Approximate Solutions of Delay Parabolic Equations with the Dirichlet Condition

Abstract

Finite difference and homotopy analysis methods are used for the approximate solution of the initial-boundary value problem for the delay parabolic partial differential equation with the Dirichlet condition. The convergence estimates for the solution of first and second orders of difference schemes in Hölder norms are obtained. A procedure of modified Gauss elimination method is used for the solution of these difference schemes. Homotopy analysis method is applied. Comparison of finite difference and homotopy analysis methods is given on the problem.

1. Introduction

Increase in interest in the theoretical aspects of numerical methods for delay differential equations points out that delay differential equations are capable of generating extensive and conceivable models for phenomena in many branches of sciences. Numerical solutions of the delay ordinary differential equations have been studied mostly for ordinary differential equations (cf., e.g., [1–14] and the references therein). Nevertheless, delay partial differential equations are less in demand than delay ordinary differential equations. Different kinds of problems for delay partial differential equations are solved by using operator approach (see, e.g., [15–17]).

In recent years, Ashyralyev and Sobolevskii considered the initial-value problem for linear delay partial differential equations of parabolic type in the spaces 𝐶(𝐸_𝛼) of functions defined on the segment [0,∞) with values in a Banach space 𝐸_𝛼 and the stability inequalities were established under stronger assumption than the necessary condition of the stability of the differential problem. The stability estimates for the solutions of difference schemes of the first- and second-order accuracy difference schemes for approximately solving this initial-value problem for delay differential equations of parabolic type were presented. They obtained the stability estimates in Hölder norms for solutions of the initial-value problem of the delay differential and difference equations of the parabolic type [15, 16]. Gabriella used extrapolation spaces to solve Banach spaces valued delay differential equations with unbounded delay operators. The author proved regularity properties of various types of solutions and investigated the existence of strong and weak solutions for a class of abstract semilinear delay equations [17].

In this paper, finite difference (see, e.g., [18–28]) and homotopy analysis methods (HAM) (see, e.g., [29–37]) for the approximate solutions of the delay differential equation of the parabolic type𝑢_𝑡(𝑡,𝑥)+−𝑎(𝑥)𝑢_𝑥𝑥(𝑡,𝑥)+𝑏(𝑥)𝑢_𝑥()(𝑥)𝑢()(𝑡)(𝑥)𝑢𝑡,𝑥+𝑐𝑡,𝑥=𝑑−𝑎_𝑥𝑥(𝑡−𝜔,𝑥)+𝑏(𝑥)𝑢_𝑥()(𝑥)𝑢((),𝑢()()[],𝑢()()𝑡−𝜔,𝑥+𝑐𝑡−𝜔,𝑥)),0<𝑡<∞,𝑥∈0,𝑙𝑡,𝑥=𝑔𝑡,𝑥,−𝜔≤𝑡≤0,𝑥∈0,𝑙𝑡,0=𝑢𝑡,𝑙=0,𝑡≥0, (1.1) are studied. Here 𝑔(𝑡,𝑥)(𝑡∈(−∞,0),𝑥∈[0,𝑙]),𝑎(𝑥),𝑏(𝑥),𝑐(𝑥)(𝑥∈(0,∞)) are given smooth bounded functions and 𝑎(𝑥)≥𝑎>0.

Difference schemes which are accurate to first and second orders for the approximate solution of problem (1.1) are presented. The convergence estimates for the solution of these difference schemes are obtained. For the numerical study, procedure of modified Gauss elimination method is used to solve these difference schemes. Homotopy analysis method is applied to find the solution of problem (1.1). The numerical results are obtained at the same points for each method. Comparison of finite difference and homotopy analysis methods is given on the problem.

2. The Finite Difference Method

In this section, the first and second orders of accuracy in 𝑡 for the approximate solution of problem (1.1) are considered. The convergence estimates for the solution of these difference schemes are established. A procedure of modified Gauss elimination method is used to solve these difference schemes.

2.1. The Difference Scheme, Convergence Estimates

The discretization of problem (1.1) is carried out in two steps. In the first step, we define the grid space[0,𝐿]_ℎ=𝑥=𝑥_𝑛∶𝑥_𝑛=𝑛ℎ,0≤𝑛≤𝑀,𝑀ℎ=𝐿. (2.1) To formulate our results, we introduce the Banach space𝐶^∘𝛼_ℎ=𝐶^∘𝛼[0,𝐿]_ℎ, 𝛼∈[0,1), of all grid functions 𝜑^ℎ={𝜑_𝑛}^𝑀−1_𝑛=1 defined on [0,𝐿]_ℎ with 𝜑₀=𝜑_𝑀=0 equipped with the norm𝜑^ℎ∘_𝐶^𝛼_ℎ=𝜑^ℎ_𝐶ℎ+sup_{1≤𝑛<𝑛+𝑟≤𝑀−1}𝜑_𝑛+𝑟−𝜑_𝑛(𝑟ℎ)^−𝛼,𝜑^ℎ_𝐶ℎ=max_{1≤𝑛≤𝑀−1}𝜑_𝑛. (2.2) Moreover, 𝐶_𝜏(𝐸)=𝐶([0,∞)_𝜏,𝐸) is the Banach space of all grid functions 𝑓^𝜏={𝑓_𝑘}^∞_𝑘=1 defined on[0,∞)_𝜏=𝑡_𝑘=𝑘𝜏,𝑘=0,1,… (2.3) with values in 𝐸 equipped with the norm𝑓^𝜏_{𝐶𝜏(𝐸)}=sup_1≤𝑘<∞𝑓_𝑘𝐸. (2.4) To the differential operator 𝐴 generated by problem (1.1), we assign the difference operators 𝐴^𝑥_ℎ,𝐵^𝑥_ℎ by the formulas𝐴^𝑥_ℎ𝜑^ℎ(𝑥)=−𝑎𝑥_𝑛𝜑_𝑛+1−2𝜑_𝑛+𝜑_𝑛−1ℎ²+𝑏𝑥_𝑛𝜑_𝑛+1−𝜑_𝑛−1𝑥2ℎ+𝑐_𝑛𝜑_𝑛1^𝑀−1,𝐵^𝑥_ℎ(𝑡)𝜑^ℎ(𝑥)=𝑑(𝑡)𝐴^𝑥_ℎ𝜑^ℎ, (2.5) acting in the space of grid functions 𝜑^ℎ(𝑥)={𝜑_𝑛}₁^𝑀−1 satisfying the conditions 𝜑₀=𝜑_𝑀=0. It is well known that 𝐴^𝑥_ℎ is a strongly positive operator in 𝐶_ℎ. With the help of 𝐴^𝑥_ℎ and 𝑑(𝑡)𝐴^𝑥_ℎ, we arrive at the initial value problem𝑑𝑢^ℎ(𝑡,𝑥)𝑑𝑡+𝐴^𝑥_ℎ𝑢^ℎ(𝑡,𝑥)=𝑑(𝑡)𝐴^𝑥_ℎ𝑢^ℎ()𝑢𝑡−𝑤,𝑥,0<𝑡<∞,0<𝑥<𝐿,_ℎ(𝑡,𝑥)=𝑔^ℎ(𝑡,𝑥),−𝜔≤𝑡≤0,0≤𝑥≤𝐿. (2.6) In the second step, we consider difference schemes of first and second orders of accuracy1𝜏𝑢^ℎ_𝑘(𝑥)−𝑢^ℎ_𝑘−1(𝑥)+𝐴^𝑥_ℎ𝑢^ℎ_𝑘(𝑥)=𝑑𝑡_𝑘𝐴^𝑥_ℎ𝑢^ℎ_𝑘−𝑁(𝑥),𝑡_𝑘𝑢=𝑘𝜏,1≤𝑘,𝑁𝜏=𝑤,_ℎ𝑘(𝑥)=𝑔^ℎ𝑡_𝑘,𝑥,𝑡_𝑘1=𝑘𝜏,−𝑁≤𝑘≤0,𝜏𝑢^ℎ_𝑘(𝑥)−𝑢^ℎ_𝑘−1(𝑥)+𝐴^𝑥_ℎ+12𝜏𝐴^𝑥_ℎ²𝑢^ℎ_𝑘(𝑥)=12𝜏𝐼+2𝐴^𝑥_ℎ𝑑𝑡_𝑘−𝜏2𝐴^𝑥_ℎ𝑢^ℎ_𝑘−𝑁(𝑥)+𝑢^ℎ_{𝑘−𝑁−1}(𝑥),𝑡_𝑘𝑢=𝑘𝜏,1≤𝑘,_ℎ𝑘=𝑔^ℎ𝑡_𝑘,𝑥,𝑡_𝑘=𝑘𝜏,−𝑁≤𝑘≤0. (2.7)(2.8)

Theorem 2.1. Assume that sup_0≤𝑡<∞|𝑑(𝑡)|≤1−𝛼𝑀2^2−𝛼. (2.9) Suppose that problem (1.1) has a smooth solution 𝑢(𝑡,𝑥) and ^∞₀max_{0≤𝑥≤𝐿}𝑢_𝑠𝑠(𝑠,𝑥)+sup_{0<𝑥<𝑥+𝑦<𝐿}𝑢_𝑠𝑠𝑠,𝑥+𝑦−𝑢_𝑠𝑠(𝑠,𝑥)y^2𝛼𝑑𝑠<∞,_∞0max_{0≤𝑥≤𝐿}𝑢_{𝑥𝑥𝑥𝑥}(𝑠,𝑥)+sup_{0<𝑥<𝑥+𝑦<𝐿}𝑢_{𝑥𝑥𝑥𝑥}𝑠,𝑥+𝑦−𝑢_{𝑥𝑥𝑥𝑥}(𝑠,𝑥)𝑦^2𝛼𝑑𝑠<∞. (2.10) Then, for the solution of difference scheme (2.7), the following convergence estimate holds: sup_𝑘𝑢^ℎ_𝑘−𝑢^ℎ𝑡_𝑘,⋅_∘𝐶ℎ^2𝛼≤𝑀₁𝜏+ℎ² (2.11) with 𝑀₁ being a real number independent of 𝜏, 𝛼, and ℎ.

Proof. Using notations of 𝐴^𝑥_ℎ and 𝐵^𝑥_ℎ, we can obtain the following formula for the solution: 𝑢^ℎ_𝑘(𝑥)=𝑅^𝑘𝑔^ℎ(0,𝑥)+^𝑘_𝑗=1𝑅^{𝑘−𝑗+1}𝐵^𝑥_𝑗𝑔^ℎ𝑡_𝑗−𝑁𝑢,𝑥𝜏,1≤𝑘≤𝑁,_ℎ𝑘(𝑥)=𝑅^{𝑘−𝑛𝑁}𝑢^ℎ_𝑛𝑁(𝑥)+^𝑘_{𝑗=𝑛𝑁+1}𝑅^{𝑘−𝑗+1}𝐵^𝑥_𝑗𝑢^ℎ_𝑗−𝑁(𝑥)()𝜏,𝑛𝑁≤𝑘≤𝑛+1𝑁, (2.12) where 𝑅=(𝐼+𝜏𝐴^𝑥_ℎ)⁻¹. The proof of Theorem 2.1 is based on the formulas (2.12), on the convergence theorem, on the difference schemes in 𝐶_𝜏(𝐸^ℎ_𝛼) (see [38]), on the estimate ‖exp−𝑡_𝑘𝐴^𝑥_ℎ‖_{𝐶ℎ→𝐶ℎ}≤𝑀,𝑘≥0, (2.13) and on the fact that in 𝐸^ℎ_𝛼=𝐸_𝛼(𝐴^𝑥_ℎ,𝐶_ℎ) the norms are equivalent to the norms in 𝐶^∘_ℎ^2𝛼 uniformly in ℎ for 0<𝛼<1/2 (see, [18]).

