# Chebyshev Collocation Method for Parabolic Partial Integrodifferential Equations

Authors: M. Sameeh; A. Elsaid
Journal: Advances in Mathematical Physics
 (2016)

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## Abstract

An efficient technique for solving parabolic partial integrodifferential equation is presented. This technique is based on Chebyshev polynomials and finite difference method.A priori error estimate for the proposed technique is deduced. Some examples are presented to illustrate the validity and efficiency of the presented method.


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## Body

## 1. Introduction

Partial integrodifferential equations (PIDEs) are the equations that combine partial differentiation and integration of the unknown function. They are used in modeling several phenomena where the effect of memory must be considered.In this work, we consider the parabolic PIDE of the following form(1)∂u∂t+αx,t∂u∂x+βx,t∂2u∂x2=gx,t+∫0tkt,sux,sds,a<x<b,0<t≤T,subject to the following initial and boundary conditions:(2)ux,0=ϕx,x∈a,b,ua,t=ht,ub,t=lt,t∈0,T,where u(x,t), g(x,t), α(x,t), and β(x,t) are continuous functions.This class of equations appears in various fields of physics and engineering such as heat conduction [1], compression of poroviscoelastic media [2], reaction diffusion problems [3], and nuclear reactor dynamics [4].PIDEs are solved by some numerical methods [5–12]. In this work, the Chebyshev polynomials are applied through finite difference method to obtain an approximate solution for problems (1) and (2).Chebyshev polynomials were proposed by the Russian mathematician Chebyshev. Many authors depend on Chebyshev collocation method to solve different types of equations such as linear differential equations [13], systems of high-order linear differential equations with variable coefficients [14], systems of high-order linear Fredholm-Volterra integrodifferential equations [15], fourth-order Sturm-Liouville problems [16], Troeschs problem [17], nonlinear differential equations [18], linear partial differential equations [19, 20], and nonlinear Fredholm-Volterra integrodifferential equations [21].This paper is organized into six sections. Section2 lists some notations and definitions of Chebyshev polynomials. Section 3 presents a description for the technique of solution to problems (1) and (2). In Section 4, we deducea priori error estimate for the approximate solution. Section 5 is devoted to present some examples that illustrate the proposed technique of solution. Finally, Section 6 presents the conclusions of this study.


## 2. Fundamental Relations

On a general intervalx∈[a,b], the shifted Chebyshev polynomials take the following form:(3)Tn∗x=cos⁡narccos2b-ax-b+a2,and the Chebyshev collocation points are given by(4)xk=b-a2a+bb-a+cos⁡kπN,k=0,1,2,…,N.The function u(x) on [a,b] is approximated using truncated shifted Chebyshev series in the form(5)ux=∑′​r=0Nar∗Tr∗x,a≤x≤b.The symbol sum with single prime indicates that the summation involves1/2a0∗ rather than a0∗. Similarly, the derivatives u(k)(x), k=0,1,2,3,…, are written as(6)ukx=∑′​r=0Nar∗kTr∗x.The function u(x) and its derivatives can be written in matrix form as (7)ux=T∗xA∗,ukx=T∗xA∗k, where (8)T∗x=T0∗xT1∗xT2∗x⋯TN∗x,A∗=12a0∗a1∗⋯aN∗τ. Matrix A∗(k) can be obtained from A∗ by the following relation [13]: (9)A∗k=4b-akMkA∗, where (10)
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## 3. Description of Chebyshev Collocation Method

The partial time-derivative in (1) is discretized using finite difference method. Denote the time step by δt and the value of the function at time iδt by ui. Then we have(13)∂u∂t=ui+1-uiδt.The integral is handled numerically using the composite trapezoidal rule, and (1) is discretized into the following form:(14)ui+1-uiδt+αi+1xui+1′+βi+1xui+1′′=gi+1x+12δtki+1t0u0+δt∑j=1iki+1tjuj+12δtki+1ti+1ui+1,which is simplified to(15)δtβi+1xui+1′′+δtαi+1xui+1′+1-12δt2ki+1ti+1ui+1=zi+1x,where (16)zi+1x=ui+12δt2ki+1t0u0+δt2∑j=1iki+1tjuj+δtgi+1x. The approximate solution of (15) is computed using truncated Chebyshev series.Theorem 1.
If the assumed approximate solution of problem (15) is described by (5), then the discrete Chebyshev system is given by(17)42δtβi+1xkT∗xkM2Ai+1∗+4δtαi+1xkT∗xkMAi+1∗+1-12δt2ki+1ti+1T∗xkAi+1∗=zi+1xk, where xk are the Chebyshev collocation points. The fundamental matrix form of the discrete Chebyshev system is given by(18)WAi+1∗=Zi+1,where (19)
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.The boundary conditions are integrated into system of (18) in the following form: (20)T∗aAi+1∗=hti+1,T∗bAi+1∗=lti+1.The final form of the system is given by(21)W~Ai+1∗=Z~.We construct the two matrices W~ and Z~ by replacing the first row and last row of the matrix [W,Z] by the corresponding row of boundary conditions.


