# Novel Cubic Trigonometric B-Spline Approach Based on the Hermite Formula for Solving the Convection-Diffusion Equation

Authors: Aatika Yousaf; Thabet Abdeljawad; Muhammad Yaseen; Muhammad Abbas
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This paper introduces a cubic trigonometric B-spline method (CuTBM) based on the Hermite formula for numerically handling the convection-diffusion equation (CDE). The method utilizes a merger of the CuTBM and the Hermite formula for the approximation of a space derivative, while the time derivative is discretized using a finite difference scheme. This combination has greatly enhanced the accuracy of the scheme. A stability analysis of the scheme is also presented to confirm that the errors do not magnify. The main advantage of the scheme is that the approximate solution is obtained as a smooth piecewise continuous function empowering us to approximate a solution at any location in the domain of interest with high accuracy. Numerical tests are performed, and the outcomes are compared with the ones presented previously to show the superiority of the presented scheme.


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## Body

## 1. Introduction

The CDE describes physical phenomena in which particles, energy, and other physical quantities are transferred within a physical system due to diffusion and convection. The CDE is given as(1)∂v∂t+α∂v∂ξ=β∂2v∂ξ2,a≤ξ≤b,t>0,where α is the coefficient of viscosity and β is the phase velocity, respectively, and both are considered positive. Equation (1) is subject to the following initial condition:(2)vξ,0=ϕξ,a≤ξ≤d,and the boundary conditions(3)va,t=g0t,vb,t=g1t,t>0.Here,ϕ, g0, and g1 are known functions of sufficient smoothness.In the literature, various numerical techniques have been developed for the one-dimensional CDE with specified initial and boundary conditions such as finite differences, finite elements, spectral methods, method of lines, and many more. Mohebbi and Dehghan [1] presented a high-order compact solution of the one-dimensional heat and advection-diffusion equation. Salkuyeh [2] used finite difference approximation to solve the CDE. Karahan [3, 4] worked on unconditional stable explicit and implicit finite difference techniques for the advection-diffusion equation using spreadsheets. Restrictive Taylor approximation was used by Ismail et al. [5] to solve the CDE. Cao et al. [6] developed a fourth-order compact finite difference scheme for solving the CDE. The generalized trapezoidal formula was used by Chawla and Al-Zanaidi [7] to solve the CDE. Dehghan [8] used weighted finite difference techniques for the one-dimensional advection-diffusion equation. Furthermore, Dehghan [9] developed a technique for the numerical solution of the three-dimensional advection-diffusion equation. A second-order space and time nodal method for the CDE was conducted by Rizwan [10]. Kara and Zhang [11] introduced an ADI method for an unsteady CDE. Feng and Tian [12] presented an alternating group explicit method for the CDE. Mittal and Jain [13] redefined the cubic B-spline collocation method for solving the CDE. Kadalbajoo and Arora [14] presented the Taylor–Galerkin B-spline finite element method for the one-dimensional advection-diffusion equation. Sari et al. [15] used a high-order finite difference scheme for solving the advection-diffusion equation. Tsai et al. [16] used a characteristics method with cubic interpolation for the advection-diffusion equation. Daig et al. [17] presented a least-squares finite element method for the advection-diffusion equation. Chawla et al. [18] presented extended one-step time-integration schemes for the CDE. Ding and Zhang [19] presented a highly accurate difference scheme for CDE. Nazir et al. [20] obtained numerical solutions of the CDE CuTBS approach. Aminikhah and Alvi [21] solved the CDE using cubic B-spline quasi-interpolation. A new Rabotnov fractional-exponential function based fractional derivative for the diffusion equation under external force was presented by Kumar et al. [22]. A modified analytical approach with existence and uniqueness was presented by Kumar et al. [23] for fractional Cauchy reaction-diffusion equations. A numerical study of modeling and analysis of fractal and fractal-fractional differential equations was initiated in [24, 25].Motivated by the boom of the spline approach in finding the numerical solutions of the partial differential equations, we have utilized a blend of the Hermite formula and the cubic B-spline for the discretization of the space derivative. This merger has significantly augmented the accuracy of the scheme. Another favorable advantage is that approximate solutions come up as a smooth piecewise continuous function permitting one to obtain approximation at any desired location in the domain. The approach used by von Neumann is utilized to confirm that the presented scheme is unconditionally stable. The scheme is applied to various test problems, and the outcomes are contrasted with those reported in [19–21].The remaining portion of the paper is organized in the following sequence. Section2 presents the proposed scheme that is derived out for the numerical treatment of the CDE. The stability analysis of the scheme is discussed in Section 3. The comparison of the numerical results is provided in Section 4. The outcomes of this study are presented in Section 5.