Theorem 2.2. Assume that assumption (2.9) of Theorem 2.1 and the following conditions hold: ^∞₀max_{0≤𝑥≤𝐿}𝑢_𝑠𝑠𝑠(𝑠,𝑥)+sup_{0<𝑥<𝑥+𝑦<L}𝑢_𝑠𝑠𝑠𝑠,𝑥+𝑦−𝑢_𝑠𝑠𝑠(𝑠,𝑥)𝑦^2𝛼𝑑𝑠<∞,_∞0max_{0≤𝑥≤𝐿}𝑢_{𝑥𝑥𝑠𝑠}(𝑠,𝑥)+sup_{0<𝑥<𝑥+𝑦<𝐿}𝑢_{𝑥𝑥𝑠𝑠}𝑠,𝑥+𝑦−𝑢_{𝑥𝑥𝑠𝑠}(𝑠,𝑥)𝑦^2𝛼𝑑𝑠<∞,_∞0max_{0≤𝑥≤𝐿}𝑢_{𝑥𝑥𝑥𝑥𝑠}(𝑠,𝑥)+sup_{0<𝑥<𝑥+𝑦<𝐿}𝑢_{𝑥𝑥𝑥𝑥𝑠}s,𝑥+𝑦−𝑢_{𝑥𝑥𝑥𝑥𝑠}(𝑠,𝑥)𝑦^2𝛼𝑑𝑠<∞. (2.14) Then for the solution of difference scheme (2.8), the following convergence estimate is satisfied: sup_𝑘𝑢^ℎ_𝑘−𝑢^ℎ𝑡_𝑘,⋅_∘𝐶ℎ^2𝛼≤𝑀₂𝜏²+ℎ² (2.15) with 𝑀₂ being a real number independent of 𝜏, 𝛼, and ℎ.

Proof. Using notations of 𝐴^𝑥_ℎ and 𝐵^𝑥_ℎ again, we can obtain the following formula for the solution: 𝑢^ℎ_𝑘(𝑥)=𝑅^𝑘𝑔^ℎ(0,𝑥)+^𝑘_𝑗=1𝑅^{𝑘−𝑗+1}𝐼+𝜏𝐴^𝑥_ℎ2𝑔^ℎ𝑡_𝑗−𝑁,𝑥−𝑔^ℎ𝑡_{𝑗−𝑁−1}𝑢,𝑥𝜏,1≤𝑘≤𝑁,_ℎ𝑘(𝑥)=𝑅^{𝑘−𝑛𝑁}𝑢^ℎ_𝑛𝑁(𝑥)+^𝑘_{𝑗=𝑛𝑁+1}𝑅^{𝑘−𝑗+1}𝐼+𝜏𝐴^𝑥_ℎ2𝐵^𝑥_𝑗12𝑢^ℎ_𝑗−𝑁(𝑥)+𝑢^ℎ_{𝑗−𝑁−1}(𝑥)()𝜏,𝑛𝑁≤𝑘≤𝑛+1𝑁, (2.16) where 𝑅=(𝐼+𝜏𝐴^𝑥_ℎ+(𝜏𝐴^𝑥_ℎ)²/2)⁻¹. The proof of Theorem 2.2 is based on the formulas (2.16), on the convergence theorem, on the difference schemes in 𝐶_𝜏(𝐸^ℎ_𝛼) (see, [38]), on the estimate (2.13), and on the equivalence of the norms as in Theorem 2.1.
Finally, the numerical methods are given in the following section for the solution of delay parabolic differential equation with the Dirichlet condition. The method is illustrated by numerical examples.

2.2. Numerical Results

We consider the initial-boundary-value problem𝜕𝑢(𝑡,𝑥)−𝜕𝜕𝑡²𝑢(𝑡,𝑥)𝜕𝑥²+()𝜕0.1²𝑢(𝑡−1,𝑥)𝜕𝑥²𝑢()=0,𝑡>0,0<𝑥<𝜋,𝑡,𝑥=𝑒^−𝑡𝑢()()sin𝑥,−1≤𝑡≤0,0≤𝑥≤𝜋,𝑡,0=𝑢𝑡,𝜋=0,𝑡≥0, (2.17) for the delay parabolic differential equation.

The exact solution of this problem for 𝑡∈[𝑛−1,𝑛], 𝑛=0,1,2,…,𝑥∈[0,𝜋] is𝑢()=𝑒𝑡,𝑥^−𝑡𝑒sin𝑥,−1≤𝑡≤0,_−𝑡{()}𝑒1+0.1𝑒𝑡sin𝑥,0≤𝑡≤1,_−𝑡())𝑒(1+0.1𝑒𝑡+[(0.1𝑡−1)]²⋮𝑒2!sin𝑥,1≤𝑡≤2,_−𝑡())𝑒(1+0.1𝑒𝑡+[(0.1𝑡−1)]²)𝑒(2!+⋯+[(0.1𝑡−𝑛)]^(𝑛+1)()!⋮𝑛+1sin𝑥,𝑛≤𝑡≤𝑛+1, (2.18) For the approximate solution of delay parabolic equation (2.17), consider the set of grid points[−1,∞]_𝜏×[0,𝜋]_ℎ=𝑡_𝑘,𝑥_𝑛∶𝑡_𝑘=𝑘𝜏,−𝑁≤𝑘≤∞,𝑥_𝑛=𝑛ℎ,0≤𝑛≤𝑀,𝑀ℎ=𝜋. (2.19) Using difference scheme accurate to first order for the approximate solutions of the initial-boundary-value problem for the delay parabolic equation (2.17), we get the following system of equations:𝑢^𝑘_𝑛−𝑢_𝑛^𝑘−1𝜏−𝑢^𝑘_𝑛+1−2𝑢^𝑘_𝑛+𝑢^𝑘_𝑛−1ℎ²+()𝑢0.1^{𝑘−𝑁𝑛+1}−2𝑢_𝑛^𝑘−𝑁+𝑢^𝑘−𝑁_𝑛−1ℎ²()𝑢=0,𝑚𝑁+1≤𝑘≤𝑚+1𝑁,𝑚=0,1,…,1≤𝑛≤𝑀−1,_𝑘𝑛=𝑒^−𝑡𝑘sin𝑥_𝑛𝑢,−𝑁≤𝑘≤0,0≤𝑛≤𝑀,_𝑘0=𝑢^𝑘_𝑀=0,𝑘≥0. (2.20) In this first step, applying difference scheme accurate to first order, we obtain a system of equations in matrix form𝐴𝑈^𝑚_𝑛+1+𝐵𝑈^𝑚_𝑛+𝐶𝑈^𝑚_𝑛−1=𝑅𝜑^𝑚_𝑛𝑈,1≤𝑛≤𝑀−1,𝑚=0,1,…,_𝑚0=0,𝑈^𝑚_𝑀=0, (2.21) where 𝐴, 𝐵, 𝐶 are (𝑁+1)×(𝑁+1) matrices defined by𝐴=00000⋅⋅⋅00000𝑎000⋅⋅⋅000000𝑎00⋅⋅⋅0000000𝑎0⋅⋅⋅00000000𝑎⋅⋅⋅0000⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅00000⋅⋅⋅𝑎00000000⋅⋅⋅0𝑎0000000⋅⋅⋅00𝑎000000⋅⋅⋅000𝑎_{(𝑁+1)×(𝑁+1)},𝐵=100⋅⋅⋅0⋅⋅⋅0⋅⋅⋅00𝑏𝑐0⋅⋅⋅0⋅⋅⋅0⋅⋅⋅000𝑏𝑐⋅⋅⋅0⋅⋅⋅0⋅⋅⋅0000𝑏⋅⋅⋅0⋅⋅⋅0⋅⋅⋅00000⋅⋅⋅0⋅⋅⋅0⋅⋅⋅00000⋅⋅⋅0⋅⋅⋅0⋅⋅⋅00000⋅⋅⋅0⋅⋅⋅0⋅⋅⋅00⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅000⋅⋅⋅0⋅⋅⋅0⋅⋅⋅00000⋅⋅⋅0⋅⋅⋅0⋅⋅⋅𝑐0000⋅⋅⋅0⋅⋅⋅0⋅⋅⋅𝑏𝑐_{(𝑁+1)×(𝑁+1)}. (2.22)𝐶=𝐴, 𝑅 is (𝑁+1)×(𝑁+1) identity matrix and 𝜑^𝑚_𝑛, 𝑈^𝑚_𝑠 are (𝑁+1)×1 column vectors as𝜑^𝑚_𝑛=𝜑_𝑛^𝑚𝑁𝜑_𝑛^𝑚𝑁+1⋮𝜑_𝑛^(𝑚+1)𝑁_{(𝑁+1)×(1)},𝑈^𝑚_𝑠=𝑈_𝑠^𝑚𝑁𝑈_𝑠^𝑚𝑁+1⋮𝑈_𝑠^(𝑚+1)𝑁_{(𝑁+1)×(1)}for𝑠=𝑛±1,𝑛, (2.23) where 𝑢_𝑛^𝑚𝑁 is given for any 𝑚=0,1,…,𝜑^𝑘_𝑛()𝑢=−0.1^{𝑘−𝑁𝑛+1}−2𝑢_𝑛^𝑘−𝑁+𝑢^𝑘−𝑁_𝑛−1ℎ²,()𝑢𝑚𝑁+1≤𝑘≤𝑚+1𝑁,𝑚=0,1,…,1≤𝑛≤𝑀−1,_𝑘𝑛=𝑒^−𝑡𝑘sin𝑥_𝑛,−𝑁≤𝑘≤0. (2.24) Here, we denote1𝑎=−ℎ²1,𝑏=−𝜏1,𝑐=𝜏+2ℎ². (2.25) So, we have second-order difference equation (2.21) with matrix coefficients. To solve this difference equation, we have applied a procedure of modified Gauss elimination method. Hence, we obtain a solution of the matrix equation in the following form:𝑈^𝑚_𝑗=𝛼_𝑗+1𝑈^𝑚_𝑗+1+𝛽^𝑚_𝑗+1𝑈,𝑗=𝑀−1,…,2,1,_𝑚𝑀=0, (2.26) where 𝛼_𝑗(𝑗=1,…,𝑀) are (𝑁+1)×(𝑁+1) square matrices and 𝛽^𝑚_𝑗(𝑗=1,…,𝑀) are (𝑁+1)×1 column matrices defined by𝛼_𝑗+1=−𝐵+𝐶𝛼_𝑗⁻¹𝛽𝐴,_𝑚𝑗+1=𝐵+𝐶𝛼_𝑗⁻¹𝑅𝜑^𝑚_𝑗−𝐶𝛽_𝑗, (2.27) where 𝑗=1,…,𝑀−1, 𝛼₁ is the (𝑁+1)×(𝑁+1) zero matrix, and 𝛽^𝑚₁ is the (𝑁+1)×1 zero matrix.

