Explicit Finite Difference Methods for the Delay Pseudoparabolic Equations

Abstract

Finite difference technique is applied to numerical solution of the initial-boundary value problem for the semilinear delay Sobolev or pseudoparabolic equation. By the method of integral identities two-level difference scheme is constructed. For the time integration the implicit rule is being used. Based on the method of energy estimates the fully discrete scheme is shown to be absolutely stable and convergent of order two in space and of order one in time. The error estimates are obtained in the discrete norm. Some numerical results confirming the expected behavior of the method are shown.

1. Introduction

We consider the initial-boundary value problem for pseudoparabolic equation with delay in the domain 𝑄=Ω∪[0,𝑇]; Ω=[0,𝑙], 𝑄=Ω∪(0,𝑇], Ω=(0,𝑙): 𝐿𝑢∶=𝜕𝑢−𝜕𝜕𝑡𝑎()𝜕𝜕𝑥𝑥,𝑡²𝑢−𝜕𝜕𝑡𝜕𝑥𝑏()𝜕𝑥𝑥,𝑡𝜕𝑢(()(,()𝑢()(),()∈𝜕𝑥=𝑓𝑥,𝑡,𝑢𝑥,𝑡,𝑢𝑥,𝑡−𝑟))𝑥,𝑡∈𝑄,𝑥,𝑡=𝜑𝑥,𝑡𝑥,𝑡[],𝑢()()(],Ω×−𝑟,00,𝑡=𝑢𝑙,𝑡=0,𝑡∈0,𝑇 (1) where 𝑟 represents the delay parameter (for simplicity we assume that 𝑇/𝑟 is an integer; i.e., 𝑇=𝑚𝑟 for some integer 𝑚>0), 𝑎(𝑥,𝑡)≥𝛼>0, |𝑏(𝑥,𝑡)|≤𝑏^∗, and 𝜑(𝑥,𝑡) and 𝑓(𝑥,𝑡,𝑢₁,𝑢₂) are given sufficiently smooth functions satisfying certain regularity conditions in 𝑄 and Ω×[−𝑟,0] and 𝑄×ℝ², respectively, to be specified, and furthermore 𝜕𝑓𝜕𝑢₁≤𝑐^∗<∞,𝜕𝑓𝜕𝑢₂≤𝑑^∗<∞. (2)

Equations of this type arise in many areas of mechanics and physics. Such equations are encountered, for example, as a model for two-phase porous media flows when dynamic effects in the capillary pressure are included [1–3]. They are used also to study heat conduction [4], homogeneous fluid flow in fissured rocks [5], shear in second order fluids [6–8], and other physical models. For a discussion of existence and uniqueness results of pseudoparabolic equations see [1, 9–11]. Various numerical treatments of equations of this type without delay have been considered in [2, 12–19] (see also the references cited in them).

In the present paper finite difference technique is applied to numerical solution of the initial-boundary value problem for the semilinear delay Sobolev or pseudoparabolic equation. By the method of integral identities with use of the piecewise linear basis functions in space and interpolating quadrature rules with weight and remainder term in integral form, two-level difference scheme is constructed (see also [12–14]) for singular perturbation cases without delay. For the time integration we use the implicit rule. The finite difference discretization is shown to be absolutely stable and convergent of order two in space and of order one in time. Based on the method of energy estimates the error analysis for approximate solution is presented. The error estimates are obtained in the discrete norm. Some numerical results confirming the expected behavior of the method are shown.

