

sup

α≤ ˜α,β∈Zⁿ

k(u^(α,β)− v^(α,β), δ(u − v)^(α,β))k_∞→ 0 as khk_∞→ 0.

Proof Denote

ω^(α,β)= u^(α,β)− v^(α,β),

γ^(α,β)= f [u]^(α,β)− f [v]^(α,β)− Θ[v, h]^(α,β). Form Lemmas 1.2 and 1.6 it follows that

kω^(α)k_∞≤ kω⁽⁰⁾k_∞+ t^α+1kγ^(α)k_∞

≤ kω⁽⁰⁾k_∞+ t^α+1{L₁kω^(α)k_∞+ L₂kδ_hω^(α)k_∞} + t^α+1Ckhk¯ _∞ and

kδω^(α)k∞≤ kδω⁽⁰⁾k∞+ ˆC

r2t^α+1

d +2(n − 1)√ 2t^α+1

√

!

kγ^(α)k∞

≤ kδω⁽⁰⁾k∞+ ˆC

r2t^α+1

d +2(n − 1)√ 2t^α+1

√

!

{L₁kω^(α)k∞+ L₂kδ_hω^(α)k∞}

+ ¯C ˆC

r2t^α+1

d +2(n − 1)√ 2t^α+1

√

! khk∞. If we denote

ζ^(α)= L₁kω^(α)k∞+ L₂kδ_hω^(α)k∞, then, since for small t^α+1i.e. t^α+1≤ b

L₁t^α+1+ ˆC

r2t^α+1

d +2(n − 1)√ 2t^α+1

√

! L₂< 1,

we get

ζ^(α)≤ ζ⁽⁰⁾+ L₁t^α+1+ ˆC

r2t^α+1

d +2(n − 1)√ 2t^α+1

√

! L₂

! ζ^(α)+

L₁t^α+1+ ˆC

r2t^α+1

d +2(n − 1)√ 2t^α+1

√

! L₂

!

Ckhk¯ ∞.

Thus kζk∞ (which means that also kωk∞ and kδωk∞) tend to 0 as khk∞ → 0,

assuming t^α+1≤ b. _

Theorem 2.3 (convergence) Suppose that Assumptions 4-6 are satisfied and k(φ^(α,β)− ¯φ^(α,β), δ(φ − ¯φ)^(α,β)k_∞→ 0 as khk_∞→ 0 on E_h⁰.

Then the solution of scheme (4),(5) converges to the solution of the differential- functional problem (1),(2).

Proof The assertion of Theorem 2.3 follows from Lemmas 2.1 and 2.2. 

3. Numerical experiments. Fix n = 2 and consider the equations (19) ∂_tu(t, x₁, x₂) −

2

X

k,l=1

a_kl∂_xkxlu(t, x₁, x₂) = f₁(t, x₁, x₂),

∂_tu(t, x₁, x₂) −

2

X

k,l=1

a_kl∂_xkxlu(t, x₁, x₂)

(20) = sin(u(t, x₁+ sin(x₁), x₂)) + f₂(t, x₁, x₂),

∂_tu(t, x₁, x₂) −

2

X

k,l=1

a_kl∂_xkxlu(t, x₁, x₂)

(21) = sin

Z x1+1 x1−1

u(t, s, x₂)ds



+ f₃(t, x₁, x₂),

∂_tu(t, x₁, x₂) −

2

X

k,l=1

a_kl∂_xkxlu(t, x₁, x₂)

(22) = sin(∂_x1u(t, x₁, x₂)) + f₄(t, x₁, x₂),

∂_tu(t, x₁, x₂) −

2

X

k,l=1

a_kl∂_xkxlu(t, x₁, x₂)

(23) = sin(∂_x1u(t, x₁+ sin(x₁), x₂)) + f₅(t, x₁, x₂),

∂_tu(t, x₁, x₂) −

2

X

k,l=1

a_kl∂_xkxlu(t, x₁, x₂)

(24) = sin(∂_x1u(t − τ₀, x₁, x₂)) + f₆(t, x₁, x₂),

∂_tu(t, x₁, x₂) −

2

X

k,l=1

a_kl∂_xkxlu(t, x₁, x₂)

(25) = sin(u(t − τ₀, x₁, x₂)) + f₇(t, x₁, x₂),

∂_tu(t, x₁, x₂) −

2

X

k,l=1

a_kl∂_xkxlu(t, x₁, x₂)

(26) = sin

Z t t−τ0

Z x1+1 x1−1

u(s, z, x₂)dzds



+ f₈(t, x₁, x₂).