Second, using the second order of accuracy difference scheme for the approximate solutions of problem (2.17) and applying formulae2𝑢(0)−5𝑢(ℎ)+4𝑢(2ℎ)−𝑢(3ℎ)ℎ²−𝑢^′′(0)=𝑂ℎ²,2𝑢(1)−5𝑢(1−ℎ)+4𝑢(1−2ℎ)−𝑢(1−3ℎ)ℎ²−𝑢^′′(1)=𝑂ℎ², (2.28) we obtain the following system of equations:𝑢^𝑘_𝑛−𝑢_𝑛^𝑘−1𝜏−𝑢^𝑘_𝑛+1−2𝑢^𝑘_𝑛+𝑢^𝑘_𝑛−1ℎ²+𝜏2𝑢^𝑘_𝑛+2−4𝑢^𝑘_𝑛+1+6𝑢^𝑘_𝑛−4𝑢^𝑘_𝑛−1+𝑢^𝑘_𝑛−2ℎ⁴+(𝑢0.1)^{𝑘−𝑁𝑛+1}−2𝑢_𝑛^𝑘−𝑁+𝑢^𝑘−𝑁_𝑛−12ℎ²+𝑢^{𝑘−1−𝑁}_𝑛+1−2𝑢_𝑛^{𝑘−1−𝑁}+𝑢^{𝑘−1−𝑁}_𝑛−12ℎ²−𝜏2𝑢^𝑘−𝑁_𝑛+2−4u^𝑘−𝑁_𝑛+1+6𝑢_𝑛^𝑘−𝑁−4𝑢^𝑘−𝑁_𝑛−1+𝑢^𝑘−𝑁_𝑛−22ℎ⁴+𝑢^{𝑘−1−𝑁}_𝑛+2−4𝑢^{𝑘−1−𝑁}_𝑛+1+6𝑢_𝑛^{𝑘−1−𝑁}−4𝑢^{𝑘−1−𝑁}_𝑛−1+𝑢^{𝑘−1−𝑁}_𝑛−22ℎ⁴()𝑢=0𝑚𝑁+1≤𝑘≤𝑚+1𝑁,𝑚=0,1,…,2≤𝑛≤𝑀−2,_𝑘𝑛=𝑒^−𝑡𝑘sin𝑥_𝑛𝑢,−𝑁≤𝑘≤0,0≤𝑛≤𝑀,_𝑘1=45𝑢^𝑘₂−15𝑢^𝑘₃𝑢,𝑘≥0,_𝑘𝑀−1=45𝑢^𝑘_𝑀−2−15𝑢^𝑘_𝑀−3𝑢,𝑘≥0,_𝑘0=𝑢^𝑘_𝑀=0,𝑘≥0. (2.29) In the second step, we apply second-order difference scheme to get the system of linear equations in matrix form𝐴𝑈^𝑚_𝑛+2+𝐵𝑈^𝑚_𝑛+1+𝐶𝑈^𝑚_𝑛+𝐷𝑈^𝑚_𝑛−1+𝐸𝑈^𝑚_𝑛−2=𝑅𝜑^𝑚_𝑛,𝑈𝑚=0,1,…,2≤𝑛≤𝑀−2,_𝑚0=0,𝑈^𝑚_𝑀=𝑈0,_𝑚1=45𝑈^𝑚₂−15𝑈^𝑚₃,𝑈^𝑚_𝑀−1=45𝑈^𝑚_𝑀−2−15𝑈^𝑚_𝑀−3, (2.30) where 𝐴, 𝐵, 𝐶 are (𝑁+1)×(𝑁+1) matrices defined by𝐴=00000⋅⋅⋅00000𝑥000⋅⋅⋅000000𝑥00⋅⋅⋅0000000𝑥0⋅⋅⋅00000000𝑥⋅⋅⋅0000⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅00000⋅⋅⋅𝑥00000000⋅⋅⋅0𝑥0000000⋅⋅⋅00𝑥000000⋅⋅⋅000𝑥_{(𝑁+1)×(𝑁+1)},𝐵=00000⋅⋅⋅00000𝑦000⋅⋅⋅000000𝑦00⋅⋅⋅0000000𝑦0⋅⋅⋅00000000𝑦⋅⋅⋅0000⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅00000⋅⋅⋅𝑦00000000⋅⋅⋅0𝑦0000000⋅⋅⋅00𝑦000000⋅⋅⋅000𝑦_{(𝑁+1)×(𝑁+1)},𝐶=10000⋅⋅⋅0000𝑧𝑡000⋅⋅⋅00000𝑧𝑡00⋅⋅⋅000000𝑧𝑡0⋅⋅⋅0000000𝑧𝑡⋅⋅⋅0000⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅00000⋅⋅⋅𝑧𝑡0000000⋅⋅⋅0𝑧𝑡000000⋅⋅⋅00𝑧𝑡_{(𝑁+1)×(𝑁+1)}, (2.31)𝐵=𝐷, 𝐸=𝐴, 𝑅 is (𝑁+1)×(𝑁+1) identity matrix, and 𝜑^𝑚_𝑛, 𝑈^𝑚_𝑠 are (𝑁+1)×1 column vectors as𝜑^𝑚_𝑛=𝜑_𝑛^𝑚𝑁𝜑_𝑛^𝑚𝑁+1⋮𝜑_𝑛^(𝑚+1)𝑁_{(𝑁+1)×(1)},𝑈^𝑚_𝑠=𝑈_𝑠^𝑚𝑁𝑈_𝑠^𝑚𝑁+1⋮𝑈_𝑠^(𝑚+1)𝑁_{(𝑁+1)×(1)}for𝑠=𝑛±1,𝑛±2,𝑛, (2.32) where 𝑢_𝑛^𝑚𝑁 is given for any 𝑚=0,1,…,𝜑^𝑘_𝑛(𝑢=−0.1)^{𝑘−𝑁𝑛+1}−2𝑢_𝑛^𝑘−𝑁+𝑢^𝑘−𝑁_𝑛−12ℎ²+𝑢^{𝑘−1−𝑁}_𝑛+1−2𝑢_𝑛^{𝑘−1−𝑁}+𝑢^{𝑘−1−𝑁}_𝑛−12ℎ²−𝜏2𝑢^𝑘−𝑁_𝑛+2−4𝑢^𝑘−𝑁_𝑛+1+6𝑢_𝑛^𝑘−𝑁−4𝑢^𝑘−𝑁_𝑛−1+𝑢^𝑘−𝑁_𝑛−22ℎ⁴+𝑢^{𝑘−1−𝑁}_𝑛+2−4𝑢^{𝑘−1−𝑁}_𝑛+1+6𝑢_𝑛^{𝑘−1−𝑁}−4𝑢^{𝑘−1−𝑁}_𝑛−1+𝑢^{𝑘−1−𝑁}_𝑛−22ℎ⁴,()𝑢𝑚𝑁+1≤𝑘≤𝑚+1𝑁,𝑚=0,1,…,1≤𝑛≤𝑀−1,_𝑘𝑛=𝑒^−𝑡𝑘sin𝑥_𝑛,−𝑁≤𝑘≤0. (2.33)

Here, we denote𝜏𝑥=2ℎ⁴1,𝑦=−ℎ²−2𝜏ℎ⁴,1𝑧=−𝜏1,𝑡=𝜏+2ℎ²+3𝜏ℎ⁴, (2.34) Hence, we have second-order difference equation (2.30) with matrix coefficients. For the solution of this matrix equation, we use the modified Gauss elimination method. We seek a solution of the matrix equation by the following form:𝑈^𝑚_𝑗=𝛼_𝑗+1𝑈^𝑚_𝑗+1+𝛽_𝑗+1𝑈^𝑚_𝑗+2+𝛾^𝑚_𝑗+1𝑈,𝑗=𝑀−2,…,2,1,0_𝑚𝑀𝑈=0,_𝑚𝑀−1=𝛽_𝑀−2+5𝐼−4𝐼−𝛼_𝑀−2𝛼_𝑀−1⁻¹4𝐼−𝛼_𝑀−2𝛾^𝑚_𝑀−1−𝛾^𝑚_𝑀−2, (2.35) where 𝛼_𝑗(𝑗=2,…,𝑀−2) and 𝛽_𝑗(𝑗=2,…,𝑀−2) are (𝑁+1)×(𝑁+1) square matrices and 𝛾^𝑚_𝑗(𝑗=2,…,𝑀−2) are column matrices defined by𝛼_𝑗+1=−𝐶+𝐷𝛼_𝑗+𝐸𝛽_𝑗−1+𝐸𝛼_𝑗−1𝛼_𝑗⁻¹𝐵+𝐷𝛽_𝑗+𝐸𝛼_𝑗−1𝛽_𝑗,𝛽_𝑗+1=−𝐶+𝐷𝛼_𝑗+𝐸𝛽_𝑗−1+𝐸𝛼_𝑗−1𝛼_𝑗⁻¹(𝐴),𝛾^𝑚_𝑗+1=−𝐶+𝐷𝛼_𝑗+𝐸𝛽_𝑗−1+𝐸𝛼_𝑗−1𝛼_𝑗⁻¹𝑅𝜑^𝑚_𝑗−𝐷𝛾^𝑚_𝑗−𝐸𝛼_𝑗−1𝛾^𝑚_𝑗−𝐸𝛾^𝑚_𝑗−1, (2.36) where 𝑗=2,…,𝑀−2, 𝛼₁ is (𝑁+1)×(𝑁+1) zero matrix, and 𝛽₁ is (𝑁+1)×(𝑁+1) zero matrix, 𝛾^𝑚₁ and 𝛾^𝑚₂ are (𝑁+1)×1 zero matrices.

We give the results of the numerical analysis. The numerical solutions are recorded for different values of 𝑁 and 𝑀 and 𝑢^𝑘_𝑛 represent the numerical solutions of these difference schemes at (𝑡_𝑘,𝑥_𝑛). Tables 1, 2, 3, and 4 are constructed for 𝑁=𝑀=20,40,80 in 𝑡∈[0,1], 𝑡∈[1,2], 𝑡∈[2,3], 𝑡∈[3,4], respectively, and the error is computed by the following formula:𝐸^𝑁_𝑀=max_{−𝑁≤𝑘≤𝑁1≤𝑛≤𝑀−1}𝑢𝑡_𝑘,𝑥_𝑛−𝑢^𝑘_𝑛. (2.37) Thus, by using the second order of accuracy difference scheme, the accuracy of solution increases faster than the first order of accuracy difference scheme.

Table 1 

Comparison of the errors of different difference schemes in 𝑡∈[0,1].

Table 2 

Comparison of the errors of different difference schemes in 𝑡∈[1,2].

Table 3 

Comparison of the errors of different difference schemes in 𝑡∈[2,3].

Table 4 

Comparison of the errors of different difference schemes in 𝑡∈[3,4].