2. Discretization and Mesh

Notation. Let a set of mesh nodes that discretises 𝑄 be given by 𝜔=𝜔_𝑁×𝜔_𝑁0, (3) with 𝜔_𝑁=𝑥_𝑖𝑙=𝑖ℎ,𝑖=1,2,…,𝑁−1,ℎ=𝑁,𝜔⁺_𝑁=𝜔_𝑁∪𝑥_𝑁=𝑙,𝜔_𝑁=𝜔_𝑁∪𝑥₀=0,𝑥_𝑁,𝜔=𝑙_𝑁0=𝑡_𝑗=𝑗𝜏,𝑗=1,2,…,𝑁₀𝑇,𝜏=𝑁₀=𝑟𝑛₀,𝜔_𝑁0=𝜔_𝑁0∪𝑡₀,𝜔=0_𝑛−0=𝑡_𝑗=𝑗𝜏,𝑗=−𝑛₀,…,0. (4)
Define the following finite differences for any mesh function 𝑣_𝑖=𝑣(𝑥_𝑖) given on 𝜔_𝑁 by 𝑣_𝑥,𝑖=𝑣_𝑖−𝑣_𝑖−1ℎ,𝑣_𝑥,𝑖=𝑣_𝑖+1−𝑣_𝑖ℎ,𝑣^∘_𝑥,𝑖=𝑣_𝑖+1−𝑣_𝑖−1,𝑣2ℎ_{𝑥𝑥,𝑖}=𝑣_𝑥,𝑖−𝑣_𝑥,𝑖ℎ=𝑣_𝑖+1−2𝑣_𝑖+𝑣_𝑖−1ℎ². (5)
Introduce the inner products for the mesh functions 𝑣_𝑖 and 𝑤_𝑖 defined on 𝜔_𝑁 as follows: (𝑣,𝑤)₀≡(𝑣,𝑤)_𝜔𝑁∶=^𝑁−1_𝑖=1ℎ𝑣_𝑖𝑤_𝑖,(𝑣,𝑤]≡(𝑣,𝑤)_𝜔⁺_𝑁∶=^𝑁_𝑖=1ℎ𝑣_𝑖𝑤_𝑖. (6)
For any mesh function 𝑣_𝑖, vanishing for 𝑖=0 and 𝑖=𝑁 we introduce the norms ‖𝑣‖²₀≡‖𝑣‖²_0,𝜔𝑁∶=(𝑣,𝑣)₀,𝑣_𝑥²₀≡𝑣_𝑥²_0,𝜔+𝑁∶=𝑣_𝑥,𝑣_𝑥,‖𝑣‖²₁=‖𝑣‖²₀+𝑣_𝑥²₀,‖𝑣‖_∞≡‖𝑣‖_∞,𝜔𝑁∶=max_{1≤𝑖≤𝑁−1}𝑣_𝑖, (7) and “negative” norm for any function 𝑤_𝑖 (1≤𝑖≤𝑁−1)‖𝑤‖₋₁∶=sup_𝑣≢0(𝑤,𝑣)₀‖𝑣‖₁. (8)
Given a function 𝑔≡𝑔^𝑗_𝑖≡𝑔(𝑥_𝑖,𝑡_𝑗);𝑔̆=𝑔_𝑖^(𝑗−1), defined on 𝜔, we will also use the notation 𝑔^𝑗_𝑡,𝑖=𝑔^𝑗_𝑖−𝑔_𝑖^𝑗−1𝜏_𝑗,𝑔^𝑗_𝑡,𝑖=𝑔_𝑖^𝑗+1−𝑔^𝑗_𝑖𝜏_𝑗+1. (9)

2.1. Difference Scheme

The approach of generating difference scheme is through the integral identity 𝜏⁻¹ℎ^−1𝑡𝑗_𝑡𝑗−1^𝑥𝑖+1_𝑥𝑖−1𝐿𝑢𝜓_𝑖(𝑥)𝑑𝑥𝑑𝑡=𝜏⁻¹ℎ^−1𝑡𝑗_𝑡𝑗−1^𝑥𝑖+1_𝑥𝑖−1𝑓(𝑥,𝑡,𝑢(𝑥,𝑡),𝑢(𝑥,𝑡−𝑟))×𝜓_𝑖(𝑥)𝑑𝑥𝑑𝑡, (10) with the usual piecewise linear basis functions for the space 𝜓_𝑖(𝑥)=ℎ⁻¹𝑥−𝑥_𝑖−1,𝑥_𝑖−1<𝑥<𝑥_𝑖,ℎ⁻¹𝑥_𝑖+1−𝑥,𝑥_𝑖<𝑥<𝑥_𝑖+1,0,𝑥∉𝑥_𝑖−1,𝑥_𝑖+1,𝑖=1,2,…,𝑁−1. (11)

Using the appropriate interpolating quadrature rules with weight and remainder term in integral form, consistent with [12–14], we obtain the precise relation ℓ𝑢∶=𝑢_𝑡−𝐴𝑢_𝑡𝑥𝑥−𝐵𝑢_𝑥𝑥(()(,()+𝑅=𝑓𝑥,𝑡−𝜏,𝑢𝑥,𝑡−𝜏,𝑢𝑥,𝑡−𝜏−𝑟))𝑥,𝑡∈𝜔_𝑁×𝜔_𝑁0, (12) where 𝐴=𝜏⁻¹ℎ^−1𝑡𝑗_𝑡𝑗−1^𝑥𝑖_𝑥𝑖−1𝑎(𝑥,𝑡)𝑑𝑥𝑑𝑡,𝐵=𝜏⁻¹ℎ^−1𝑡𝑗_𝑡𝑗−1^𝑥𝑖_𝑥𝑖−1𝑏(𝑥,𝑡)𝑑𝑥𝑑𝑡. (13)