## 2. Derivation of the Scheme

For positive integersM and N, let k=T/N and h=b−a/M be the time and the space step sizes, respectively. The time domain is discretized as tn=nk,n=0,1,2,…,N. The spatial domain a,b is partitioned as ξj=jh,j=0,1,2,…,M, where a=ξ0<ξ1<⋯<ξn−1<ξM=b. The procedure for finding the approximate solution of (1) involves determination of the approximate solution Vξ,t to the exact solution vξ,t as follows [26]:(4)Vξ,t=∑j=−3M−1σjtTBj4ξ,where σjt are time-dependent quantities to be determined and TBj4ξ are cubic trigonometric basis functions given in [26] as follows:(5)TBj4ξ=1py3ξj,ξ∈ξj,ξj+1,yξjyξjzξj+2+zξj+3yξj+1+zξj+4l2ξj+1,ξ∈ξj+1,ξj+2,zξj+4yξj+1zξj+3+zξj+4yξj+2+yξjm2ξj+3,ξ∈ξj+2,ξj+3,m3ξj+4,ξ∈ξj+3,ξj+4,where(6)yξj=sinξ−ξj2,zξj=sinξj−ξ2,p=sinh2sinhsin3h2.By applying the local support property ofTBj4ξ, it is observed that only TBj−34ξ,TBj−24ξ, and TBj−14ξ are survived. Consequently, the approximation vjn at ξj,tn becomes(7)vξj,tn=vjn=∑w=j−3j−1σjntTBj4ξ.Now,vjn and its necessary derivatives are approximated by applying the collocation conditions on Bj3ξ. The obtained approximations are given by(8)vjn=ϑ1σj−3n+ϑ2σj−2n+ϑ1σj−1n,vξjn=−ϑ3σj−3n+ϑ4σj−2n+ϑ3σj−1n,vξξjn=ϑ5σj−3n+ϑ6σj−2n+ϑ5σj−1n,where(9)ϑ1=cschcsc3h2sin2h2,ϑ2=21+2cosh,ϑ3=34csc3h2,ϑ4=0,ϑ5=3+9cosh4cosh/2−4cos5h/2,ϑ6=−3cot2h/22+4cosh.Consider the Hermite formula at the knotξj,tn [27] given by(10)vξξj−1n+10vξξjn+vξξj+1n−12h2vj−1n−2vjn+vj+1n=0.Substituting (8) in (10), we obtain(11)vξξjn=ω1σj−4n+ω2σj−3n+ω3σj−2n+ω2σj−1n+ω1σjn,where(12)ω1=−ϑ510+6ϑ15h2,ω2=−ϑ610−12ϑ15h2+6ϑ25h2,ω3=−ϑ55+12ϑ15h2−12ϑ25h2.Note that (11) provides new approximation of the second derivative. Now, applying the θ-weighted scheme to (1), we obtain(13)vtjn=θhjn+1+1−θhjn,n=0,1,2,…,where hjn=βvξξjn−αvξjn. Using the difference scheme vtjn=vjn+1−vjn/k in (13), we obtain(14)vjn+1+kαθvξjn+1−kβθvξξjn+1=vjn−kα1−θvξjn+kβ1−θvξξjn.For the Crank–Nicolson approach, we chooseθ=0.5 so that (14) reduces to(15)vjn+1+12kαvξjn+1−12kβvξξjn+1=vjn−12kαvξjn+12kβvξξjn.Inserting (8) in (15) and replacing j with j−1, we obtain(16)ϑ1−12kαϑ3−12kβϑ5σj−4n+1+ϑ2+12kαϑ4−12kβϑ6σj−3n+1+ϑ1+12kαϑ3−12kβϑ5σj−2n+1=ϑ1+12kαϑ3+12kβϑ5σj−4n+ϑ2−12kαϑ4+12kβϑ6σj−3n+ϑ1−12kαϑ3+12kβϑ5σj−2n,j=0,M.Inserting (8) and (11) in (15), we obtain(17)−12kβω1σj−4n+1+ϑ1−12kαϑ3−12kβω2σj−3n+1+ϑ2+12kαϑ4−12kβω3σj−2n+1+ϑ1+12kαϑ3−12kβω2σj−1n+1−12kβω1σjn+1=12kβω1σj−4n+ϑ1+12kαϑ3+12kβω2σj−3n+ϑ2−12kαϑ4+12kβω3σj−2n+ϑ1−12kαϑ3+12kβω2σj−1n+12kβω1σjn,j=1,2,3,…,M−1.Note that equations (16) and (17) together produce an inconsistent system of M+1 equations in M+3 unknowns. To obtain a consistent system, we need two additional equations which can be obtained using the given boundary conditions. Consequently, a consistent system of dimension M+3×M+3 is obtained which can be solved using any Gaussian elimination-based numerical algorithm.