3. Homotopy Analysis Method

In this section, we consider homotopy analysis method for the solution of problem (1.1). We study the initial-boundary-value problem for the delay parabolic equation (1.1). To illustrate the basic idea of homotopy analysis method (HAM) developed by Liao (see, e.g., [29–35]), the following differential equation is considered:𝑁[𝑢(𝑡,𝑥)]=𝑓(𝑡,𝑥), (3.1) where 𝑁 is a linear operator for problem (1.1), 𝑡 and 𝑥 denote independent variables, 𝑢(𝑡,𝑥) is an unknown function, and 𝑓(𝑡,𝑥) is a known analytical function. Liao constructs the so-called zero-order deformation equation1−𝑞𝐿𝜙𝑡,𝑥;𝑞−𝑢₀(𝑡,𝑥)=𝑞ℏ𝑁𝜙𝑡,𝑥;𝑞−𝑓(𝑡,𝑥), (3.2) where 𝑞∈[0,1] is an embedding parameter, ℏ is a nonzero auxiliary parameter, 𝐿 is an auxiliary linear operator, 𝑢₀(𝑡,𝑥) is an initial guess of 𝑢(𝑡,𝑥), and 𝜙(𝑡,𝑥;𝑞) is an unknown function. When 𝑞=0 and 𝑞=1, it holds𝜙(𝑡,𝑥;0)=𝑢₀(𝑡,x),𝜙(𝑡,𝑥;1)=𝑢(𝑡,𝑥), (3.3) respectively. As 𝑞 increases from 0 to 1, the solution 𝜙(𝑡,𝑥;𝑞) varies from the initial guess 𝑢₀(𝑡,𝑥) to the solution 𝑢(𝑡,𝑥). Expanding 𝜙(𝑡,𝑥;𝑞) in Taylor series with respect to 𝑞, we get𝜙𝑡,𝑥;𝑞=𝑢₀(𝑡,𝑥)+^+∞_𝑚=1𝑢_𝑚(𝑡,𝑥)𝑞^𝑚, (3.4) where𝑢_𝑚()=1𝑡,𝑥𝜕𝑚!^𝑚𝜙𝑡,𝑥;𝑞𝜕𝑞^𝑚_𝑞=0, (3.5) when the initial guess 𝑢₀(𝑡,𝑥), the auxiliary linear operator 𝐿 and the auxiliary parameter ℏ are chosen properly, the series (3.4) converges at 𝑞=1. We get𝑢(𝑡,𝑥)=𝑢₀(𝑡,𝑥)+^+∞_𝑚=1𝑢_𝑚(𝑡,𝑥). (3.6) Then define the vectors𝑢_𝑛=𝑢₀(𝑡,𝑥),𝑢₁(𝑡,𝑥),…,𝑢_𝑛(𝑡,𝑥). (3.7) Differentiating the zero-order deformation equation (3.2) 𝑚 times with respect to the embedding parameter 𝑞 and dividing them by 𝑚!, we obtain the 𝑚th-order deformation equation𝐿𝑢_𝑚(𝑡,𝑥)−𝜒_𝑚𝑢_𝑚−1(𝑡,𝑥)=ℏℜ_𝑚𝑢⃗_𝑚−1, (3.8) whereℜ_𝑚𝑢⃗_𝑚−1=1()!𝜕𝑚−1^𝑚−1𝑁𝜙𝑡,𝑥;𝑞−𝑓(𝑡,𝑥)𝜕𝑞^𝑚−1_𝑞=0,𝜒_𝑚=0,𝑚≤1,1,𝑚>1. (3.9) with the initial condition𝑢_𝑚(0,𝑥)=0,𝑚≥1. (3.10) High-order deformation equation (3.8) is governed by the linear operator 𝐿. ℜ_𝑚(𝑢⃗_𝑚−1) can be represented by 𝑢₁(𝑡,𝑥),𝑢_𝑚(𝑡,𝑥),𝑢₂(𝑡,𝑥),…,𝑢_𝑚−1(𝑡,𝑥) and high-order deformation equation can be solved consecutively. The 𝑁th-order approximation of 𝑢(𝑡,𝑥) is given by𝑢(𝑡,𝑥)≈𝑢₀(𝑡,𝑥)+^𝑁_𝑚=1𝑢_𝑚(𝑡,𝑥). (3.11)

3.1. Homotopy Analysis Solution

For the approximate solution of the delay parabolic differential equation with the Dirichlet condition, we consider the delay parabolic equation (2.17) and rewrite the equation for 𝑡∈[0,1] in the following form:𝜕𝑢(𝑡,𝑥)−𝜕𝜕𝑡²𝑢(𝑡,𝑥)𝜕𝑥²=(0.1)𝑒^{−(𝑡−1)}𝑢()𝑢()()sin𝑥,0<𝑡≤1,0<𝑥<𝜋,0,𝑥=sin𝑥,0≤𝑥≤𝜋𝑡,0=𝑢𝑡,𝜋=0,0≤𝑡≤1. (3.12) To solve the initial-boundary-value problem (3.12) by means of HAM, we choose the initial approximation𝑢₀(𝑡,𝑥)=sin𝑥, (3.13) and the linear operator𝐿𝜙𝑡,𝑥;𝑞=𝜕𝜙𝑡,𝑥;𝑞,𝜕𝑡 (3.14) with the property𝐿[𝑐]=0, (3.15) where 𝑐 is constant of integration. From (3.12), we define a linear operator as𝑁𝜙𝑡,𝑥;𝑞=𝜕𝜙𝑡,𝑥;𝑞−𝜕𝜕𝑡²𝜙𝑡,𝑥;𝑞𝜕𝑥². (3.16) Firstly, we construct the zero-order deformation equation1−𝑞𝐿𝜙𝑡,𝑥;𝑞−𝑢₀(𝑡,𝑥)=𝑞ℏ𝑁𝜙𝑡,𝑥;𝑞−𝑓(𝑡,𝑥), (3.17) when 𝑞=0 and 𝑞=1,𝜙(𝑡,𝑥;0)=𝑢₀(𝑡,𝑥)=𝑢(0,𝑥),𝜙(𝑡,𝑥;1)=𝑢(𝑡,𝑥). (3.18) Then, we get 𝑚th-order deformation equations (3.8) for 𝑚≥1 with the initial conditions𝑢_𝑚(0,𝑥)=0, (3.19) whereℜ_𝑚𝑢⃗_𝑚−1=𝜕𝑢_𝑚−1(𝑡,𝑥)−𝜕𝜕𝑡²𝑢_𝑚−1(𝑡,𝑥)𝜕𝑥²−1−𝜒_𝑚(0.1)𝑒^{−(𝑡−1)}sin𝑥. (3.20) The solution of the 𝑚th-order deformation equations (3.20) for 𝑚≥1 is𝑢_𝑚(𝑡,𝑥)=𝜒_𝑚𝑢_𝑚−1(𝑡,𝑥)+ℏ𝐿⁻¹ℜ_𝑚𝑢⃗_𝑚−1. (3.21) From (3.12) and (3.21), we obtain𝑢₀()𝑢𝑡,𝑥=sin𝑥,₁(𝑡,𝑥)=−ℏ(0.1)𝑒^∞_𝑘=1(−1)^𝑘+1𝑡^𝑘𝑢𝑘!sin𝑥+ℏ𝑡sin𝑥,₂(𝑡,𝑥)=ℏ(ℏ+1)𝑡sin𝑥−ℏ(ℏ+1)^∞_𝑘=2(−1)^𝑘𝑡^𝑘−1(𝑘−1)!sin𝑥+ℏ²(0.1)𝑒^∞_𝑘=2(−1)^𝑘+1𝑡^𝑘𝑘!sin𝑥+ℏ²𝑡²𝑢2!sin𝑥,₃(𝑡,𝑥)=ℏ(ℏ+1)𝑡sin𝑥+3ℏ²()𝑡ℏ+1²()2!sin𝑥−ℏℏ+1^2∞_𝑘=3(−1)^𝑘+1𝑡^𝑘−2(𝑘−2)!sin𝑥−2ℏ²(ℏ+1)^∞_𝑘=3(−1)^𝑘+1𝑡^𝑘−1(𝑘−1)!sin𝑥−ℏ³(0.1)𝑒^∞_𝑘=3(−1)^𝑘+1𝑡^𝑘𝑘!sin𝑥+ℏ³𝑡³⋮𝑢3!sin𝑥,_𝑛(𝑡,𝑥)=𝑓₁(ℏ+1,𝑡,𝑥)+(−1)^𝑛ℏ^𝑛(0.1)𝑒^∞_𝑘=𝑛(−1)^𝑘+1𝑡^𝑘𝑘!sin𝑥+ℏ^𝑛𝑡^𝑛⋮𝑛!sin𝑥, (3.22) and so on. Then for ℏ=−1, we get𝑢₀()𝑢𝑡,𝑥=sin𝑥,₁(𝑡,𝑥)=(0.1)𝑒^∞_𝑘=1(−1)^𝑘+1𝑡^𝑘𝑢𝑘!sin𝑥−𝑡sin𝑥,₂(𝑡,𝑥)=(0.1)𝑒^∞_𝑘=2(−1)^𝑘+1𝑡^𝑘𝑡𝑘!sin𝑥+²𝑢2!sin𝑥,₃(𝑡,𝑥)=(0.1)𝑒^∞_𝑘=3(−1)^𝑘+1𝑡^𝑘𝑡𝑘!sin𝑥−³⋮𝑢3!sin𝑥,_𝑛(𝑡,𝑥)=(0.1)𝑒^∞_𝑘=𝑛(−1)^𝑘+1𝑡^𝑘+()𝑘!−1^𝑛𝑡^𝑛⋮𝑛!sin𝑥, (3.23) and so on.

From (3.6), when we take ℏ=−1, the solution of (3.12) can be obtained as𝑢(𝑡,𝑥)=((0.1)𝑒𝑡+1)^∞_𝑘=0(−1)^𝑘𝑡^𝑘𝑘!sin𝑥. (3.24) Equation (3.24) has the closed form𝑢(𝑡,𝑥)=((0.1)𝑒𝑡+1)𝑒^−𝑡sin𝑥, (3.25) which is the exact solution of (3.12).

Second, we consider the solution of the delay parabolic equation (2.17) for 𝑡∈[1,2] and rewrite this equation in the following form:𝜕𝑢(𝑡,𝑥)−𝜕𝜕𝑡²𝑢(𝑡,𝑥)𝜕𝑥²=(0.1)𝑒^{−(𝑡−1)})𝑒()]𝑢()[(0.1𝑡−1+1sin𝑥,1<𝑡≤2,0<𝑥<𝜋,1,𝑥=𝑒⁻¹)]𝑢()()[(0.1𝑒+1sin𝑥,0≤𝑥≤𝜋𝑡,0=𝑢𝑡,𝜋=0,1≤𝑡≤2. (3.26) Now, we choose the initial approximation𝑢₀(𝑡,𝑥)=𝑒⁻¹[(0.1)𝑒+1]sin𝑥. (3.27) We take the linear operator (3.14) with the property (3.15), and we define the operator (3.16) from (3.26).

