In recent years, more and more mathematicians have been studying nonlocal problems for ordinary differential equations and partial differential equations because of their existence in many applied problems included in applied sciences. Theory and numerical methods of solutions of the nonlocal boundary value problems for these partial differential equations were investigated by many researchers (see, e.g., 1 2 3 4 5 6 7 8 9 10 11 12 13 and the references therein). Several types of problems in fluid mechanics, other areas of physics, and mathematical biology led to partial differential equations of elliptic-parabolic type (see, 14 15 16 17 18 ). The purpose of this paper is to study the second order of accuracy difference schemes of elliptic-parabolic problem with nonlocal boundary value problems.

In 19 , we established the well-posedness of multipoint nonlocal boundary value problem

{ − d 2 u ( t ) d t 2 + A u ( t ) = g ( t ) ( 0 ≤ t ≤ 1 ) , d u ( t ) d t − A u ( t ) = f ( t ) ( − 1 ≤ t ≤ 0 ) , u ( 1 ) = ∑ i = 1 J α i u ( λ i ) + φ , − 1 ≤ λ 1 < ⋯ < λ J ≤ 0

in a Hilbert space H with the self-adjoint positive definite operator A under assumption

∑ i = 1 J | α i | ≤ 1 .

The well-posedness of multipoint nonlocal boundary value problem (1) in Hölder spaces with a weight was established. In applications, coercivity inequalities for the solutions of nonlocal boundary value problems for elliptic-parabolic equations were obtained.

In 20 , we studied the well-posedness of the first order of accuracy difference scheme for the approximate solution of boundary value problem (1) under assumption (2).

In the present paper, we consider the second order of accuracy difference scheme generated by Crank-Nicholson difference scheme

{ − τ − 2 ( u k + 1 − 2 u k + u k − 1 ) + A u k = g k , g k = g ( t k ) , t k = k τ , 1 ≤ k ≤ N − 1 , N τ = 1 , τ − 1 ( u k − u k − 1 ) − 1 2 ( A u k − 1 + A u k ) = f k , f k = f ( t k − 1 2 ) , t k − 1 2 = ( k − 1 2 ) τ , − ( N − 1 ) ≤ k ≤ 0 , u 2 − 4 u 1 + 3 u 0 = − 3 u 0 + 4 u − 1 − u − 2 , u N = ∑ k = 1 J α i ( u [ λ i τ ] + ( λ i − [ λ i τ ] τ ) ( f [ λ i τ ] + A u [ λ i τ ] ) ) + φ

for the approximate solution of boundary value problem (1) under assumption (2).

The well-posedness of difference scheme (3) in Hölder spaces is established. In applications, the stability, almost coercivity stability, coercivity stability estimates for solutions of the second order of accuracy difference scheme for elliptic-parabolic equations are obtained. Furthermore, the theoretical statements for the solution of the first and second order of accuracy schemes for one-dimensional elliptic-parabolic differential equation are supported by the results of a numerical example.

Let us give some auxiliary lemmas we need below. Throughout the paper, H is a Hilbert space and we denote B = 1 2 ( τ A + A ( 4 + τ 2 A ) ) , where A is a self-adjoint positive definite operator. Then, it is clear that B is a self-adjoint positive definite operator and B ≥ δ 1 2 I , where δ > δ 0 > 0 , and R = ( I + τ B ) − 1 which is defined on the whole space H is a bounded operator. The following operators

P = ( I − τ A 2 ) G , G = ( I + τ A 2 ) − 1 , R = ( I + τ B ) − 1 ,

and

exist and are bounded for a self-adjoint positive operator A. Here,

B = 1 2 ( τ A + A ( 4 + τ 2 A ) ) , K = ( I + 2 τ A + 5 4 ( τ A ) 2 ) − 1 ,

and I is the identity operator.

Lemma 2.1 For any g k , 1 ≤ k ≤ N − 1 and f k , − N + 1 ≤ k ≤ 0 , the solution of problem (3) exists and the following formulas hold:

Proof Clearly, the solution formula of the problem

τ − 1 ( u k − u k − 1 ) − 1 2 ( A u k − 1 + A u k ) = f k , − ( N − 1 ) ≤ k ≤ 0 , u 0 = γ

is 22 :

u k = P − k γ − τ ∑ s = k + 1 0 P s − k − 1 G f s , − N ≤ k ≤ − 1

for any { f k } k = − N − 1 and γ. Equation (9) and the fact that u 0 = γ yield Equation (6).