## 4. Error Analysis

In (1), the time is discretized using finite difference method. From Taylor series, we have (22)∂u∂tti+1=ui+1-uiδt+Oδt. Approximation of integral term using composite trapezoidal integration rule leads to an error of order Oδt2. So (15) can be written in the form(23)δtβi+1xui+1′′+δtαi+1xui+1′+1-12δt2ki+1ti+1ui+1=zi+1x+Oδt2.Equation (23) shows that the truncation error due to time discretization of (1) is of order Oδt2.Ifu(x,t) is the exact solution of (1), then u(x,ti+1) is the exact solution of (23). Let u1(x,ti+1) and u2(x,ti+1) be the Chebyshev series solution of (15) and (23), respectively. If u1(x,ti+1)=T(x)A¯i+1∗, then the discrete Chebyshev system is given by(24)WA¯i+1∗=Zi+1+Oδt2.By subtracting (18) from (24), the following relation is obtained:(25)A¯i+1∗-Ai+1∗=W-1Oδt2=Oδt2.Let INX(x,ti+1) be the Lagrange interpolating polynomial of order N of u(x,ti+1) on the grid X of Chebyshev collocation points. Then INX(x,ti+1) is given by(26)INXx,ti+1=∑k=0Nuxk,ti+1lix,where(27)lixj=δij=1,i=j,0,i≠j.Theorem 2 (see [22]).
Iff∈CN+1a,b, then for any grid X of N+1 nodes and for any x∈a,b the interpolation error at x is given by(28)fx,ti+1-INXx,ti+1=fN+1ξ,ti+1N+1!ωN+1Xx,where ξ∈[a,b] and ωN+1X(x) are the nodal polynomial given by(29)ωN+1Xx=∏j=0Nx-xj.Let u(x,ti+1)=IN(x,ti+1)+HN(x,ti+1). Then IN(x,ti+1) is the solution of the following equation:(30)δtβi+1xIN′′x,ti+1+δtαi+1xIN′x,ti+1+1-12δt2ki+1ti+1INx,ti+1=zi+1x+Δzi+1x+Oδt2,where(31)Δzi+1x=δtβi+1xHN′′x,ti+1+δtαi+1xHN′x,ti+1+1-12δt2ki+1ti+1HNx,ti+1.IN(x,ti+1) can be written as IN(x,ti+1)=T(x)Ai+1∗′. So the discrete Chebyshev system of (30) is given by(32)WAi+1∗′=Z+ΔZ+Oδt2;by subtracting (18) from (32), we obtain (33)Ai+1∗′-Ai+1∗=W-1ΔZ+Oδt2.Theorem 3.
Letu1(x,ti+1) be the Chebyshev series solution of (15) and u(x,ti+1) be the exact solution of (23). If u(x,ti+1) is sufficiently smooth, then(34)ux,ti+1-u1x,ti+1≤HNx,ti+1+TxW-1ΔZ+Oδt2.Proof.
Adding and subtracting the polynomialu2(x,ti+1) yield (35)ux,ti+1-u1x,ti+1≤ux,ti+1-u2x,ti+1+u2x,ti+1-u1x,ti+1.The upper bound for the second term can be found using some properties of norm(36)u2x,ti+1-u1x,ti+1=TxA¯i+1∗-TxAi+1∗≤TxA¯i+1∗-Ai+1∗≤TxOδt2≤Oδt2.The upper bound for the second term can be found by adding and subtracting the polynomial IN(x,ti+1) following the work in [20].(37)ux,ti+1-u2x,ti+1≤ux,ti+1-INx,ti+1+INx,ti+1-u2x,ti+1≤HNx,ti+1+TxAi+1∗′-TxA¯i+1∗≤HNx,ti+1+TxAi+1∗′-A¯i+1∗≤HNx,ti+1+TxW-1ΔZ+Oδt2. Summing up these two upper bounds yields (34).