Firstly, we construct the zero-order deformation equation (3.2) and then, we obtain 𝑚th-order deformation equations (3.8) for 𝑚≥1 with the initial conditions𝑢_𝑚(1,𝑥)=0, (3.28) whereℜ_𝑚𝑢⃗_𝑚−1=𝜕𝑢_𝑚−1(𝑡,𝑥)−𝜕𝜕𝑡²𝑢_𝑚−1(𝑡,𝑥)𝜕𝑥²−1−𝜒_𝑚(0.1)𝑒^{−(𝑡−1)}[(0.1)𝑒(𝑡−1)+1]sin𝑥. (3.29) The solution of the 𝑚th-order deformation equations (3.29) for 𝑚≥1 is𝑢_𝑚(𝑡,𝑥)=𝜒_𝑚𝑢_𝑚−1(𝑡,𝑥)+ℏ𝐿⁻¹ℜ_𝑚𝑢⃗_𝑚−1. (3.30) From (3.26) and the 𝑚th-order deformation equations (3.30), we get 𝑢₀(𝑡,𝑥)=𝑒⁻¹)]𝑢[(0.1𝑒+1sin𝑥,₁(𝑡,𝑥)=ℏ𝑒⁻¹)]()()[(0.1𝑒+1sin𝑥𝑡−1−ℏ0.1^∞_𝑘=1(−1)^𝑘+1(𝑡−1)^𝑘()𝑘!−ℏ0.1²𝑒^∞_𝑘=1(−1)^𝑘+1(𝑡−1)^𝑘+1()!𝑢𝑘+1)(𝑘−1sin𝑥,₂(𝑡,𝑥)=ℏ(ℏ+1)𝑒⁻¹))()[(0.1𝑒+1](𝑡−1sin𝑥−ℏℏ+1)(0.1^∞_𝑘=2(−1)^𝑘(𝑡−1)^𝑘−1()!()𝑘−1sin𝑥−ℏℏ+1)(0.1²𝑒^∞_𝑘=2(−1)^𝑘(𝑡−1)^𝑘𝑘(𝑘−2)!sin𝑥+ℏ²𝑒⁻¹)]()[(0.1𝑒+1sin𝑥𝑡−1²2!+ℏ²(0.1)^∞_𝑘=2(−1)^𝑘+1(𝑡−1)^𝑘𝑘!sin𝑥+ℏ²(0.1)²𝑒^∞_𝑘=2(−1)^𝑘+1(𝑡−1)^𝑘+1()𝑘()!𝑢𝑘+1𝑘−2sin𝑥,₃(𝑡,𝑥)=ℏ(ℏ+1)²𝑒⁻¹[(0.1)𝑒+1]sin𝑥(𝑡−1)+2ℏ²(ℏ+1)𝑒⁻¹)]()[(0.1𝑒+1sin𝑥𝑡−1²()2!+ℏℏ+1²(0.1)^∞_𝑘=3(−1)^𝑘+1(𝑡−1)^𝑘−2()!()𝑘−2sin𝑥−ℏℏ+1²(0.1)²𝑒^∞_𝑘=3(−1)^𝑘+1(𝑡−1)^𝑘(𝑘−1)(𝑘−3)!sin𝑥−2ℏ²(ℏ+1)(0.1)^∞_𝑘=3(−1)^𝑘(𝑡−1)^𝑘−1()!()𝑘−1sin𝑥−ℏℏ+1)(0.1²𝑒^∞_𝑘=3(−1)^𝑘(𝑡−1)^𝑘𝑘(𝑘−1)(𝑘−3)!sin𝑥+ℏ²(ℏ+1)(0.1)²𝑒^∞_𝑘=3(−1)^𝑘(𝑡−1)^𝑘𝑘(𝑘−1)(𝑘−2)!sin𝑥+ℏ³𝑒⁻¹)]()[(0.1𝑒+1sin𝑥𝑡−1³3!−ℏ³(0.1)^∞_𝑘=3(−1)^𝑘+1(𝑡−1)^𝑘𝑘!sin𝑥−ℏ³(0.1)²𝑒^∞_𝑘=3(−1)^𝑘+1(𝑡−1)^𝑘+1()𝑘()!⋮𝑢𝑘+1𝑘−1)(𝑘−3sin𝑥,_𝑛(𝑡,𝑥)=𝑓₂(ℏ+1,𝑡,𝑥)+ℏ^𝑛𝑒⁻¹)]()[(0.1𝑒+1sin𝑥𝑡−1^𝑛𝑛!−ℏ^𝑛(0.1)^∞_𝑘=𝑛(−1)^𝑘+1(𝑡−1)^𝑘+()𝑘!sin𝑥−1^𝑛ℏ^𝑛(0.1)²𝑒^∞_𝑘=𝑛(−1)^𝑘+1(𝑡−1)^𝑘+1()𝑘()⋯(()!⋮𝑘+1𝑘−1𝑘−𝑛−2))(𝑘−𝑛sin𝑥, (3.31) and so on. When we choose ℏ=−1, we obtain 𝑢₀(𝑡,𝑥)=𝑒⁻¹)]𝑢[(0.1𝑒+1sin𝑥,₁(𝑡,𝑥)=−𝑒⁻¹)]()+()[(0.1e+1sin𝑥𝑡−10.1^∞_𝑘=1(−1)^𝑘+1(𝑡−1)^𝑘()𝑘!sin𝑥+0.1²𝑒^∞_𝑘=1(−1)^𝑘+1(𝑡−1)^𝑘+1()!𝑢𝑘+1)(𝑘−1sin𝑥,₂(𝑡,𝑥)=𝑒⁻¹)]()[(0.1𝑒+1sin𝑥𝑡−1²+()2!0.1^∞_𝑘=2(−1)^𝑘+1(𝑡−1)^𝑘()𝑘!sin𝑥+0.1²𝑒^∞_𝑘=2(−1)^𝑘+1(𝑡−1)^𝑘+1()𝑘()!𝑢𝑘+1𝑘−2sin𝑥,₃(𝑡,𝑥)=−𝑒⁻¹)]()[(0.1𝑒+1sin𝑥𝑡−1³+()3!0.1^∞_𝑘=3(−1)^𝑘+1(𝑡−1)^𝑘()𝑘!sin𝑥+0.1²𝑒^∞_𝑘=3(−1)^𝑘+1(𝑡−1)^𝑘+1()𝑘()!⋮𝑢𝑘+1𝑘−1)(𝑘−3sin𝑥,_𝑛(𝑡,𝑥)=(−1)^𝑛𝑒⁻¹)]()[(0.1𝑒+1sin𝑥𝑡−1^𝑛+()𝑛!0.1^∞_𝑘=𝑛(−1)^𝑘+1(𝑡−1)^𝑘+()𝑘!sin𝑥0.1²𝑒^∞_𝑘=𝑛(−1)^𝑘+1(𝑡−1)^𝑘+1()𝑘()⋯(()!⋮𝑘+1𝑘−1𝑘−𝑛−2))(𝑘−𝑛sin𝑥, (3.32) and so on.

From (3.6), the solution of (3.26) for ℏ=−1 can be obtained as𝑢()=)𝑒(𝑡,𝑥[(0.1𝑡−1)]²+()2!0.1𝑒𝑡+1^∞_𝑘=0(−1)^𝑘𝑡^𝑘𝑘!sin𝑥. (3.33) Equation (3.33) has the closed form𝑢()=)𝑒(𝑡,𝑥[(0.1𝑡−1)]²+()𝑒2!0.1𝑒𝑡+1^−𝑡sin𝑥, (3.34) which is the exact solution of the (3.26).

Now, we consider the solution of the delay parabolic equation (2.17) for 𝑡∈[2,3] and rewrite the equation in the following form:𝜕𝑢(𝑡,𝑥)−𝜕𝜕𝑡²𝑢(𝑡,𝑥)𝜕𝑥²=(0.1)𝑒^{−(𝑡−1)}[(0.1)𝑒(𝑡−2)]²+()𝑒()𝑢()2!0.1𝑡−1+1sin𝑥,2<𝑡≤3,0<𝑥<𝜋,2,𝑥=𝑒⁻²[(0.1)𝑒]²()𝑢()()2!+20.1𝑒+1sin𝑥,0≤𝑥≤𝜋𝑡,0=𝑢𝑡,𝜋=0,2≤𝑡≤3. (3.35) The initial approximation is𝑢₀(𝑡,𝑥)=𝑒⁻²[(0.1)𝑒]²()2!+20.1𝑒+1sin𝑥. (3.36)