### 2.1. Initial State

The initial condition and the derivatives of initial condition are used to find initial vectorσ0 as follows:(18)vξj0=ϕ′ξj,j=0,vj0=ϕξj,j=0,1,…,M,vξj0=ϕ′ξj,j=M.System (18) produces an M+3×M+3 matrix system of the following form:(19)Hσ0=b,where(20)H=−ϑ3ϑ4ϑ30⋯00ϑ1ϑ2ϑ10⋯000ϑ1ϑ2ϑ10⋯0⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮00⋯⋯ϑ1ϑ2ϑ100⋯⋯−ϑ3ϑ4ϑ3,σ0=σ−40,σ−20,σ−10,…,σM−20T,b=ϕ′ξ0,ϕξ0,…,ϕξM,ϕ′ξMT.


## 2.1. Initial State

The initial condition and the derivatives of initial condition are used to find initial vectorσ0 as follows:(18)vξj0=ϕ′ξj,j=0,vj0=ϕξj,j=0,1,…,M,vξj0=ϕ′ξj,j=M.System (18) produces an M+3×M+3 matrix system of the following form:(19)Hσ0=b,where(20)H=−ϑ3ϑ4ϑ30⋯00ϑ1ϑ2ϑ10⋯000ϑ1ϑ2ϑ10⋯0⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮00⋯⋯ϑ1ϑ2ϑ100⋯⋯−ϑ3ϑ4ϑ3,σ0=σ−40,σ−20,σ−10,…,σM−20T,b=ϕ′ξ0,ϕξ0,…,ϕξM,ϕ′ξMT.