The remainder term 𝑅 has the form 𝑅=𝑅⁽⁰⁾_𝑥+𝑅⁽¹⁾,for(𝑥,𝑡)∈𝜔_𝑁×𝜔_𝑁0 (14) with 𝑅⁽⁰⁾=−𝜏⁻¹ℎ^−1𝑡𝑗_𝑡𝑗−1𝑑𝑡^𝑥𝑖_𝑥𝑖−1𝑑𝑥𝜕𝑎×𝜕𝑥_{𝑥𝑖𝑥𝑖−1}𝜕³𝑢𝜕𝑡𝜕𝑥²(𝜉,𝑡)𝐾₁(𝑥,𝜉)𝑑𝜉−𝜏⁻¹ℎ^−1𝑡𝑗_𝑡𝑗−1𝑑𝑡^𝑥𝑖_𝑥𝑖−1𝜕𝑎()×𝜕𝑡𝑥,𝑡𝑑𝑥_{𝑡𝑗𝑡𝑗−1}𝐾₀𝜕𝜂,𝑡𝑢𝜕𝑡^𝑥,𝑖𝑥,𝜂𝑑𝜂−𝜏⁻¹ℎ^−1𝑡𝑗_𝑡𝑗−1𝑑𝑡^𝑥𝑖_𝑥𝑖−1𝑑𝑥𝜕𝑏()×𝜕𝑥𝑥,𝑡_{𝑥𝑖𝑥𝑖−1}𝜕²𝑢𝜕𝑥²(𝜉,𝑡)𝐾₁(𝑥,𝜉)𝑑𝜉+𝜏⁻¹ℎ^−1𝑡𝑗_𝑡𝑗−1𝑑𝑡^𝑥𝑖_𝑥𝑖−1𝑏()×𝑥,𝑡𝑑𝑥_{𝑡𝑗𝑡𝑗−1}𝑇₀()𝜕𝑡−𝜉𝑢𝜕𝑡^𝑥,𝑖()𝑅𝑥,𝜉𝑑𝜉,₍₁₎=−𝜏⁻¹ℎ^−1𝑡𝑗_𝑡𝑗−1𝑑𝑡^𝑥𝑖+1_𝑥𝑖−1𝑑𝑥𝜑_𝑖(𝑥)×^𝑥𝑖+1_𝑥𝑖−1𝜕³𝑢𝜕𝑡𝜕𝑥²(𝜉,𝑡)𝐾^∗₁(𝑥,𝜉)𝑑𝜉+𝜏⁻¹ℎ^−1𝑡𝑗_𝑡𝑗−1𝑑𝑡^𝑥𝑖+1_𝑥𝑖−1𝑑𝑥𝜑_𝑖(𝑥)×^𝑥𝑖+1_𝑥𝑖−1𝑑²𝑑𝑥²𝑓(𝜉,𝑡,𝑢(𝜉,𝑡),𝑢(𝜉,𝑡−𝑟))×𝐾^∗₁(𝑥,𝜉)𝑑𝜉+𝜏^−1𝑡𝑗_𝑡𝑗−1𝑡_𝑗𝑑−𝑡𝑥𝑑𝑡×𝑓_𝑖,𝑡,𝑢𝑥_𝑖,𝑡,𝑢𝑥_𝑖𝐾,𝑡−𝑟𝑑𝑡,₁(𝑥,𝜉)=𝑇₁(𝑥−𝜉)−ℎ⁻¹𝑥−𝑥_𝑖−1𝑥_𝑖,𝐾−𝜉₀𝑡,𝜂=𝑇₀𝑡−𝜂−𝜏⁻¹𝑡−𝑡_𝑗−1,𝐾^∗₁(𝑥,𝜉)=𝑇₁(𝑥−𝜉)−(2ℎ)⁻¹𝑥−𝑥_𝑖−1𝑥_𝑖+1,𝑇−𝜉_𝑠(𝜆)=𝜆^𝑠/𝑠!,𝜆>0,𝑇_𝑠(𝜆)=0,𝜆<0. (15)(16) Based on (12), we propose the following difference scheme for approximating (1): ()(),()ℓ𝑦=𝑓𝑥,𝑡−𝜏,𝑦𝑥,𝑡−𝜏,𝑦𝑥,𝑡−𝜏−𝑟𝑥,𝑡∈𝜔_𝑁×𝜔_𝑁0,𝑦(0,𝑡)=𝑦(𝑙,𝑡)=0,𝑡∈𝜔_𝑁0,𝑦(𝑥,0)=𝜑(𝑥,0),𝑥∈𝜔_𝑁, (17) where ℓ𝑦 is defined by (12).