We take the linear operator (3.14) with the property (3.15). From parabolic equation (3.35), we define a linear operator (3.16) and obtain the zero-order deformation equation (3.2). Thus, we get the 𝑚th-order deformation equations (3.8) for 𝑚≥1 with the initial conditions𝑢_𝑚(2,𝑥)=0, (3.37) whereℜ_𝑚𝑢⃗_𝑚−1=𝜕𝑢_𝑚−1(𝑡,𝑥)−𝜕𝜕𝑡²𝑢_𝑚−1(𝑡,𝑥)𝜕𝑥²−1−𝜒_𝑚×(0.1)𝑒^{−(𝑡−1)}[(0.1)𝑒(𝑡−2)]²+()𝑒()2!0.1𝑡−1+1sin𝑥. (3.38) The solution of the 𝑚th-order deformation equations (3.38) for 𝑚≥1 is𝑢_𝑚(𝑡,𝑥)=𝜒_𝑚𝑢_𝑚−1(𝑡,𝑥)+ℏ𝐿⁻¹ℜ_𝑚𝑢⃗_𝑚−1. (3.39) From (3.35) and (3.39), we obtain 𝑢₀(𝑡,𝑥)=𝑒⁻²[(0.1)𝑒]²()𝑢2!+20.1𝑒+1sin𝑥,₁(𝑡,𝑥)=ℏ𝑒⁻²[(0.1)𝑒]²()()2!+20.1𝑒+1sin𝑥𝑡−2−ℏ𝑒⁻¹(0.1)^∞_𝑘=1(−1)^𝑘+1(𝑡−2)^𝑘()𝑘!sin𝑥−ℏ0.1^2∞_𝑘=1(−1)^𝑘+1(𝑡−2)^𝑘+1()!()𝑘+1)(𝑘−1sin𝑥−ℏ0.1^2∞_𝑘=1(−1)^𝑘+1(𝑡−2)^𝑘()𝑘!sin𝑥−ℏ𝑒0.1³2!^∞_𝑘=1(−1)^𝑘+1(𝑡−2)^𝑘+2()!𝑢𝑘+2)(𝑘−1sin𝑥,₂(𝑡,𝑥)=ℏ(ℏ+1)𝑒⁻²[(0.1)𝑒]²()()()𝑒2!+20.1𝑒+1sin𝑥𝑡−2−ℏℏ+1⁻¹(0.1)^∞_𝑘=2(−1)^𝑘(𝑡−2)^𝑘−1()!()𝑘−1sin𝑥−ℏℏ+1)(0.1^2∞_𝑘=2(−1)^𝑘(𝑡−2)^𝑘−1()!()𝑘−1sin𝑥−ℏℏ+1)(0.1^2∞_𝑘=2(−1)^𝑘(𝑡−2)^𝑘𝑘()!()𝑘−2sin𝑥−ℏℏ+1^∞_𝑘=2(−1)^𝑘(𝑡−2)^𝑘+1(𝑘+1)(𝑘−2)!sin𝑥+ℏ²𝑒⁻²[(0.1)𝑒]²()()2!+20.1𝑒+1sin𝑥𝑡−2²2!+ℏ²𝑒⁻¹(0.1)^∞_𝑘=2(−1)^𝑘+1(𝑡−2)^𝑘𝑘!sin𝑥+ℏ²(0.1)^2∞_𝑘=2(−1)^𝑘+1(𝑡−2)^𝑘+1(𝑘+1)𝑘(𝑘−2)!sin𝑥+ℏ²(0.1)^2∞_𝑘=2(−1)^𝑘+1(𝑡−2)^𝑘𝑘!sin𝑥+ℏ²𝑒(0.1)³2!^∞_𝑘=2(−1)^𝑘+1(𝑡−2)^𝑘+2()!𝑢𝑘+2)(𝑘+1)(𝑘−2sin𝑥,₃(𝑡,𝑥)=ℏ(ℏ+1)²𝑒⁻²[(0.1)𝑒]²()()2!+20.1𝑒+1sin𝑥𝑡−2+2ℏ²(ℏ+1)𝑒⁻²[(0.1)𝑒]²()()2!+20.1𝑒+1sin𝑥𝑡−2²()2!−ℏℏ+1²(0.1)𝑒^∞−1_𝑘=3(−1)^𝑘+1(𝑡−2)^𝑘−2()!()𝑘−2sin𝑥−ℏℏ+1²(0.1)^2∞_𝑘=3(−1)^𝑘+1(𝑡−2)^𝑘−2()!()𝑘−2sin𝑥−3ℏℏ+1)(0.1^2∞_𝑘=3(−1)^𝑘+1(𝑡−2)^𝑘−1(𝑘−1)!sin𝑥−ℏ²(ℏ+1)(0.1)^2∞_𝑘=3(−1)^𝑘+1(𝑡−2)^𝑘−1(𝑘−1)!sin𝑥−2ℏ²(ℏ+1)𝑒⁻¹(0.1)^∞_𝑘=3(−1)^𝑘+1(𝑡−2)^𝑘−1()!()𝑘−1sin𝑥−ℏℏ+1²(0.1)³𝑒2!^∞_𝑘=3(−1)^𝑘+1(𝑡−2)^𝑘𝑘(𝑘−3)!sin𝑥−2ℏ²(ℏ+1)(0.1)^2∞_𝑘=3(−1)^𝑘+1(𝑡−2)^𝑘𝑘(𝑘−1)(𝑘−3)!sin𝑥−2ℏ²()()ℏ+10.1³𝑒2!^∞_𝑘=3(−1)^𝑘+1(𝑡−2)^𝑘+1(𝑘+1)𝑘(𝑘−3)!sin𝑥+ℏ³𝑒⁻²[(0.1)𝑒]²()()2!+20.1𝑒+1sin𝑥𝑡−2³3!−ℏ³𝑒⁻¹(0.1)^∞_𝑘=3(−1)^𝑘+1(𝑡−2)^𝑘𝑘!sin𝑥−ℏ³(0.1)^2∞_𝑘=3(−1)^𝑘+1(𝑡−2)^𝑘+1(𝑘+1)𝑘(𝑘−1)(𝑘−3)!sin𝑥−ℏ³(0.1)^2∞_𝑘=3(−1)^𝑘+1(𝑡−2)^𝑘𝑘!sin𝑥−ℏ³𝑒(0.1)³2!^∞_𝑘=3(−1)^𝑘+1(𝑡−2)^𝑘+2()𝑘()!⋮𝑢𝑘+2)(𝑘+1𝑘−3sin𝑥,_𝑛(𝑡,𝑥)=𝑓₃(ℏ+1,𝑡,𝑥)+ℏ^𝑛𝑒⁻²[(0.1)𝑒]²()()2!+20.1𝑒+1sin𝑥𝑡−2^𝑛+()𝑛!−1^𝑛ℏ^𝑛𝑒⁻¹(0.1)^∞_𝑘=𝑛(−1)^𝑘+1(𝑡−2)^𝑘+()𝑘!sin𝑥−1^𝑛ℏ^𝑛(0.1)^2∞_𝑘=𝑛(−1)^𝑘+1(𝑡−2)^𝑘+1()𝑘()⋯(()!+()𝑘+1𝑘−1𝑘−𝑛−2))(𝑘−𝑛sin𝑥−1^𝑛ℏ^𝑛(0.1)^2∞_𝑘=𝑛(−1)^𝑘+1(𝑡−2)^𝑘+()𝑘!sin𝑥−1^𝑛ℏ^𝑛𝑒(0.1)³2!^∞_𝑘=𝑛(−1)^𝑘+1(𝑡−2)^𝑘+2()(()!⋮𝑘+2)(𝑘+1𝑘⋯𝑘−𝑛−3))(𝑘−𝑛sin𝑥, (3.40) and so on. When we choose ℏ=−1, we obtain 𝑢₀(𝑡,𝑥)=𝑒⁻²[(0.1)𝑒]²()𝑢2!+20.1𝑒+1sin𝑥,₁(𝑡,𝑥)=−𝑒⁻²[(0.1)𝑒]²()()2!+20.1𝑒+1sin𝑥𝑡−2+𝑒⁻¹(0.1)^∞_𝑘=1(−1)^𝑘+1(𝑡−2)^𝑘()𝑘!sin𝑥+0.1^2∞_𝑘=1(−1)^𝑘+1(𝑡−2)^𝑘+1()!+()𝑘+1)(𝑘−1sin𝑥0.1^2∞_𝑘=1(−1)^𝑘+1(𝑡−2)^𝑘()𝑘!sin𝑥+𝑒0.1³2!^∞_𝑘=1(−1)^𝑘+1(𝑡−2)^𝑘+2()!𝑢𝑘+2)(𝑘−1sin𝑥,₂(𝑡,𝑥)=𝑒⁻²[(0.1)𝑒]²()()2!+20.1𝑒+1sin𝑥𝑡−2²2!+𝑒⁻¹(0.1)^∞_𝑘=2(−1)^𝑘+1(𝑡−2)^𝑘()𝑘!sin𝑥+0.1^2∞_𝑘=2(−1)^𝑘+1(𝑡−2)^𝑘+1()𝑘()!+()𝑘+1𝑘−2sin𝑥0.1^2∞_𝑘=2(−1)^𝑘+1(𝑡−2)^𝑘()𝑘!sin𝑥+𝑒0.1³2!^∞_𝑘=2(−1)^𝑘+1(𝑡−2)^𝑘+2()!𝑢𝑘+2)(𝑘+1)(𝑘−2sin𝑥,₃(𝑡,𝑥)=−𝑒⁻²[(0.1)𝑒]²()()2!+20.1𝑒+1sin𝑥𝑡−2³3!+𝑒⁻¹(0.1)^∞_𝑘=3(−1)^𝑘+1(𝑡−2)^𝑘()𝑘!sin𝑥+0.1^2∞_𝑘=3(−1)^𝑘+1(𝑡−2)^𝑘+1()𝑘()!+()𝑘+1𝑘−1)(𝑘−3sin𝑥0.1^2∞_𝑘=3(−1)^𝑘+1(𝑡−2)^𝑘()𝑘!sin𝑥+𝑒0.1³2!^∞_𝑘=3(−1)^𝑘+1(𝑡−2)^𝑘+2()𝑘()!⋮𝑢𝑘+2)(𝑘+1𝑘−3sin𝑥,_𝑛(𝑡,𝑥)=(−1)^𝑛𝑒⁻²[(0.1)𝑒]²()()2!+20.1𝑒+1sin𝑥𝑡−2^𝑛𝑛!+𝑒⁻¹(0.1)^∞_𝑘=𝑛(−1)^𝑘+1(𝑡−2)^𝑘+()𝑘!sin𝑥0.1^2∞_𝑘=𝑛(−1)^𝑘+1(𝑡−2)^𝑘+1()𝑘()⋯(()!+()𝑘+1𝑘−1𝑘−𝑛−2))(𝑘−𝑛sin𝑥0.1^2∞_𝑘=𝑛(−1)^𝑘+1(𝑡−2)^𝑘()𝑘!sin𝑥+𝑒0.1³2!^∞_𝑘=𝑛(−1)^𝑘+1(𝑡−2)^𝑘+2()(()!⋮𝑘+2)(𝑘+1𝑘⋯𝑘−𝑛−3))(𝑘−𝑛sin𝑥, (3.41) and so on.

From (3.6), the solution of (3.35) for ℏ=−1 is𝑢()=)𝑒(𝑡,𝑥[(0.1𝑡−2)]³+)𝑒(3![(0.1𝑡−1)]²+()2!0.1𝑒𝑡+1^∞_𝑘=0(−1)^𝑘𝑡^𝑘𝑘!sin𝑥. (3.42) This series has the closed form𝑢()=)𝑒(𝑡,𝑥[(0.1𝑡−2)]³+)𝑒(3![(0.1𝑡−1)]²+()𝑒2!0.1𝑒𝑡+1^−𝑡sin𝑥, (3.43) which is the exact solution of the (3.35).

Finally, we consider the solution of the delay parabolic equation (2.17) for 𝑡∈[3,4] and rewrite this equation in the following form:𝜕𝑢(𝑡,𝑥)−𝜕𝜕𝑡²𝑢(𝑡,𝑥)𝜕𝑥²=(0.1)𝑒^{−(𝑡−1)}[(0.1)𝑒(𝑡−3)]³+)𝑒(3![(0.1𝑡−2)]²+()𝑒()𝑢()2!0.1𝑡−1+1sin𝑥,3<𝑡≤4,0<𝑥<𝜋,3,𝑥=𝑒⁻³[(0.1)𝑒]³+[2()𝑒]3!0.1²()𝑢()()2!+30.1𝑒+1sin𝑥,0≤𝑥≤𝜋,𝑡,0=𝑢𝑡,𝜋=0,3≤𝑡≤4. (3.44) We take the initial approximation𝑢₀(𝑡,𝑥)=𝑒⁻³[(0.1)𝑒]³+[2()𝑒]3!0.1²()2!+30.1𝑒+1sin𝑥 (3.45) and the linear operator (3.14) with the property (3.15). From (3.44), we define linear operator (3.16).