## 3. Stability Analysis

The von Neumann stability technique is applied in this section to explore the stability of the given scheme. Consider the growth of error in a single Fourier mode,Ωjn=δneiηhj, where η is the mode number, h is the step size, and i=−1. Inserting the Fourier mode into equation (15) yields(21)−ρ1δn+1eiηj−4h+ρ2δn+1eiηj−3h+ρ3δn+1eiηj−2h+ρ4δn+1eiηj−1h−ρ1δn+1eiηjh=ρ1δneiηj−4h+ρ5δneiηj−3h+ρ6δneiηj−2h+ρ7δneiηj−1h+ρ1δneiηjh,where(22)ρ1=12kβω1,ρ2=ϑ1−12kαϑ3−12kβω2,ρ3=ϑ2+12kαϑ4−12kβω3,ρ4=ϑ1+12kαϑ3−12kβω2,ρ5=ϑ1+12kαϑ3+12kβω2,ρ6=ϑ2−12kαϑ4+12kβω3,ρ7=ϑ1−12kαϑ3+12kβω2.Dividing equation (21) by δneiηj−2h and rearranging the equation, we obtain(23)δ=ρ1e−2iηh+ρ5e−iηh+ρ6+ρ7eiηh+ρ1e2iηh−ρ1e−2iηh+ρ2e−iηh+ρ3+ρ4eiηh−ρ1e2iηh.Usingcosηh=eiηh+e−iηh/2 and sinηh=eiηh−e−iηh/2i in equation (23) and simplifying, we obtain(24)δ=2ρ1cos2ηh+ρ6+2A1cosηh−i2B1sinηh−2ρ1cos2ηh+ρ3+2A2cosηh+i2B2sinηh,where(25)A1=ϑ1+12kβω2,B1=12kαϑ3,A2=ϑ1−12kβω2,B2=12kαϑ3.Note thatη∈−π,π. Without loss of generality, we can assume that η=0 so that equation (24) takes the following form:(26)δ=kβω1+ϑ2−1/2kαϑ4+1/2kβω3+2ϑ1+kβω2−kβω1+ϑ2+1/2kαϑ4−1/2kβω3+2ϑ1−kβω2,=2ϑ1+ϑ2+kβω1+ω2+1/2ω32ϑ1+ϑ2−kβω1+ω2+1/2ω3≤1,which proves that the present computational scheme is unconditionally stable.