Table 1: The maximum error for (t, x, y) ∈ [0, 1] × [−1, 1]²with m=10 (number of added rectangles from each side), h₁= h₂= 1/n and h₀= 1/4n²

step 1/5 1/10 1/20

time t ∈ [0, 0.5] t ∈ [0, 1] t ∈ [0, 0.5] t ∈ [0, 1] t ∈ [0, 0.5] t ∈ [0, 1]

Eq. (19) 1.74e-03 5.02e-03 4.34e-04 1.26e-03 1.08e-04 3.14e-04 Eq. (20) 1.98e-03 6.08e-03 4.94e-04 1.53e-03 1.23e-04 3.81e-04 Eq. (21) 1.54e-03 4.52e-03 3.82e-04 1.13e-03 9.53e-05 2.83e-04 Eq. (22) 2.09e-03 6.01e-03 5.19e-04 1.50e-03 1.30e-04 3.75e-04 Eq. (23) 1.90e-03 5.54e-03 4.73e-04 1.41e-03 1.19e-04 3.55e-04 Eq. (24)

(τ0= 0.5) 1.74e-03 5.09e-03 4.34e-04 1.27e-03 1.08e-04 3.17e-04 Eq. (24)

(τ0= 0.25) 1.66e-03 4.72e-03 4.15e-04 1.18e-03 1.04e-04 2.95e-04 Eq. (25)

(τ0= 0.5) 1.74e-03 4.98e-03 4.34e-04 1.25e-03 1.08e-04 3.12e-04 Eq. (25)

(τ0= 0.25) 1.72e-03 4.92e-03 4.29e-04 1.23e-03 1.07e-04 3.07e-04 Eq. (26)

(τ0= 0.5) 2.54e-03 5.20e-03 6.35e-04 1.31e-03 1.59e-04 3.27e-04 Eq. (26)

(τ0= 0.25) 2.54e-03 5.21e-03 6.36e-04 1.31e-03 1.59e-04 3.27e-04

We perform numerical experiments for equations (19)-(26) with a₁₁ = a₂₂= 1, a₁₂= a₂₁= −0.25. The results were compared with the prescribed solution

u(t, x₁, x₂) = cos(t sin(t + x₁+ x₂)).

The right-hand sides f₁(t, x, y), . . . , f₈(t, x, y) are designed to obtain the prescribed solution. Since the computations require a large amount of time and computer memory (especially for small h₀), the domain considered in computations is cut-off to [0, 1] × [−11, 11]²with boundary values equal to the initial values at t = 0. This may cause large errors near the cut boundary, so we present the maximal errors on a smaller domain, namely [0, 1] × [−1, 1]²instead of [0, 1] × [−11, 11]². The maximal errors on [0, 1] × [−1, 1]²are presented in Tables 1 and 2. All numerical experiments confirm convergence of a discrete function to the exact solution.

Table 2: The maximum error for u_x(t, x, y) with the same parameters as in Table 1

step 1/5 1/10 1/20

time t ∈ [0, 0.5] t ∈ [0, 1] t ∈ [0, 0.5] t ∈ [0, 1] t ∈ [0, 0.5] t ∈ [0, 1]

Eq. (22) 3.42e-03 1.01e-02 8.55e-04 2.52e-03 2.14e-04 6.29e-04 Eq. (23) 4.31e-03 1.31e-02 1.12e-03 3.37e-03 2.82e-04 8.54e-04 Eq. (24)

(τ0= 0.5) 3.23e-03 8.92e-03 8.25e-04 2.28e-03 2.06e-04 5.69e-04 Eq. (24)

(τ0= 0.25) 3.12e-03 9.31e-03 7.90e-04 2.33e-03 1.97e-04 5.84e-04

References

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[3] Z. Kamont and H. Leszczyński, Stability of difference equations generated by parabolic differential-functional problems, Rend. Mat. Appl. 16 (1996), 265-287.

[4] D. Kleitman, On a lemma of Littlewood and Offord on the distribution of certain sums, Math.

Zeitschrift 90 (1965), 251-259.

[5] H. Leszczyński, Finite difference schemes for a non-linear heat equation with functional de- pendence, ZAMM 79 (1999), 53-64.

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Artur Poliński

Faculty of Applied Physics and Mathematics, Gdańsk University of Technology Narutowicza 11/12, 80-952 Gdańsk

E-mail: apoli@mif.pg.gda.pl

(Received: 3.01.05; revised: 3.05.05)