We construct the zero-order deformation equation (3.2) and the 𝑚th-order deformation equations (3.8) for 𝑚≥1 with the initial conditions𝑢_𝑚(3,𝑥)=0, (3.46) whereℜ_𝑚𝑢⃗_𝑚−1=𝜕𝑢_𝑚−1(𝑡,𝑥)−𝜕𝜕𝑡²𝑢_𝑚−1(𝑡,𝑥)𝜕𝑥²−1−𝜒_𝑚×(0.1)𝑒^{−(𝑡−1)}[(0.1)𝑒(𝑡−3)]³+)𝑒(3![(0.1𝑡−2)]²+()𝑒()2!0.1𝑡−1+1sin𝑥. (3.47)

The solution of the 𝑚th-order deformation equations (3.47) for 𝑚≥1 is𝑢_𝑚(𝑡,𝑥)=𝜒_𝑚𝑢_𝑚−1(𝑡,𝑥)+ℏ𝐿⁻¹ℜ_𝑚𝑢⃗_𝑚−1. (3.48)

From (3.44) and (3.48), we get 𝑢₀(𝑡,𝑥)=𝑒⁻³[(0.1)𝑒]³+[2()𝑒]3!0.1²()𝑢2!+30.1𝑒+1sin𝑥,₁(𝑡,𝑥)=ℏ𝑒⁻³[(0.1)𝑒]³+[2()𝑒]3!0.1²()()2!+30.1𝑒+1sin𝑥𝑡−3−ℏ𝑒⁻²(0.1)^∞_𝑘=1(−1)^𝑘+1(𝑡−3)^𝑘𝑘!sin𝑥−ℏ𝑒⁻¹(0.1)^2∞_𝑘=1(−1)^𝑘+1(𝑡−3)^𝑘+1(𝑘+1)(𝑘−1)!sin𝑥−2ℏ𝑒⁻¹(0.1)^2∞_𝑘=1(−1)^𝑘+1(𝑡−3)^𝑘()𝑘!sin𝑥−ℏ0.1³2!^∞_𝑘=1(−1)^𝑘+1(𝑡−3)^𝑘+2()!()𝑘+2)(𝑘−1sin𝑥−ℏ0.1^3∞_𝑘=1(−1)^𝑘+1(𝑡−3)^𝑘+1()!()𝑘+1)(𝑘−1sin𝑥−ℏ0.1³2!^∞_𝑘=1(−1)^𝑘+1(𝑡−3)^𝑘()𝑘!sin𝑥−ℏ𝑒0.1⁴3!^∞_𝑘=1(−1)^𝑘+1(𝑡−3)^𝑘+3()!𝑢𝑘+3)(𝑘−1sin𝑥,₂(𝑡,𝑥)=ℏ(ℏ+1)𝑒⁻³[(0.1)𝑒]³+[2()𝑒]3!0.1²()()()2!+30.1𝑒+1sin𝑥𝑡−3−ℏℏ+1)(0.1^∞_𝑘=2(−1)^𝑘(𝑡−3)^𝑘−1()!()𝑘−2sin𝑥−ℏℏ+1)(0.1²𝑒^∞−1_𝑘=2(−1)^𝑘(𝑡−3)^𝑘()𝑘!sin𝑥−2ℏℏ+1)(0.1²𝑒^∞−1_𝑘=2(−1)^𝑘(𝑡−3)^𝑘−1()!()()𝑘−1sin𝑥−ℏℏ+10.1³2!^∞_𝑘=2(−1)^𝑘(𝑡−3)^𝑘−1()!()𝑘−1sin𝑥−ℏℏ+1)(0.1^3∞_𝑘=2(−1)^𝑘(𝑡−3)^𝑘𝑘()!()𝑘−2sin𝑥−ℏℏ+1)(0.1^3∞_𝑘=2(−1)^𝑘(𝑡−3)^𝑘+1()!()()𝑘+1)(𝑘−2sin𝑥−ℏℏ+10.1⁴𝑒3!^∞_𝑘=2(−1)^𝑘(𝑡−3)^𝑘+2(𝑘+2)(𝑘−2)!sin𝑥+ℏ²𝑒⁻³[(0.1)𝑒]³+[2()𝑒]3!0.1²()()2!+30.1𝑒+1sin𝑥𝑡−3²2!+ℏ²𝑒⁻²(0.1)^∞_𝑘=2(−1)^𝑘+1(𝑡−3)^𝑘𝑘!sin𝑥+ℏ²𝑒⁻¹(0.1)^2∞_𝑘=2(−1)^𝑘+1(𝑡−3)^𝑘+1(𝑘+1)𝑘(𝑘−2)!sin𝑥+2ℏ²𝑒⁻¹(0.1)^2∞_𝑘=2(−1)^𝑘+1(𝑡−3)^𝑘𝑘!sin𝑥+ℏ²(0.1)³2!^∞_𝑘=2(−1)^𝑘+1(𝑡−3)^𝑘+2(𝑘+2)(𝑘+1)(𝑘−2)!sin𝑥+ℏ²(0.1)^3∞_𝑘=2(−1)^𝑘+1(𝑡−3)^𝑘+1(𝑘+1)𝑘(𝑘−2)!sin𝑥+ℏ²(0.1)³2!^∞_𝑘=2(−1)^𝑘+1(𝑡−3)^𝑘𝑘!sin𝑥+ℏ²𝑒(0.1)⁴3!^∞_𝑘=2(−1)^𝑘+1(𝑡−3)^𝑘+3()!𝑢𝑘+3)(𝑘+2)(𝑘−2sin𝑥,₃(𝑡,𝑥)=ℏ(ℏ+1)²𝑒⁻³[(0.1)𝑒]³+[2()𝑒]3!0.1²()()()2!+30.1𝑒+1sin𝑥𝑡−3−ℏℏ+1²(0.1)𝑒^∞−2_𝑘=3(−1)^𝑘+1(𝑡−3)^𝑘−2()!()𝑘−2sin𝑥−ℏℏ+1²(0.1)²𝑒^∞−1_𝑘=3(−1)^𝑘+1(𝑡−3)^𝑘−1()!()𝑘−1)(𝑘−3sin𝑥−2ℏℏ+1²(0.1)²𝑒^∞−1_𝑘=3(−1)^𝑘+1(𝑡−3)^𝑘−2()!()𝑘−2sin𝑥−ℏℏ+1²(0.1)³2!^∞_𝑘=3(−1)^𝑘+1(𝑡−3)^𝑘𝑘()!()𝑘−3sin𝑥−ℏℏ+1²(0.1)^3∞_𝑘=3(−1)^𝑘+1(𝑡−3)^𝑘−1()!()𝑘−1)(𝑘−3sin𝑥−ℏℏ+1²(0.1)³2!^∞_𝑘=3(−1)^𝑘+1(𝑡−3)^𝑘−2()!()𝑘−2sin𝑥−ℏℏ+1²𝑒(0.1)⁴3!^∞_𝑘=3(−1)^𝑘+1(𝑡−3)^𝑘+1(𝑘+1)(𝑘−3)!sin𝑥+2ℏ²(ℏ+1)𝑒⁻³[(0.1)𝑒]³+[2()𝑒]3!0.1²()()2!+30.1𝑒+1sin𝑥𝑡−3²2!−2ℏ²(ℏ+1)𝑒⁻²(0.1)^∞_𝑘=3(−1)^𝑘+1(𝑡−3)^𝑘−1(𝑘−1)!sin𝑥−ℏ²(ℏ+1)𝑒⁻¹(0.1)^2∞_𝑘=3(−1)^𝑘+1(𝑡−3)^𝑘𝑘(𝑘−1)(𝑘−3)!sin𝑥−4ℏ²(ℏ+1)𝑒⁻¹(0.1)^2∞_𝑘=3(−1)^𝑘+1(𝑡−3)^𝑘−1(𝑘−1)!sin𝑥−2ℏ²()()ℏ+10.1³2!^∞_𝑘=3(−1)^𝑘+1(𝑡−3)^𝑘+1(𝑘+1)𝑘(𝑘−3)!sin𝑥−2ℏ²(ℏ+1)(0.1)^3∞_𝑘=3(−1)^𝑘+1(𝑡−3)^𝑘𝑘(𝑘−1)(𝑘−3)!sin𝑥−2ℏ²()()ℏ+10.1³2!^∞_𝑘=3(−1)^𝑘+1(𝑡−3)^𝑘−1(𝑘−1)!sin𝑥−2ℏ²()𝑒()ℏ+10.1⁴3!^∞_𝑘=3(−1)^𝑘+1(𝑡−3)^𝑘+2(𝑘+2)(𝑘+1)(𝑘−3)!sin𝑥+ℏ³𝑒⁻³[(0.1)𝑒]³+[2()𝑒]3!0.1²()()2!+30.1𝑒+1sin𝑥𝑡−3³3!−ℏ³𝑒⁻²(0.1)^∞_𝑘=3(−1)^𝑘+1(𝑡−3)^𝑘𝑘!sin𝑥−ℏ³𝑒⁻¹(0.1)^2∞_𝑘=3(−1)^𝑘+1(𝑡−3)^𝑘+1(𝑘+1)𝑘(𝑘−1)(𝑘−3)!sin𝑥−2ℏ³𝑒⁻¹(0.1)^2∞_𝑘=3(−1)^𝑘+1(𝑡−3)^𝑘𝑘!sin𝑥−ℏ³(0.1)³2!^∞_𝑘=3(−1)^𝑘+1(𝑡−3)^𝑘+2(𝑘+2)(𝑘+1)𝑘(𝑘−3)!sin𝑥−ℏ³(0.1)^3∞_𝑘=3(−1)^𝑘+1(𝑡−3)^𝑘+1(𝑘+1)𝑘(𝑘−1)(𝑘−3)!sin𝑥−ℏ³(0.1)³2!^∞_𝑘=3(−1)^𝑘+1(𝑡−3)^𝑘𝑘!sin𝑥−ℏ³𝑒(0.1)⁴3!^∞_𝑘=3(−1)^𝑘+1(𝑡−3)^𝑘+3()!⋮𝑢𝑘+3)(𝑘+2)(𝑘+1)(𝑘−3sin𝑥,_𝑛(𝑡,𝑥)=𝑓₄(ℏ+1,𝑡,𝑥)+ℏ^𝑛𝑒⁻³[(0.1)𝑒]³+[2()𝑒]3!0.1²()()2!+30.1𝑒+1sin𝑥𝑡−3^𝑛+()𝑛!−1^𝑛ℏ^𝑛𝑒⁻²(0.1)^∞_𝑘=𝑛(−1)^𝑘+1(𝑡−3)^𝑘+()𝑘!sin𝑥−1^𝑛ℏ^𝑛𝑒⁻¹(0.1)^2∞_𝑘=𝑛(−1)^𝑘+1(𝑡−3)^𝑘+1()𝑘()⋯(()!+()𝑘+1𝑘−1𝑘−𝑛−2))(𝑘−𝑛sin𝑥−1^𝑛ℏ^𝑛2𝑒⁻¹(0.1)^2∞_𝑘=𝑛(−1)^𝑘+1(𝑡−3)^k+()𝑘!sin𝑥−1^𝑛ℏ^𝑛(0.1)³2!^∞_𝑘=𝑛(−1)^𝑘+1(𝑡−3)^𝑘+2()𝑘()⋯(()!+()𝑘+2)(𝑘+1𝑘−3𝑘−𝑛−3))(𝑘−𝑛sin𝑥−1^𝑛ℏ^𝑛(0.1)^3∞_𝑘=𝑛(−1)^𝑘+1(𝑡−3)^𝑘+1()𝑘()⋯(()!+()𝑘+1𝑘−1𝑘−𝑛−2))(𝑘−𝑛sin𝑥−1^𝑛ℏ^𝑛(0.1)³2!^∞_𝑘=𝑛(−1)^𝑘+1(𝑡−3)^𝑘+()𝑘!sin𝑥−1^𝑛ℏ^𝑛𝑒(0.1)⁴3!^∞_𝑘=𝑛(−1)^𝑘+1(𝑡−3)^𝑘+3()(()!⋮𝑘+2)(𝑘+1𝑘⋯𝑘−𝑛−4))(𝑘−𝑛sin𝑥, (3.49) and so on. For ℏ=−1, we get 𝑢₀(𝑡,𝑥)=𝑒⁻³[(0.1)𝑒]³+[2()𝑒]3!0.1²()𝑢2!+30.1𝑒+1sin𝑥,₁(𝑡,𝑥)=−𝑒⁻³[(0.1)𝑒]³+[2()𝑒]3!0.1²()()2!+30.1𝑒+1sin𝑥𝑡−3+𝑒⁻²(0.1)^∞_𝑘=1(−1)^𝑘+1(𝑡−3)^𝑘𝑘!sin𝑥+𝑒⁻¹(0.1)^2∞_𝑘=1(−1)^𝑘+1(𝑡−3)^𝑘+1(𝑘+1)(𝑘−1)!sin𝑥+2𝑒⁻¹(0.1)^2∞_𝑘=1(−1)^𝑘+1(𝑡−3)^𝑘()𝑘!sin𝑥+0.1³2!^∞_𝑘=1(−1)^𝑘+1(𝑡−3)^𝑘+2()!+()𝑘+2)(𝑘−1sin𝑥0.1^3∞_𝑘=1(−1)^𝑘+1(𝑡−3)^𝑘+1()!()𝑘+1)(𝑘−1sin𝑥+0.1³2!^∞_𝑘=1(−1)^𝑘+1(𝑡−3)^𝑘()𝑘!sin𝑥+𝑒0.1⁴3!^∞_𝑘=1(−1)^𝑘+1(𝑡−3)^𝑘+3()!𝑢𝑘+3)(𝑘−1sin𝑥,₂(𝑡,𝑥)=𝑒⁻³[(0.1)𝑒]³+[2()𝑒]3!0.1²()()2!+30.1𝑒+1sin𝑥𝑡−3²2!+𝑒⁻²(0.1)^∞_𝑘=2(−1)^𝑘+1(𝑡−3)^𝑘𝑘!sin𝑥+𝑒⁻¹(0.1)^2∞_𝑘=2(−1)^𝑘+1(𝑡−3)^𝑘+1(𝑘+1)𝑘(𝑘−2)!sin𝑥+2𝑒⁻¹(0.1)^2∞_𝑘=2(−1)^𝑘+1(𝑡−3)^𝑘()𝑘!sin𝑥+0.1³2!^∞_𝑘=2(−1)^𝑘+1(𝑡−3)^𝑘+2()!+()𝑘+2)(𝑘+1)(𝑘−2sin𝑥0.1^3∞_𝑘=2(−1)^𝑘+1(𝑡−3)^𝑘+1()𝑘()!()𝑘+1𝑘−2sin𝑥+0.1³2!^∞_𝑘=2(−1)^𝑘+1(𝑡−3)^𝑘()𝑘!sin𝑥+𝑒0.1⁴3!^∞_𝑘=2(−1)^𝑘+1(𝑡−3)^𝑘+3()!𝑢𝑘+3)(𝑘+2)(𝑘−2sin𝑥,₃(𝑡,𝑥)=−𝑒⁻³[(0.1)𝑒]³+[2()𝑒]3!0.1²()()2!+30.1𝑒+1sin𝑥𝑡−3³3!+𝑒⁻²(0.1)^∞_𝑘=3(−1)^𝑘+1(𝑡−3)^𝑘𝑘!sin𝑥+𝑒⁻¹(0.1)^2∞_𝑘=3(−1)^𝑘+1(𝑡−3)^𝑘+1(𝑘+1)𝑘(𝑘−1)(𝑘−3)!sin𝑥+2𝑒⁻¹(0.1)^2∞_𝑘=3(−1)^𝑘+1(𝑡−3)^𝑘()𝑘!sin𝑥+0.1³2!^∞_𝑘=3(−1)^𝑘+1(𝑡−3)^𝑘+2()𝑘()!+()𝑘+2)(𝑘+1𝑘−3sin𝑥0.1^3∞_𝑘=3(−1)^𝑘+1(𝑡−3)^𝑘+1()𝑘()!()𝑘+1𝑘−1)(𝑘−3sin𝑥+0.1³2!^∞_𝑘=3(−1)^𝑘+1(𝑡−3)^𝑘()𝑘!sin𝑥+𝑒0.1⁴3!^∞_𝑘=3(−1)^𝑘+1(𝑡−3)^𝑘+3()!⋮𝑢𝑘+3)(𝑘+2)(𝑘+1)(𝑘−3sin𝑥,_𝑛(𝑡,𝑥)=(−1)^𝑛𝑒⁻³[(0.1)𝑒]³+[2()𝑒]3!0.1²()()2!+30.1𝑒+1sin𝑥𝑡−3^𝑛𝑛!+𝑒⁻²(0.1)^∞_𝑘=𝑛(−1)^𝑘+1(𝑡−3)^𝑘𝑘!sin𝑥+𝑒⁻¹(0.1)^2∞_𝑘=𝑛(−1)^𝑘+1(𝑡−3)^𝑘+1(𝑘+1)𝑘(𝑘−1)⋯(𝑘−(𝑛−2))(𝑘−𝑛)!sin𝑥+2𝑒⁻¹(0.1)^2∞_𝑘=𝑛(−1)^𝑘+1(𝑡−3)^𝑘+()𝑘!sin𝑥0.1³2!^∞_𝑘=𝑛(−1)^𝑘+1(𝑡−3)^𝑘+2()𝑘()⋯(()!+()𝑘+2)(𝑘+1𝑘−3𝑘−𝑛−3))(𝑘−𝑛sin𝑥0.1^3∞_𝑘=𝑛(−1)^𝑘+1(𝑡−3)^𝑘+1()𝑘()⋯(()!+()𝑘+1𝑘−1𝑘−𝑛−2))(𝑘−𝑛sin𝑥0.1³2!^∞_𝑘=𝑛(−1)^𝑘+1(𝑡−3)^𝑘()𝑘!sin𝑥+𝑒0.1⁴3!^∞_𝑘=𝑛(−1)^𝑘+1(𝑡−3)^𝑘+3()(()!⋮𝑘+2)(𝑘+1𝑘⋯𝑘−𝑛−4))(𝑘−𝑛sin𝑥, (3.50) and so on.