## 4. Numerical Experiments and Discussion

In this section, some numerical calculations are performed to test the accuracy of the offered scheme. In all examples, we use the following error norms:(27)L∞=maxjVnumzj,t−vexactzj,t,L2=h∑j=1M+1hVnumzj,t−vexactzj,t.Example 1.
Consider the CDE(28)∂v∂t+0.1∂v∂ξ=0.01∂2v∂ξ2,0≤ξ≤1,t>0,with the initial condition(29)vξ,0=exp5ξsinπξand the boundary conditions(30)v0,t=0,v1,t=0.
The analytic solution of the given problem isvξ,t=exp5ξ−0.25+0.01π2tsinπξ. The numerical results are obtained by utilizing the presented scheme. In Table 1, the absolute errors are compared with those obtained in [19] at various time stages. Figure 1 illustrates the comparison between the exact and numerical solutions at various time stages. Figure 2 shows the 2D and 3D error profiles at T=1. A 3D comparison between the exact and numerical solutions is presented to exhibit the exactness of the scheme in Figure 3. The approximate solution when t=1,k=0.01, and h=0.05 is given by(31)Vξ,1=10.9843cosξ2−10.9843cos3ξ2+54.9966sin3ξ2−41.2474cscξ2sin2ξ+sinξ2169.428+16.4765sinξ,ξ∈0,120,10.0258cosξ2−10.0266cos3ξ2+67.7423sin3ξ2−50.8067cscξ2sin2ξ+sinξ2207.761+15.0399sinξ,ξ∈120,110,7.76173cosξ2−7.77007cos3ξ2+82.6729sin3ξ2−62.0047cscξ2sin2ξ+sinξ2253.005+11.6551sinξ,ξ∈110,320,⋮⋮4135.9cosξ2−3377.74cos3ξ2−1565.64sinξ23+1174.23cscξ2sin2ξ+sinξ2−10328.2+5066.61sinξ,ξ∈1720,910,5855.3cosξ2−4663.41cos3ξ2−1854.21sin3ξ2+1390.66cscξ2sin2ξ+sinξ2−13887.7+6995.11sinξ,ξ∈910,1920,7812.87cosξ2−6075.06cos3ξ2−2061.5sin3ξ2+1546.12cscξ2sin2ξ+sinξ2−17694.2+9112.59sinξ,ξ∈1920,1.Table 1
Absolute errors whenk=0.001 at h=0.005 for Example 1.
tξ=0.1ξ=0.3ξ=0.5ξ=0.7ξ=0.9PresentCNM [19]PresentCNM [19]PresentCNM [19]PresentCNM [19]PresentCNM [19]0.29.19×10−71.88×10−57.52×10−83.61×10−56.83×10−61.39×10−53.06×10−52.05×10−48.13×10−59.67×10−40.41.52×10−63.23×10−51.39×10−76.76×10−51.27×10−52.61×10−55.71×10−53.84×10−41.41×10−41.64×10−30.61.88×10−64.15×10−51.86×10−79.45×10−51.78×10−53.66×10−57.97×10−55.37×10−41.84×10−42.10×10−30.82.10×10−64.81×10−51.99×10−71.17×10−42.22×10−54.56×10−59.89×10−56.64×10−42.16×10−42.42×10−31.02.22×10−65.26×10−51.68×10−71.35×10−42.58×10−55.31×10−51.15×10−47.68×10−42.39×10−42.63×10−3106.19×10−87.17×10−61.90×10−62.87×10−51.12×10−52.31×10−53.52×10−51.11×10−44.78×10−52.86×10−4201.57×10−82.57×10−71.58×10−71.13×10−66.85×10−71.41×10−61.85×10−62.10×10−62.27×10−67.60×10−6Figure 1
The approximate (stars, circles, and triangles) and exact (solid lines) solutions for various time stages whenh=0.005,k=0.001 for Example 1.Figure 2
2D and 3D error profiles whenT=1,h=k=0.01 for Example 1.
(a)(b)Figure 3
The exact and approximate solutions whenT=1,h=k=0.01 for Example 1.
(a)(b)Example 2.
Consider the CDE(32)∂v∂t+0.22∂v∂ξ=0.5∂2v∂ξ2,0≤ξ≤1,t>0,with the initial condition(33)vξ,0=exp0.22ξsinπξand the boundary conditions(34)v0,t=0,v1,t=0.