3. The Error Estimates and Convergence

To estimate the convergence of this method, note that the error function 𝑧=𝑦−𝑢 is the solution of the discrete problem, ()()(()(()ℓ𝑧=𝑓𝑥,𝑡−𝜏,𝑦𝑥,𝑡−𝜏,𝑦𝑥,𝑡−𝜏−𝑟−𝑓𝑥,𝑡−𝜏,𝑢𝑥,𝑡−𝜏,𝑢𝑥,𝑡−𝜏−𝑟))+𝑅,𝑥,𝑡∈𝜔_𝑁×𝜔_𝑁0,𝑧(𝑥,𝑡)=0,(𝑥,𝑡)∈𝜔_𝑁×𝜔_𝑛⁻₀,𝑧(0,𝑡)=𝑧(𝑙,𝑡)=0,𝑡∈𝜔_𝑁0. (18)

Before obtaining the estimate for the solution (18) we give the following Lemma.

Lemma 1. Let the mesh function 𝛿≥0, defined on 𝜔_𝑁0, satisfy 𝛿_𝑗≤𝛼+𝜏^𝑗_𝑘=1𝑐₁𝛿_k−1+𝑐_∗𝛿_{𝑗−𝑛0−1}+𝜌_𝑘,𝑗=1,2,…,𝑁₀,𝛿_𝑗≤𝜑_𝑗,𝑗=−𝑛₀,…,0,𝜑₀≤𝛼, (19) where 0≤𝛼,𝑐₁,𝑐_∗=𝑐𝑜𝑛𝑠𝑡,𝜌_𝑗≥0,𝜑_𝑗 given, 𝑛₀≥0 is an integer. Then 𝛿_𝑗≤𝛼+𝑐_∗𝜑_{𝐿1(𝜔𝑛−0)}𝑒^{𝑐1𝑡𝑗}+𝜏^𝑗_𝑘=1𝑒^{(𝑐1+𝑐∗)𝑡𝑗−𝑘}𝜌_𝑘, (20) where 𝜑_{𝐿1(𝜔𝑛−0)}=⁰_{𝑗=−𝑛0}𝜏𝜑_𝑗. (21)

Proof. For 1≤𝑗≤𝑛₀+1,−𝑛₀≤𝑘−1−𝑛₀≤0 and inequality (19) reduces to 𝛿_𝑗≤𝛼+𝜏^𝑗_𝑘=1𝑐₁𝛿_𝑘−1+𝜌_𝑘+𝑐_∗𝜑_{𝐿1(𝜔𝑛−0)}. (22) Applying now the difference analogue of the Gronwall's inequality we get 𝛿_𝑗≤𝛼+𝑐_∗𝜑_{𝐿1(𝜔𝑛−0)}𝑒^{𝑐1𝑡𝑗}+𝜏^𝑗_𝑘=1𝑒^{𝑐1𝑡𝑗−𝑘}𝜌_𝑘,1≤𝑗≤𝑛₀+1. (23)
For 𝑗>𝑛₀+1, after replacing in (19) 𝑘−𝑛₀=𝑝, we have 𝛿_𝑗≤𝛼+𝜏^𝑗_𝑘=1𝑐₁𝛿_𝑘−1+𝜌_𝑘+𝜏^𝑗−𝑛0_{𝑝=1−𝑛0}𝑐_∗𝛿_𝑘−1=𝛼+𝜏^𝑗_𝑘=1𝑐₁𝛿_𝑘−1+𝜌_𝑘+𝜏¹_{𝑝=1−𝑛0}𝑐_∗𝛿_𝑘−1+𝜏^𝑗−𝑛0_𝑝=2𝑐_∗𝛿_𝑘−1≤𝛼+𝜏^𝑗_𝑘=1𝑐₁+𝑐_∗𝛿_𝑘−1+𝜌_𝑘+𝑐_∗𝜑_{𝐿1(𝜔𝑛−0)}, (24) which by virtue of difference analogue of the Gronwall's inequality leads to (20), immediately.