## 5. Numerical Examples

Example 1 (see [12]).
Consider the parabolic PIDE (1) with the initial condition (38)ux,0=sin⁡x0≤x≤π and with boundary conditions (39)u0,t=uπ,t=0,0≤t≤1 and with α=0, β=-κ, kx,t=λexp⁡(x-t), and g(x,t)=0. The exact solution of this example is ux,t=e-θtcos⁡νt+1-θ/νsin⁡νtsin⁡(x), where θ=(κ+1)/2 and ν=-κ2+2κ-1-4λ/2.
The maximum absolute error is tabulated in Table1 for Chebyshev polynomial together with the results of [12] at κ=1 and λ=-1. The graphs of exact and approximate solutions for t=0.5 are illustrated in Figure 1.
The maximum absolute error is tabulated in Table2 for Chebyshev polynomial at κ=2 and λ=-2.Table 1
l ∞  error of Example 1 at δt=0.0025, t=0.5, and κ=1, λ=-1.
x Chebyshev collocation method Backward-Euler scheme [12] N = 10 N = 100 0.31 0.5545 E - 04 0.632306 E - 04 0.63 0.1070 E - 03 0.120272 E - 03 0.94 0.1467 E - 03 0.165540 E - 03 1.26 0.1730 E - 03 0.194604 E - 03 1.57 0.1817 E - 03 0.204619 E - 03Table 2
l ∞ error of Example 1 at δt=10-3, N=10, κ=2, and λ=-2.
t l ∞ error 0.2 1.72 E - 04 0.4 2.75 E - 04 0.6 3.09 E - 04 0.8 2.88 E - 04 1 2.33 E - 04Figure 1
Exact and Chebyshev approximated solutions of Example1 at t=0.5,κ=1,λ=-1.Example 2.
Consider the parabolic PIDE (1) with the initial condition (40)ux,0=e-kx-0.520≤x≤1 and with boundary conditions (41)u0,t=u1,t=cos⁡te-k/4,0≤t≤1and with α=0, β=1, k(x,t)=(t2-x2), and g(x,t)=cos⁡te-k(x-0.5)22t+4k2x-0.52-2k. The exact solution of this example is given by u(x,t)=cos⁡(t)e-k(x-0.5)2.
The maximum absolute error is tabulated in Table3 for Chebyshev polynomial at different times. The graphs of exact and approximate solutions for t=1 and k=100 are illustrated in Figure 2.Table 3
l ∞ error of Example 2 at δt=10-4, N=75, and k=100.
t l ∞ error 0.2 2.59 E - 06 0.4 3.29 E - 06 0.6 3.33 E - 06 0.8 3.03 E - 06 1 2.514 E - 06Figure 2
Exact and Chebyshev approximated solutions of Example2 at t=1,k=100.Example 3 (see [10]).
Consider the parabolic PIDE (1) with the initial condition (42)ux,0=x0≤x≤1 and with boundary conditions (43)u0,t=0,u1,t=e-t,0≤t≤1and with α=0, β=-x2/2, k(x,t)=e(x-t), and g(x,t) which can be chosen such that the exact solution in this example is u(x,t)=xe-tx.
The maximum absolute error is tabulated in Table4 for Chebyshev polynomial together with the results of [10]. The graphs of exact and approximate solutions for t=1 are illustrated in Figure 3.Table 4
l ∞ error of Example 3 at δt=10-3.
t Chebyshev collocation method Radial basis functions [10] N = 10 N = 40 0.1 1.5265 E - 5 1.8818 E - 5 0.2 2.0744 E - 5 2.6480 E - 5 0.3 2.4945 E - 5 3.0188 E - 5 0.4 2.7769 E - 5 3.1915 E - 5 0.5 2.9610 E - 5 3.2470 E - 5 0.6 3.08267 E - 5 3.2421 E - 5 0.7 3.1628 E - 5 3.2001 E - 5 0.8 3.2145 E - 5 3.1393 E - 5 0.9 3.2466 E - 5 3.0699 E - 5 1 3.2649 E - 5 2.9994 E - 5Figure 3
Exact and Chebyshev approximated solutions of Example3 at t=1.Example 4.
Consider the parabolic PIDE (1) with the initial condition (44)ux,0=e-kxsin⁡ωπx0≤x≤1and with boundary conditions (45)u0,t=0,u1,t=0,0≤t≤1and with α=0, β=-1, k(x,t)=e-xt, and g(x,t) which can be chosen such that the exact solution in this example is ux,t=e-te-kxsin⁡(ωπx).
The maximum absolute error is tabulated in Table5 for Chebyshev polynomial at different values of k and ω. The graphs of exact and approximate solutions for t=1, at k=10,ω=5, and k=10,ω=50 are illustrated in Figures 4 and 5, respectively.Table 5
l ∞ error of Example 4 at δt=10-3.
t l ∞ error l ∞ error k = 10, ω=5,  N=15 k = 10,  ω=50, N=85 0.2 3.20 E - 06 1.66 E - 05 0.4 2.62 E - 06 1.38  E - 05 0.6 2.15 E - 06 1.13 E - 05 0.8 1.76 E - 06 9.27 E - 06 1 1.43 E - 06 7.60 E - 06Figure 4
Exact and Chebyshev approximated solutions of Example4 at t=1,k=10,ω=5.Figure 5
Exact and Chebyshev approximated solutions of Example4 at t=1,k=10,ω=50.


## 6. Conclusions

Chebyshev collocation method is successfully used for solving parabolic partial integrodifferential equation. This method reduced the considered problem into linear system of algebraic equations that can be solved successively to obtain a numerical solution at varied time levels. Numerical examples show that the results of above scheme are in good agreement with the exact ones. Comparisons with the results obtained by using radial basis functions and backward-Euler scheme show that the Chebyshev method yields good results with fewer number of iterations. The method also renders good results for problems where solution exhibits fast changes but with larger number of iterations as anticipated by the error estimate. Moreover, the above scheme can be developed to solve nonlinear parabolic partial integrodifferential equation.



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