The analytic solution of the given problem isvξ,t=exp0.22ξ−0.0242+0.5π2tsinπξ. By utilizing the proposed scheme, the numerical results are acquired. An excellent comparison between absolute errors computed by our scheme and the scheme in [19] is presented in Table 2. A close comparison between the exact and numerical solutions at different time stages is depicted in Figure 4. Figure 5 plots 2D and 3D absolute errors at T=1. Figure 6 deals with the 3D comparison that occurs between the exact and numerical solutions. The approximate solution when t=1,k=0.01, and h=0.05 is given as(35)Vξ,1=0.00705027cosξ2−0.00705027cos3ξ2−0.107735sin3ξ2+sinξ2−0.279207+0.0105754sinξ+0.0808013cscξ2sin2ξ,ξ∈0,120,0.00365488cosξ2−0.00365771cos3ξ2−0.0625858sin3ξ2+sinξ2−0.143419+0.00548657sinξ+0.0469394cscξ2sin2ξ,ξ∈120,110,0.00555436cosξ2−0.00555086cos3ξ2−0.075112sin3ξ2+sinξ2−0.181377+0.00832629sinξ+0.056334cscξ2sin2ξ,ξ∈110,320,⋮⋮−0.0808988cosξ2+0.0700979cos3ξ2−0.00330615sin3ξ2+sinξ20.130845−0.105147sinξ+0.00247962cscξ2sin2ξ,ξ∈1720,910,−0.114779cosξ2+0.0954314cos3ξ2+0.00238008sin3ξ2+sinξ20.200982−0.143147sinξ−0.00178506cscξ2sin2ξ,ξ∈910,1920,−0.0214135cosξ2+0.0281036cos3ξ2−0.0075062sin3ξ2+sinξ20.0194332−0.0421554sinξ+0.00562965cscξ2sin2ξ,ξ∈1920,1.Table 2
Absolute errors whenk=0.002 and h=0.002 for Example 2.
ξt=0.8t=1.0Present schemeCNM [19]Present schemeCNM [19]0.11.84×10−72.12×10−78.52×10−81.79×10−70.35.02×10−75.85×10−72.33×10−74.92×10−70.56.47×10−77.62×10−73.00×10−76.40×10−70.75.46×10−76.49×10−72.53×10−75.45×10−70.92.18×10−72.61×10−71.01×10−72.19×10−7Figure 4
The approximate (stars, circles, and triangles) and exact (solid lines) solutions for various time stages whenh=k=0.002 for Example 2.Figure 5
2D and 3D error profile whenT=1,h=k=0.01 for Example 2.
(a)(b)Figure 6
The exact and approximate solutions withT=1,h=k=0.01 for Example 2.
(a)(b)Example 3.
Consider the CDE(36)∂v∂t+0.1∂v∂ξ=0.2∂2v∂ξ2,0≤ξ≤1,t>0,with the initial condition(37)vξ,0=exp0.25ξsinπξand the boundary conditions(38)v0,t=0,v1,t=0.
The analytic solution of the given problem isvξ,t=exp0.25ξ−0.012ξ+0.2π2tsinπξ. The numerical outcomes are obtained utilizing the proposed scheme. An excellent comparison between absolute errors computed by our scheme and the scheme in [19] is discussed in Table 3. Figure 7 deals with the behavior of exact and approximate solutions at various time stages. Figure 8 plots 2D and 3D absolute errors at T=1. In Figure 9, a tremendous 3D contrast between the exact and numerical solutions is shown. The approximate solution when t=1,k=0.01, and h=0.05 is given as(39)Vξ,1=0.110325cosξ2−0.110325cos3ξ2−1.42017sin3ξ2+sinξ2−3.3983+0.165487sinξ+1.06513cscξ2sin2ξ,ξ∈0,120,0.109556cosξ2−0.109557cos3ξ2−1.40995sin3ξ2+sinξ2−3.36757+0.164336sinξ+1.05747cscξ2sin2ξ,ξ∈120,110,0.113008cosξ2−0.112997cosξ23−1.43272sin3ξ2+sinξ2−3.43655+0.169496sinξ+1.07454cscξ2sin2ξ,ξ∈110,320,⋮⋮−1.70311cosξ2+1.4605cos3ξ2−0.0651712sin3ξ2+sinξ22.76626−2.19076sinξ+0.0488784cscξ2sin2ξ,ξ∈1720,910,−1.90731cosξ2+1.6132cos3ξ2−0.0308979sin3ξ2+sinξ23.189−2.4198sinξ+0.0231734cscξ2sin2ξ,ξ∈910,1920,−1.95402cosξ2+1.64688cos3ξ2−0.0259522sin3ξ2+sinξ23.27983−2.47032sinξ+0.0194642cscξ2sin2ξ,ξ∈1920,1.Table 3
Absolute errors whenk=0.005 at h=0.01 for Example 3.