The solution of the problem

{ − τ − 2 ( u k + 1 − 2 u k + u k − 1 ) + A u k = g k , g k = g ( t k ) , t k = k τ , 1 ≤ k ≤ N − 1 , u 0 = γ , u N = ψ

satisfies the following formula 21 :

u k = ( I − R 2 N ) − 1 { [ R k − R 2 N − k ] γ + [ R N − k − R N + k ] ψ − [ R N − k − R N + k ] ( I + τ B ) ( 2 I + τ B ) − 1 B − 1 ∑ s = 1 N − 1 [ R N − s − R N + s ] g s τ } + ( I + τ B ) ( 2 I + τ B ) − 1 B − 1 ∑ s = 1 N − 1 [ R | k − s | − R k + s ] g s τ , 1 ≤ k ≤ N .

Equation (5) follows from Equations (9) and (11), initial condition u 0 = γ , and

ψ = ∑ k = 1 J α i ( u [ λ i τ ] + ( λ i − [ λ i τ ] τ ) ( f [ λ i τ ] + A u [ λ i τ ] ) ) + φ .

Finally, let us obtain formula (7). Combining (5), (6), and the condition

u 2 − 4 u 1 + 3 u 0 = − 3 u 0 + 4 u − 1 − u − 2 ,

we get

From Equation (4), it follows that

u 0 = 1 2 T τ K G − 2 × { ( 2 I − τ 2 A ) { ( 2 + τ B ) R N × [ ∑ i = 1 n α i ( I + ( λ i − [ λ i τ ] τ ) A ) ( − τ ∑ s = [ λ i τ ] + 1 0 P s − [ λ i τ ] G f s ) + ( λ i − [ λ i τ ] τ ) f [ λ i τ ] + φ ] − R N − 1 B − 1 ∑ s = 1 N − 1 [ R N − s − R N + s ] g s τ + ( I − R 2 N ) B − 1 ∑ s = 1 N − 1 R s − 1 g s τ } + ( I − R 2 N ) ( I + τ B ) ( τ B − 1 g 1 − 4 G B − 1 f 0 + P G B − 1 f 0 + G B − 1 f − 1 ) } .

This finishes the proof of Lemma 2.1. □

Here, we study well-posedness of problem (3). First, we give some necessary estimates for P k , R k and T τ .

Lemma 2.2 For a self-adjoint positive operator A the following estimates are satisfied 21 22 24 :

{ ∥ P k ∥ H → H ≤ 1 , k τ ∥ A P k G 2 ∥ H → H ≤ M ( δ ) , k τ ∥ B R k ∥ H → H ≤ M ( δ ) , ∥ R k ∥ H → H ≤ M ( δ ) ( 1 + δ τ ) − k , ∥ ( I − R 2 N ) − 1 ∥ H → H ≤ M ( δ ) , ∥ G ∥ H → H ≤ 1 , ∥ P k − e − k τ A ∥ H → H ≤ M ( δ ) τ k τ , ∥ R k − e − k τ A 1 2 ∥ H → H ≤ M ( δ ) τ k τ , k ≥ 1 , δ > 0 ,

where M ( δ ) is independent of τ.

From these estimates, it follows that

Now, we study well-posedness of problem (3). Let F τ ( H ) = F ( [ a , b ] τ , H ) be the linear space of mesh functions φ τ = { φ k } N ˜ N ˜ ˜ defined on [ a , b ] τ = { t k = k h , N ˜ ≤ k ≤ N ˜ ˜ , N ˜ τ = a , N ˜ ˜ τ = b } with values in the Hilbert space H. Next, on F τ ( H ) we denote C ( [ a , b ] τ , H ) , C 0 , 1 α ( [ − 1 , 1 ] τ , H ) , C 0 , 1 α ( [ − 1 , 0 ] τ , H ) , C 0 α ( [ 0 , 1 ] τ , H ) , C ˜ 0 , 1 α ( [ − 1 , 1 ] τ , H ) , and C ˜ 0 α ( [ − 1 , 0 ] τ , H ) , 0 < α < 1 Banach spaces with the norms