Theorem 2. Let the derivatives 𝜕𝑓/𝜕𝑡,𝜕^𝑠𝑓/𝜕𝑥^𝑠, 𝜕^1+𝑘𝑓/𝜕𝑢_𝑠𝜕𝑥^𝑘(𝑠=1,2;𝑘=0,1),   and  𝜕²𝑓/𝜕𝑢^𝑘₁𝜕𝑢₂^2−𝑘(𝑘=0,1,2) be continuous and bounded on 𝑄×ℝ²,𝑎, 𝑏∈𝐶(𝑄),(𝜕²/𝜕𝑥²)𝜑(𝑥,𝑡),(𝜕/𝜕𝑡)𝜑(𝑥,𝑡)∈𝐶(Ω×[−𝑟,0]), and 𝜕𝑢/𝜕𝑡, 𝜕𝑢/𝜕𝑥, 𝜕²𝑢/𝜕𝑥²,  and  𝜕³𝑢/𝜕𝑥²𝜕𝑡∈𝐶(𝑄). Then for the discrete problem (17) the following error estimate holds: 𝑦−𝑢₁≤𝐶ℎ²+𝜏,𝑡∈𝜔_𝑁0. (25)

Proof. Consider identity ℓ𝑧,𝑧_𝑡𝜔𝑁=𝑓𝑥,𝑡−𝜏,𝑦𝑥,𝑡−𝜏−𝑛₀𝜏−𝑓𝑥,𝑡−𝜏,𝑢𝑥,𝑡−𝜏−𝑛₀𝜏,𝑧_𝑡𝜔𝑁+𝑅,𝑧_𝑡𝜔𝑁. (26) After some manipulations, we get 𝑧_𝑡²₀+𝛼𝑧_𝑡𝑥²₀≤𝑏^∗𝑧̆_𝑥0𝑧_𝑡𝑥0+𝑐^∗𝑧_𝑡0‖𝑧‖̆₀+𝑑^∗𝑧_𝑡0𝑧𝑡−𝜏−𝑛₀𝜏₀+𝑧_𝑡𝑥0‖𝑅‖₋₁,12𝑧_𝑡²₀+𝛼2𝑧_𝑡𝑥²₀≤𝛼⁻¹𝑏^∗2𝑧̆_𝑥²₀+𝑐^∗2‖𝑧‖̆²₀+𝑑^∗2𝑧𝑡−𝜏−𝑛₀𝜏²₀+𝛼⁻¹‖𝑅‖²₋₁. (27) Multiplying this inequality by 𝜏 and summing it up from 𝑘=1 to 𝑘=𝑗, also, using here the inequality 𝑣²_𝑗≤𝑡_𝑗𝜏^𝑗_𝑘=1𝑣²_𝑡,𝑘≤𝑇𝜏^𝑗_𝑘=1𝑣²_𝑡,𝑘,𝑣₀=0, (28) we obtain 𝑧^𝑗2₀+𝛼𝑧^𝑗_𝑥²₀≤𝜏^𝑗_𝑘=12𝛼⁻¹𝑇𝑏^∗2𝑧̆_𝑥²₀+2𝑐^∗2𝑇‖𝑧‖̆²₀+2𝑑^∗2𝑇‖𝑧(̆𝑡−𝑟)‖²₀+2𝑇𝛼⁻¹‖𝑅‖²₋₁. (29) Denoting 𝛿_𝑗=𝑧^𝑗2₀+𝛼𝑧^𝑗_𝑥²₀, (30) we have 𝛿_𝑗≤𝜏^𝑗_𝑘=1𝑐₁𝛿_𝑘−1+𝑐_∗𝛿_{𝑘−𝑛0−1}+𝜌_𝑘,𝑗⩾1, (31) where 𝑐₁=2𝑇max𝛼⁻¹𝑏^∗2,𝑐^∗2,𝑐_∗=2𝑇𝑑^∗2,𝜌=2𝑇𝛼⁻¹‖𝑅‖²₋₁. (32) Applying now Lemma 1 we obtain 𝑧^𝑗2₀+𝛼𝑧^𝑗_𝑥²₀⩽2𝑇𝛼⁻¹𝜏^𝑗_𝑘=1𝑒^{(𝑐1+𝑐∗)𝑡𝑘−𝑗}𝑅^𝑘2₋₁. (33) Further, in view of the fact that (𝑅,𝑧)₀≤𝑅_𝑥⁽⁰⁾,𝑧₀+𝑅⁽¹⁾,𝑧₀=𝑅⁽⁰⁾,𝑧_𝑥+𝑅⁽¹⁾,𝑧₀,𝑡∈𝜔_𝑁0, (34) we obtain ‖𝑅‖₋₁≤𝑅⁽⁰⁾_0,𝜔+𝑁+𝑅⁽¹⁾_0,𝜔𝑁,𝑡∈𝜔_𝑁0, (35) where 𝑅⁽⁰⁾ and 𝑅⁽¹⁾are given by (15). From (35), under the assumed smoothness, we have 𝜏^𝑁0_𝑘=1𝑅^𝑘2₋₁=𝑂ℎ²+𝜏, (36) which together with (33) completes the proof of the theorem.