tξ=0.1ξ=0.3ξ=0.5ξ=0.7ξ=0.9PresentCNM [19]PresentCNM [19]PresentCNM [19]PresentCNM [19]PresentCNM [19]0.21.52×10−73.72×10−62.47×10−71.02×10−51.05×10−61.32×10−51.52×10−61.12×10−59.06×10−74.48×10−60.42.16×10−82.80×10−65.85×10−77.68×10−61.42×10−69.93×10−61.76×10−68.41×10−69.42×10−73.36×10−60.61.32×10−71.57×10−67.34×10−74.29×10−61.43×10−65.55×10−61.62×10−64.69×10−68.14×10−71.88×10−60.81.75×10−77.77×10−77.34×10−72.13×10−61.28×10−62.75×10−61.37×10−62.33×10−66.60×10−79.29×10−71.01.77×10−73.61×10−76.58×10−79.88×10−71.08×10−61.28×10−61.10×10−61.08×10−65.18×10−74.31×10−7Figure 7
The approximate (stars, circles, and triangles) and exact (solid lines) solutions for various time stages whenh=0.01,k=0.005 for Example 3.Figure 8
2D and 3D error profiles whenT=1,M=100,k=0.005 for Example 3.
(a)(b)Figure 9
The exact and approximate solutions withT=1,h=0.01,k=0.005 for Example 3.
(a)(b)Example 4.
Consider the CDE(40)∂v∂t+0.8∂v∂ξ=0.1∂2v∂ξ2,0≤ξ≤1,t>0,with the initial condition(41)vξ,0=exp−ξ−2280and the boundary conditions(42)v0,t=2020+texp−−2−0.8t20.420+t,v1,t=2020+texp−−1−0.8t20.420+t.
The analytic solution isvξ,t=20/20+texp−ξ−2−0.8t2/0.420+t. In Table 4, the comparative analysis of absolute errors with those in [20] is provided. Figure 10 illustrates the behavior of numerical solutions at various stages of time. Figure 11 depicts the 2D and 3D graphs of absolute errors. Figure 12 shows the rattling accuracy that exists between the exact and numerical solutions. The approximate solution when t=1,k=0.01, and h=0.05 is given as(43)Vξ,1=0.471308cosξ2−0.0875439cos3ξ2+0.000784654sin3ξ2+sinξ20.514041+0.131316sinξ−0.000588491cscξ2sin2ξ,ξ∈0,120,0.470916cosξ2−0.0871524cos3ξ2+0.00599438sin3ξ2+sinξ20.529709+0.130729sinξ−0.00449579cscξ2sin2ξ,ξ∈120,110,0.470136cosξ2−0.0863755cos3ξ2+0.0111351sin3ξ2+sinξ20.545287+0.129563sinξ−0.00835133cscξ2sin2ξ,ξ∈110,320,⋮⋮0.419092cosξ2−0.0414767cos3ξ2+0.0611033sin3ξ2+sinξ20.750213+0.062215sinξ−0.0458274cscξ2sin2ξ,ξ∈1720,910,0.413674cosξ2−0.0374249cos3ξ2+0.0620127sin3ξ2+sinξ20.761431+0.0561374sinξ−0.0465095cscξ2sin2ξ,ξ∈910,1920,0.407727cosξ2−0.0331368cos3ξ2+0.0626424sin3ξ2+sinξ20.772993+0.0497052sinξ−0.0469818cscξ2sin2ξ,ξ∈1920,1.Table 4
Error norms whent=1.0 and k=0.001 for Example 4.
hPresent schemeCuTBS [20]L2L∞L2L∞1/48.17×10−61.40×10−51.20×10−41.09×10−41/82.87×10−64.14×10−62.98×10−52.35×10−51/167.95×10−71.12×10−67.56×10−65.76×10−61/322.05×10−72.89×10−71.91×10−61.43×10−61/645.32×10−87.49×10−84.77×10−73.55×10−71/1281.49×10−82.12×10−81.17×10−78.65×10−8Figure 10
The approximate (stars, circles, and triangles) and exact (solid lines) solutions for various time stages whenh=0.01,k=0.001 for Example 4.Figure 11
2D and 3D error profile whenT=1,h=k=0.01 for Example 4.
(a)(b)Figure 12
The exact and approximate solutions whenT=1,h=k=0.01 for Example 4.
(a)(b)Example 5.
Consider the CDE(44)∂v∂t+0.1∂v∂ξ=0.02∂2v∂ξ2,0≤ξ≤1,t>0,with the initial condition(45)vξ,0=exp1.17712434446770ξand the boundary conditions(46)v0,t=exp−0.09t,v1,t=exp1.17712434446770−0.09t.