∥ φ τ ∥ C ( [ a , b ] τ , H ) = max N a ≤ k ≤ N b ∥ φ k ∥ H , ∥ φ τ ∥ C 0 , 1 α ( [ − 1 , 1 ] τ , H ) = ∥ φ τ ∥ C ( [ − 1 , 1 ] τ , H ) + sup − N ≤ k < k + r ≤ 0 ∥ φ k + r − φ k ∥ E ( − k ) α r − α ∥ φ τ ∥ C 0 , 1 α ( [ − 1 , 1 ] τ , H ) = + sup 1 ≤ k < k + r ≤ N − 1 ∥ φ k + r − φ k ∥ E ( ( k + r ) τ ) α ( N − k ) α r − α , ∥ φ τ ∥ C 0 α ( [ − 1 , 0 ] τ , H ) = ∥ φ τ ∥ C ( [ − 1 , 0 ] τ , H ) + sup − N ≤ k < k + r ≤ 0 ∥ φ k + r − φ k ∥ E ( − k ) α r − α , ∥ φ τ ∥ C 0 , 1 α ( [ 0 , 1 ] τ , H ) = ∥ φ τ ∥ C ( [ 0 , 1 ] τ , H ) ∥ φ τ ∥ C 0 , 1 α ( [ 0 , 1 ] τ , H ) = + sup 1 ≤ k < k + r ≤ N − 1 ∥ φ k + r − φ k ∥ E ( ( k + r ) τ ) α ( N − k ) α r − α , ∥ φ τ ∥ C ˜ 0 , 1 α ( [ − 1 , 1 ] τ , H ) = ∥ φ τ ∥ C ( [ − 1 , 1 ] τ , H ) + sup − N ≤ k < k + 2 r ≤ 0 ∥ φ k + 2 r − φ k ∥ E ( − k ) α ( 2 r ) − α ∥ φ τ ∥ C ˜ 0 , 1 α ( [ − 1 , 1 ] τ , H ) = + sup 1 ≤ k < k + r ≤ N − 1 ∥ φ k + r − φ k ∥ E ( ( k + r ) τ ) α ( N − k ) α r − α , ∥ φ τ ∥ C ˜ 0 α ( [ − 1 , 0 ] τ , H ) = ∥ φ τ ∥ C ( [ − 1 , 0 ] τ , H ) + sup − N ≤ k < k + 2 r ≤ 0 ∥ φ k + 2 r − φ k ∥ E ( − k ) α ( 2 r ) − α .

Theorem 2.1 Nonlocal boundary value problem (3) is stable in C ( [ − 1 , 1 ] τ , H ) norm.

Proof By 21 , we have

∥ { u k } 1 N − 1 ∥ C ( [ 0 , 1 ] τ , H ) ≤ M [ ∥ g τ ∥ C ( [ 0 , 1 ] τ , H ) + ∥ u 0 ∥ H + ∥ u N ∥ H ]

for the solution of boundary value problem (10).

By 22 , we have

∥ { u k } − N 0 ∥ C ( [ − 1 , 0 ] τ , H ) ≤ M [ ∥ f τ ∥ C ( [ − 1 , 0 ] τ , H ) + ∥ u 0 ∥ H ]

for the solution of inverse Cauchy difference problem (8).

Then, the proof of Theorem 2.1 is based on stability inequalities (14), (15), and on estimates

for the solution of boundary value problem (3). Estimates (16) and (17) follow from formula (7) and estimates (12), (13). Theorem 2.1 is proved. □

Theorem 2.2 Assume that φ ∈ D ( A ) and f 0 , f − 1 , g 1 ∈ D ( I + τ B ) . Then, for the solution of difference problem (3), we have the following almost coercivity inequality:

∥ { τ − 2 ( u k + 1 − 2 u k + u k − 1 ) } 1 N − 1 ∥ C ( [ 0 , 1 ] τ , H ) + ∥ { τ − 1 ( u k − u k − 1 ) } − N + 1 0 ∥ C ( [ − 1 , 0 ] τ , H ) + ∥ { A u k } 1 N − 1 ∥ C ( [ 0 , 1 ] τ , H ) + ∥ { 1 2 ( A u k + A u k − 1 ) } − N + 1 0 ∥ C ( [ − 1 , 0 ] τ , H ) ≤ M ( δ ) [ min { ln 1 τ , 1 + | ln ∥ A ∥ H → H | } [ ∥ f τ ∥ C ( [ − 1 , 0 ] τ , H ) + ∥ g τ ∥ C ( [ 0 , 1 ] τ , H ) ] + ∥ A φ ∥ H + ∥ ( I + τ B ) f 0 ∥ H + ∥ ( I + τ B ) g 1 ∥ H + ∥ ( I + τ B ) f − 1 ∥ H ] ,

where M ( δ ) is independent not only of f τ , g τ , φ but also of τ.

Proof By 24 , we have

for the solution of inverse Cauchy difference problem (8).

By 21 , we have

for the solution of boundary value problem (10). Then, the proof of Theorem 2.2 is based on almost coercivity inequalities (18), (19), and on the estimates

∥ A u 0 ∥ H ≤ M ( δ ) [ ∥ A φ ∥ H + ∥ ( I + τ B ) f 0 ∥ H + min { ln 1 τ , 1 + | ln ∥ A ∥ H → H | } [ ∥ f τ ∥ C ( [ − 1 , 0 ] τ , H ) + ∥ g τ ∥ C ( [ 0 , 1 ] τ , H ) ] ] , ∥ A u N ∥ H ≤ M ( δ ) [ [ ∥ A φ ∥ H + ∥ ( I + τ B ) f 0 ∥ H ] + min { ln 1 τ , 1 + | ln ∥ A ∥ H → H | } [ ∥ f τ ∥ C ( [ − 1 , 0 ] τ , H ) + ∥ g τ ∥ C ( [ 0 , 1 ] τ , H ) ] ]

for the solution of boundary value problem (3). The proof of these estimates follows the scheme of papers 21 24 and relies on both formula (7) and estimates (12), (13). This finalizes the proof of Theorem 2.2. □

Theorem 2.3 Let the assumptions of Theorem 2.2 be satisfied. Then, boundary value problem (3) is well-posed in Hölder spaces C 0 , 1 α ( [ − 1 , 1 ] τ , H ) , C ˜ 0 , 1 α ( [ − 1 , 1 ] τ , H ) and the following coercivity inequalities hold:

∥ { τ − 2 ( u k + 1 − 2 u k + u k − 1 ) } 1 N − 1 ∥ C 0 , 1 α ( [ 0 , 1 ] τ , H ) + ∥ { τ − 1 ( u k − u k − 1 ) } − N + 1 0 ∥ C ˜ 0 α ( [ − 1 , 0 ] τ , H ) + ∥ { A u k } 1 N − 1 ∥ C 0 , 1 α ( [ 0 , 1 ] τ , H ) + ∥ { 1 2 ( A u k + A u k − 1 ) } − N + 1 0 ∥ C ˜ 0 α ( [ − 1 , 0 ] τ , H ) ≤ M ( δ ) [ 1 α ( 1 − α ) [ ∥ f τ ∥ C 0 α ( [ − 1 , 0 ] τ , H ) + ∥ g τ ∥ C 0 , 1 α ( [ 0 , 1 ] τ , H ) ] + ∥ A φ ∥ H + ∥ ( I + τ B ) f 0 ∥ H + ∥ ( I + τ B ) g 1 ∥ H + ∥ ( I + τ B ) f − 1 ∥ H ] , ∥ { τ − 2 ( u k + 1 − 2 u k + u k − 1 ) } 1 N − 1 ∥ C 0 , 1 α ( [ 0 , 1 ] τ , H ) + ∥ { τ − 1 ( u k − u k − 1 ) } − N + 1 0 ∥ C ˜ 0 α ( [ − 1 , 0 ] τ , H ) + ∥ { A u k } 1 N − 1 ∥ C 0 , 1 α ( [ 0 , 1 ] τ , H ) + ∥ { 1 2 ( A u k + A u k − 1 ) } − N + 1 0 ∥ C ˜ 0 α ( [ − 1 , 0 ] τ , H ) ≤ M ( δ ) [ 1 α ( 1 − α ) [ ∥ f τ ∥ C ˜ 0 α ( [ − 1 , 0 ] τ , H ) + ∥ g τ ∥ C 0 , 1 α ( [ 0 , 1 ] τ , H ) ] + ∥ A φ ∥ H + ∥ ( I + τ B ) f 0 ∥ H + ∥ ( I + τ B ) g 1 ∥ H + ∥ ( I + τ B ) f − 1 ∥ H ] ,

where M is independent of f τ , g τ , φ, τ and α.