From (3.6), the solution of (3.44) is obtained as𝑢()=)𝑒(𝑡,𝑥[(0.1𝑡−3)]⁴+)𝑒(4![(0.1𝑡−2)]³+)𝑒(3![(0.1𝑡−1)]²+()2!0.1𝑒𝑡+1^∞_𝑘=0(−1)^𝑘𝑡^𝑘𝑘!sin𝑥. (3.51) This series has the closed form𝑢()=)𝑒(𝑡,𝑥[(0.1𝑡−3)]⁴+)𝑒(4![(0.1𝑡−2)]³+)𝑒(3![(0.1𝑡−1)]²+()𝑒2!0.1𝑒𝑡+1^−𝑡sin𝑥, (3.52) which is the exact solution of (3.44).

We give the HAM solutions in 𝑡∈[0,1], 𝑡∈[1,2], 𝑡∈[2,3], 𝑡∈[3,4]. We use four terms for evaluating the approximate solution 𝑢_app=∑³_𝑘=0𝑢_𝑘(𝑡,𝑥). According to the ℏ-curve of 𝑢_𝑥𝑡(0,0), the solution series is convergent when −1.48≤ℏ≤0.48, −1.41≤ℏ≤2.10, −1.19≤ℏ≤0.12, and −1.02≤ℏ≤0, respectively, in 𝑡∈[0,1], 𝑡∈[1,2], 𝑡∈[2,3], 𝑡∈[3,4]. We take ℏ=−1 to determine how much the approximate solution is accurate and compute the absolute errors 𝑢_err=|𝑢_exact−𝑢_app| at the points (0.5,𝜋/2), (1.5,𝜋/2), (2.5,𝜋/2), (3.5,𝜋/2) in Table 5.

Table 5 

The absolute error at 𝑥=𝜋/2 when ℏ=−1.

4. Conclusion

The numerical solutions of first order of difference scheme (2.20) and second order of difference scheme (2.29) for different values of 𝑁 and 𝑀 and the approximate solutions obtained by HAM for 𝑁=3 in (3.11) when ℏ=−1 are given at the same points (0.5,𝜋/2), (1.5,𝜋/2), (2.5,𝜋/2), (3.5,𝜋/2) in Tables 6, 7, 8, and 9, respectively. The absolute errors computed show that, with homotopy analysis method, the results are more accurate for the parabolic delay equation (2.17).

Table 6 

Comparison of the absolute error in HAM when ℏ=−1 and the errors of different difference schemes at (0.5,𝜋/2).

Table 7 

Comparison of the absolute error in HAM when ℏ=−1 and the errors of different difference schemes at (1.5,𝜋/2).

Table 8 

Comparison of the absolute error in HAM when ℏ=−1 and the errors of different difference schemes at (2.5,𝜋/2).

Table 9 

Comparison of the absolute error in HAM when ℏ=−1 and the errors of different difference schemes at (3.5,𝜋/2).

Although HAM seems to be more rapid than finite difference method, the series solutions obtained by HAM are convergence only for the regions determined by convergence control parameter ℏ. So, convergence region is limited for HAM. The comparison of two methods of finite difference and homotopy analysis shows that latter is more rapid and more accurate in the cases that series solutions are convergence. When we take ℏ out of the convergence region determined by ℏ curves, it is shown that finite difference method is faster and more accurate than HAM. The approximate solutions obtained by HAM for different values of ℏ chosen from out of the convergence region of the series solutions and the numerical solutions of first and second order of difference schemes (2.20) and (2.29) for 𝑁=𝑀=4 are given in Tables 10, 11, 12, and 13, respectively at the same points (0.5,𝜋/2), (1.5,𝜋/2), (2.5,𝜋/2), (3.5,𝜋/2).

Table 10 

Comparison of the absolute error in HAM when ℏ=−2 and the errors of different difference schemes at (0.5,𝜋/2).

Table 11 

Comparison of the absolute error in HAM when ℏ=−2.1 and the errors of different difference schemes at (1.5,𝜋/2).

Table 12 

Comparison of the absolute error in HAM when ℏ=1.5 and the errors of different difference schemes at (2.5,𝜋/2).

Table 13 

Comparison of the absolute error in HAM when ℏ=2 and the errors of different difference schemes at (3.5,𝜋/2).

Despite HAM, by finite difference method, we can guarantee the convergence in the whole domain that (2.17) is defined in. Therefore finite difference method is more efficient than HAM.

Acknowledgments

This work is supported by Trakya University Scientific Research Projects Unit (Project number: 2010-91). The author is grateful to Professor Allaberen Ashyralyev (Fatih University, Turkey) for his valuable suggestions for the improvement of this paper.