The analytic solution isvξ,t=exp1.17712434446770ξ−0.09t. The numerical outcomes are acquired by using the introduced scheme. The error norms are computed and compared with those obtained in [21] in Tables 5 and 6. Figure 13 shows the behavior of exact and numerical solutions at various stages of time. Figure 14 depicts the 2D and 3D absolute error profiles at T=1. In Figure 15, a 3D contrast between the exact and numerical solutions is presented and all the graphs are in good agreement. The approximate solution when t=1,k=0.01, and h=0.05 is given as(47)Vξ,1=1.66078cosξ2−0.746853cos3ξ2+0.632987sin3ξ2−0.47474cscξ2sin2ξ+sinξ24.05063+1.12028sinξ,ξ∈0,120,1.65375cosξ2−0.739828cos3ξ2+0.726471sin3ξ2−0.544853cscξ2sin2ξ+sinξ24.33178+1.10974sinξ,ξ∈120,110,1.63814cosξ2−0.724265cos3ξ2+0.829444sin3ξ2−0.622083cscξ2sin2ξ+sinξ24.64382+1.0864sinξ,ξ∈110,320,⋮⋮−0.420769cosξ2+1.04963cos3ξ2+2.44322sin3ξ2+sinξ211.8844−1.57444sinξ−1.83242cscξ2sin2ξ,ξ∈1720,910,−0.768804cosξ2+1.30987cos3ξ2+2.50164sin3ξ2+sinξ212.6049−1.9648sinξ−1.87623cscξ2sin2ξ,ξ∈910,1920,−1.14283cosξ2+1.57959cos3ξ2+2.54124sin3ξ2+sinξ213.3322−2.36938sinξ−1.90593cscξ2sin2ξ,ξ∈1920,1.Table 5
Absolute errors whenh=0.01 and k=0.001 for Example 5.
t = 1t = 2CuBQI [21]Present methodCuBQI [21]Present methodξ=0.12.1506×10−61.0675×10−72.8217×10−61.4143×10−7ξ=0.57.0601×10−63.2442×10−71.2276×10−55.6490×10−7ξ=0.97.6594×10−63.3157×10−71.1643×10−54.9986×10−7Table 6
Error norms whenh=0.01 and k=0.001 for Example 5.
t = 1t = 2CuBQI [21]Present methodCuBQI [21]Present methodL2-norm6.4790×10−72.9393×10−71.0719×10−64.8607×10−7L∞-norm9.1107×10−64.1329×10−71.5204×10−56.8951×10−7Figure 13
The approximate (stars, circles, and triangles) and exact (solid lines) solutions for various time stages whenh=0.01,k=0.001 for Example 5.Figure 14
2D and 3D error profile whenT=1,M=100,k=0.01 for Example 5.
(a)(b)Figure 15
The exact and approximate solutions whenT=1,h=k=0.01 for Example 5.
(a)(b)


## 5. Concluding Remarks

In this study, a cubic trigonometric B-spline collocation method based on the Hermite formula is developed for the convection-diffusion equation. The smooth piecewise cubic B-spline has been used to approximate derivatives in space, whereas a standard finite difference has been used to discretize the time derivative. A combination of the Hermite formula and the cubic trigonometric B-splines for approximating the space derivative has considerably augmented the accuracy of the scheme. The solution comes up as a smooth piecewise continuous function so that one can find an approximate solution at any wanted location in the domain of interest. A special attention is devoted to the stability analysis of the scheme to confirm that the errors do not amplify. The numerical results are contrasted with some current numerical techniques. It is inferred that the presented scheme is more precise and provides better accuracy. It is also worthwhile to mention that the offered scheme is applicable to a variety of problems of applied nature in science and engineering.



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