Proof By 24 ,

for the solution of inverse Cauchy difference problem (8).

By 21 , we have

for the solution of boundary value problem (10). Then, the proof of Theorem 2.3 is based on coercivity inequalities (20)-(22) and estimates

for the solution of difference scheme (3). The proof of these estimates follows the scheme of the papers 21 24 and relies on both formula (7) and estimates (12), (13). This is the end of the proof of Theorem 2.3. □

Now, the application of the abstract result is considered. In [ − 1 , 1 ] × Ω , let us consider the boundary value problem for multi-dimensional elliptic-parabolic equation

{ − u t t − ∑ r = 1 n ( a r ( x ) u x r ) x r = g ( t , x ) , 0 < t < 1 , x ∈ Ω , u t + ∑ r = 1 n ( a r ( x ) u x r ) x r = f ( t , x ) , − 1 < t < 0 , x ∈ Ω , u ( t , x ) = 0 , x ∈ S , − 1 ≤ t ≤ 1 ; u ( 1 , x ) = ∑ i = 1 J α i u ( λ i , x ) + φ ( x ) , ∑ i = 1 J | α i | ≤ 1 , − 1 ≤ λ 1 < λ 2 < ⋯ < λ i < ⋯ < λ J ≤ 0 , u ( 0 + , x ) = u ( 0 − , x ) , u t ( 0 + , x ) = u t ( 0 − , x ) , x ∈ Ω ¯ ,

where a r ( x ) ( x ∈ Ω ), φ ( x ) ( φ ( x ) = 0 , x ∈ S ), g ( t , x ) ( t ∈ ( 0 , 1 ) , x ∈ Ω ¯ ), and f ( t , x ) ( t ∈ ( − 1 , 0 ) , x ∈ Ω ¯ ) are given smooth functions. Here, Ω is the unit open cube in the n-dimensional Euclidean space R n ( 0 < x k < 1 , 1 ≤ k ≤ n ) with boundary S, Ω ¯ = Ω ∪ S , and a r ( x ) ≥ a > 0 .

The discretization of problem (23) is carried out in two steps. In the first step, the grid sets

are defined. To the differential operator A generated by problem (23), we assign the difference operator A h x by formula

A h x u h = − ∑ r = 1 n ( a r ( x ) u x ¯ r h ) x r , m r

acting in the space of grid functions u h ( x ) , satisfying the conditions u h ( x ) = 0 for all x ∈ S h . With the help of A h x , we arrive at the nonlocal boundary value problem

{ − d 2 u h ( t , x ) d t 2 + A h x u h ( t , x ) = g h ( t , x ) , 0 < t < 1 , x ∈ Ω h , d u h ( t , x ) d t − A h x u h ( t , x ) = f h ( t , x ) , − 1 < t < 0 , x ∈ Ω h , u h ( 1 , x ) = u h ( − 1 , x ) + φ h ( x ) , x ∈ Ω ˜ h , u h ( 0 + , x ) = u h ( 0 − , x ) , d u h ( 0 + , x ) d t = d u h ( 0 − , x ) d t , x ∈ Ω ˜ h

for an infinite system of ordinary differential equations (see 21 ).