Remark 3. Under sufficiently smoothness of 𝑎(𝑥,𝑡) and 𝑏(𝑥,𝑡) for calculations of 𝐴 and 𝐵 appropriate numerical quadrature formulae can be applied; for example, 𝐴=𝑎(𝑥_𝑖−ℎ/2,𝑡_𝑗), 𝐴=(1/2)[𝑎(𝑥_𝑖,𝑡_𝑗)+𝑎(𝑥_𝑖−1,𝑡_𝑗)], and so forth.

4. Numerical Results

In this section, we present numerical results obtained by applying the numerical method (17) to the particular problems.

Example 1. Consider the following linear problems: 𝜕𝑢𝜕𝜕𝑡−2³𝑢𝜕𝑡𝜕𝑥²−𝜕²𝑢𝜕𝑥²()(),()∈[]×[]𝑢()+𝑢𝑥,𝑡−1=𝑓𝑥,𝑡𝑥,𝑡0,10,2𝑥,𝑡=𝑒^−𝑡((𝑥)(1,()∈[]×[]𝑢()()(],sinh−𝑥sinh))𝑥,𝑡0,1−1,00,𝑡=0,𝑢1,𝑡=0,𝑡∈0,2 (37) where 𝑓(𝑥,𝑡)=𝑒^−𝑡𝑥sinh(1)+𝑒^1−𝑡(sinh(𝑥)−𝑥sinh(1)). (38)

The exact solution of this problem is 𝑢(𝑥,𝑡)=𝑒^−𝑡(sinh(𝑥)−𝑥sinh(1)). The computational results are presented in Tables 1 and 2.

Table 1 

The numerical results on (0,1)×(0,1).

Table 2 

The numerical results on (0,1)×(1,2).

Example 2. Now consider the following nonlinear problem: 𝜕𝑢𝜕𝜕𝑡−3³𝑢𝜕𝑡𝜕𝑥²𝜕−2²𝑢𝜕𝑥²(𝑢((),()∈[]×[],𝑢()+tanh𝑥,𝑡−1))=𝑓𝑥,𝑡𝑥,𝑡0,10,2𝑥,𝑡=𝑒^−𝑡((𝑥)(1)),()∈[]×[],𝑢()()(],cosh−𝑥cosh+𝑥−1𝑥,𝑡0,1−1,00,𝑡=0,𝑢1,𝑡=0,𝑡∈0,2 (39) where 𝑓(𝑥,𝑡)=𝑒^−𝑡((1))𝑒𝑥cosh−𝑥+1+tanh^1−𝑡(cosh(𝑥)−𝑥cosh(1)+𝑥−1). (40)

The exact solution of this problem is 𝑢(𝑥,𝑡)=𝑒^−𝑡(cosh(𝑥)−𝑥cosh(1)+𝑥−1). The computational results are presented in Tables 3 and 4.

Table 3 

The numerical results on (0,1)×(0,1).

Table 4 

The numerical results on (0,1)×(1,2).

It can be observed that the obtained results are essentially in agreement with the theoretical analysis described above.

5. Conclusion

In this paper, we proposed an efficient numerical method for solving initial-boundary value problem for the semilinear pseudoparabolic equation. The proposed finite difference method was constructed, and based on the method of energy estimates the fully discrete scheme was shown to be absolutely stable and convergent of order two in space and of order one in time. The error estimates were obtained in discrete norm. Numerical results were presented, which numerically validate this theoretical result.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.