Secondly, problem (25) is replaced by difference scheme (3), so that the following second order of accuracy difference scheme

{ − u k + 1 h ( x ) − 2 u k h ( x ) + u k − 1 h ( x ) τ 2 + A h x u k h ( x ) = g k h ( x ) , g k h ( x ) = g h ( t k , x ) , t k = k τ , 1 ≤ k ≤ N − 1 , N τ = 1 , x ∈ Ω h , u k h ( x ) − u k − 1 h ( x ) τ − A h x 2 ( u k h ( x ) + u k − 1 h ( x ) ) = f k h ( x ) , f k h ( x ) = f h ( t k − 1 2 , x ) , t k − 1 2 = ( k − 1 2 ) τ , − N + 1 ≤ k ≤ 0 , x ∈ Ω h , − u 2 h ( x ) + 4 u 1 h ( x ) − 3 u 0 h ( x ) = 3 u 0 h ( x ) − 4 u − 1 h ( x ) + u − 2 h ( x ) , x ∈ Ω ˜ h , u N h ( x ) = ∑ k = 1 J α i ( u h [ λ i τ ] ( x ) + ( λ k − [ λ i τ ] τ ) ( f h [ λ i τ ] + A h x u h [ λ i τ ] ( x ) ) ) + φ h ( x ) , x ∈ Ω ˜ h .

is obtained (see 21 , 22 ).

To formulate the results, we introduce the spaces L 2 h = L 2 ( Ω ¯ h ) , W 2 h 1 = W 2 1 ( Ω ¯ h ) , and W 2 h 2 = W 2 2 ( Ω ¯ h ) of the grid functions φ h ( x ) = { φ ( h 1 m 1 , … , h n m n ) } defined on Ω ¯ h , equipped with the norms

∥ φ h ∥ L 2 h = ( ∑ x ∈ Ω ¯ h | φ h ( x ) | 2 h 1 ⋯ h n ) 1 / 2 , ∥ φ h ∥ W 2 h 1 = ∥ φ h ∥ L 2 h + ( ∑ x ∈ Ω ¯ h ∑ r = 1 n | ( φ h ) x r | 2 h 1 ⋯ h n ) 1 / 2 ,

and

∥ φ h ∥ W 2 h 2 = ∥ φ h ∥ L 2 h + ( ∑ x ∈ Ω ¯ h ∑ r = 1 n | ( φ h ) x r | 2 h 1 ⋯ h n ) 1 / 2 + ( ∑ x ∈ Ω ¯ h ∑ r = 1 n | ( φ h ) x r x ¯ r , m r | 2 h 1 ⋯ h n ) 1 / 2 .

Theorem 3.1 Let τ and | h | = h 1 2 + ⋯ + h n 2 be sufficiently small positive numbers. Then, solutions of difference scheme (26) satisfy the following stability and almost coercivity estimates:

where M is independent not only of τ, h, φ h ( x ) but also of f k h , − N + 1 ≤ k ≤ 0 and g k h ( x ) , 1 ≤ k ≤ N − 1 .

The proof of Theorem 3.1 is based on Theorem 2.1, Theorem 2.2, the symmetry properties of the difference operator A h x defined by formula (24) in L 2 h , the estimate

min { ln 1 τ , 1 + | ln ∥ A h x ∥ L 2 h → L 2 h | } ≤ M ln 1 τ + | h | ,

and the following theorem in L 2 h :

Theorem 3.2 For the solution of the elliptic difference problem A h x u h ( x ) = ω h ( x ) , x ∈ Ω h , u h ( x ) = 0 , x ∈ S h

the following coercivity inequality holds 23 :

∑ r = 1 n ∥ ( u h ) x ¯ r x r , m r ∥ L 2 h ≤ M ∥ ω h ∥ L 2 h .

Here M is independent of h and ω h .

Theorem 3.3 Let τ and | h | be sufficiently small positive numbers. Then, the solutions of difference scheme (26) satisfy the following coercivity stability estimates:

∥ { τ − 2 ( u k + 1 h − 2 u k h + u k − 1 h ) } 1 N − 1 ∥ C 0 , 1 α ( [ 0 , 1 ] τ , L 2 h ) + ∥ { τ − 1 ( u k h − u k − 1 h ) } − N + 1 0 ∥ C ˜ 0 α ( [ − 1 , 0 ] τ , L 2 h ) + ∥ { u k h } 1 N − 1 ∥ C 0 , 1 α ( [ 0 , 1 ] τ , W 2 h 2 ) + ∥ { u k h + u k − 1 h 2 } − N + 1 0 ∥ C ˜ 0 α ( [ − 1 , 0 ] τ , W 2 h 2 ) ≤ M [ ∥ φ h ∥ W 2 h 2 + τ ∥ f 0 h ∥ W 2 h 1 + τ ∥ f − 1 h ∥ W 2 h 1 + τ ∥ g 1 h ∥ W 2 h 1 + 1 α ( 1 − α ) [ ∥ { f k h } − N + 1 − 1 ∥ C 0 α ( [ − 1 , 0 ] τ , L 2 h ) + ∥ { g k h } 1 N − 1 ∥ C 0 , 1 α ( [ 0 , 1 ] τ , L 2 h ) ] ] , ∥ { τ − 2 ( u k + 1 h − 2 u k h + u k − 1 h ) } 1 N − 1 ∥ C 0 , 1 α ( [ 0 , 1 ] τ , L 2 h ) + ∥ { u k h + u k − 1 h 2 } − N + 1 0 ∥ C ˜ 0 α ( [ − 1 , 0 ] τ , W 2 h 2 ) + ∥ { τ − 1 ( u k h − u k − 1 h ) } − N + 1 0 ∥ C ˜ 0 α ( [ − 1 , 0 ] τ , L 2 h ) + ∥ { u k h } 1 N − 1 ∥ C 0 , 1 α ( [ 0 , 1 ] τ , W 2 h 2 ) ≤ M [ ∥ φ h ∥ W 2 h 2 + τ ∥ f 0 h ∥ W 2 h 1 + τ ∥ f − 1 h ∥ W 2 h 1 + τ ∥ g 1 h ∥ W 2 h 1 + 1 α ( 1 − α ) [ ∥ { f k h } − N + 1 − 1 ∥ C ˜ 0 α ( [ − 1 , 0 ] τ , L 2 h ) + ∥ { g k h } 1 N − 1 ∥ C 0 , 1 α ( [ 0 , 1 ] τ , L 2 h ) ] ] .

Here, M is independent not only of τ, h, φ h ( x ) but also of f k h , − N + 1 ≤ k ≤ 0 and g k h ( x ) , 1 ≤ k ≤ N − 1 .

The proof of Theorem 3.3 is based on the abstract Theorem 2.3, Theorem 3.2, and the symmetry properties of the difference operator A h x defined by formula (24).

The theoretical statements for the solution of these difference schemes are supported by the results of numerical experiments of the nonlocal boundary value problem

{ ∂ u ∂ t + ∂ u ∂ x ( ( 1 + x ) ∂ u ∂ x ) = f ( t , x ) , f ( t , x ) = ( − 2 e − t + 1 − t ) sin x + ( e − t + t ) ( cos x − x sin x ) , − 1 < t ≤ 0 , 0 < x < π , ∂ 2 u ∂ t 2 + ∂ u ∂ x ( ( 1 + x ) ∂ u ∂ x ) = g ( t , x ) , g ( t , x ) = − t sin x + ( e − t + t ) ( cos x − x sin x ) , 0 < t < 1 , 0 < x < π , u ( 1 , x ) = 1 2 u ( − 1 , x ) + 1 2 u ( − 1 2 , x ) + φ ( x ) , φ ( x ) = ( e − 1 − e 2 − 1 2 e 1 2 + 7 4 ) sin x , 0 ≤ x ≤ π , u ( t , 0 ) = u ( t , π ) = 0 , − 1 ≤ t ≤ 1

for the elliptic-parabolic equation. The exact solution of this problem is

u ( t , x ) = ( e − t + t ) sin x .

For the comparison, the errors computed by the following formula

E M N = max − N ≤ k ≤ N 1 ≤ n ≤ M − 1 | u ( t k , x n ) − u n k |

are recorded for different values of N and M, where u ( t k , x n ) represents the exact solution and u n k represents the numerical solution at ( t k , x n ) . The results are shown in Table 1 for N = M = 30 , 60 and 90 respectively.

Therefore, the results indicate that the second order of accuracy difference scheme is more accurate than the first order of accuracy difference scheme.

The authors declare that they have no competing interests.

The manuscript was drafted by OG and it is based on his PhD thesis. AA is the supervisor of the thesis and gave detailed comments on the manuscript. All authors read and approved the final manuscript.