# Collocation Approach Based on an Extended CubicB-Spline for a Second-Order Volterra Partial Integrodifferential Equation

Authors: Reny George; Muhammad Yaseen; Sana Khan
Journal: Journal of Function Spaces
 (2022)

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## Abstract

This paper focuses on an efficient spline-based numerical technique for numerically addressing a second-order Volterra partial integrodifferential equation. The time derivative is discretized using a finite difference scheme, while the space derivative is approximated using the extended cubicB-spline basis. The scheme is also tested for stability study to ensure that the errors do not accumulate. The convergence of the proposed scheme is also investigated. The scheme’s key benefit is that the approximate solution is produced as a smooth piecewise continuous function allowing us to approximate the solution at any location in the domain. Numerical study is performed, and the comparison of results is made to previously reported results in the literature to show the efficiency of the suggested scheme.


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## Body

## 1. Introduction

Integro-Differential Equations (IDEs) are equations that involve both integrals and derivatives of an unknown function. These equations appear very commonly as mathematical models in various fields. Abel, Lotka, Fredholm, Malthus, Verhulst, and Volterra utilized the integral equations and IDEs [1] to study the problems of physics, economics, and mathematical biology. A vast number of research papers and books are devoted to the ongoing phase of the initiative and growth of IDEs over the last few decades. Special implementation of IDEs to deal with statistical models of spatial-temporal development of epidemics was discussed in [2].Many methods have been used to approximate IDEs previously. The Jacobi-spectral method was used by Ali [3] to approximate the integrodelay differential equations with a weakly singular kernel. Ogunlaran and Oke [4] presented the numerical solution of first order IDEs. Chrysafinos [5] used the method of wavelet-Galerkin to solve IDEs numerically. Abbas et al. [6] approximated IDEs using the direct method of multiwavelet. The first order linear Fredholm IDEs has recently been solved using the rationalized form of Haar functions by Bhrawy et al. [7]. For the numerical solution of Fredholm IDEs, Behiry and Hashish [8] utilized the wavelet technique. The finite element method was used by Chen et al. [9] to approximate the parabolic IDEs. The parabolic Volterra IDEs were approximated by Fakhar and Dehghan [10] using the spectral technique.In several problems of applied sciences, partial integrodifferential equations (PIDEs) are used to represent the complex systems in physical, chemical, and biological sciences and population dynamics [11–19]. Such systems have been solved analytically as well as numerically. Several researchers have contributed to present the numerical solutions of PIDEs using different numerical schemes such as finite differences, Sinc-collocation method, finite element method, spectral collocation method, Legendre method, Galerkin method, and quasiwavelet-based method. Tang [20] approximated PIDEs by using a finite difference scheme. Dehghan [21] gave an approximate solution to a PIDE arising in viscoelasticity. Zarebnia [22] approximated PIDEs by using Sinc-collocation method. Quasiwavelet methods were used by Long et al. [23] to solve PIDEs numerically. Yang et al. [24] used the Crank-Nicolson/quasiwavelet-based numerical method to approximate a class of PIDEs. Izadi and Dehghan [10] developed spectral methods for parabolic Volterra IDEs. Legendre multiwavelet collocation method was proposed for the numerical solution of PIDEs by Aziz and Khan [25]. Numerical solution of Volterra partial integrodifferential equations based on Sinc-collocation method was presented by Fahim et al. [26]. Izadi and Dehghan [27] developed an effective pseudospectral Legendre-Galerkin technique to solve a nonlinear PIDE emerging in population dynamics. A piecewise polynomial function of degree n−1 is a spline function of order n. B-spline-based numerical methods for curves and surfaces were first proposed in the 1940s but were strengthened in the 1970s by various experts. B-splines come in a variety of shapes and sizes, including uniform, nonuniform, rational, and nonrational. Cubic B-spline is a fourth-order B-spline of degree three. The extended cubic B-spline also has a free parameter that allows for local control of this form of B-spline. Collocation techniques based on B-splines proven to be quite effective at approximating the IDEs. Amir and Shakibi [28] used B-spline interpolation to numerically solve IDEs. Exponential splines were used to find the numerical solutions of linear Fredholm IDEs by Tahernezhad and Jalilian [29]. For approximating linear stochastic IDE of fractional order, Mirzaee and Alipour [30] used a cubic B-spline-based collocation method. For a class of hyperbolic PIDE, Fairweather [31] utilized spline-based collocation technique. Gholamian and Saberi-Nadjafi [32] proposed a cubic B-spline collocation technique for a class of PIDEs. Ali et al. [33] developed a quartic B-spline collocation approach for solving PIDEs with a weakly singular kernel. Trigonometric cubic B-spline-based collocation method was used to solve PIDEs by Ali et al. [34].In this paper, we consider the following second order Volterra PIDE:(1)∂vx,t∂t=∫0tt−z−γ∂2vx,z∂x2dz+gx,t,a≤x≤b,t≥0,0<γ<1,subjected to initial condition,
(2)vx,0=φx,and the boundary conditions,
(3)va,t=0,vb,t=0,t≥0,where a,b,φx,aregivenand0<γ<1. Motivated by the popularity of the spline approach, we have utilized the extended cubic B-spline to numerically study the above second order Volterra PIDE.The rest of the paper is organized as follows. In Section2, extended cubic B-spline-based collocation method is derived in detail. In Sections 3 and 4, the stability and convergence of the proposed scheme are discussed, respectively. Section 4 compares numerical results with some other numerical techniques available in literature. Section 5 summarizes the conclusions of this study.


## 2. Derivation of the Scheme

LetΔt=t/Q denotes the time, and h=b−a/N denotes the space step sizes, with Q and N being positive integers. Set the partitions, tq=qΔt0≤q≤Q and xn=nh0≤n≤N of both the temporal and spatial domain. The knots xj evenly discretize the spatial domain a≤x≤b, and the interval a,b is divided into N subintervals, xj,xj+1 of equal length h, j=0,1,2,⋯,N−1, where a=x0<x1<⋯<xn−1<xN=b. The numerical solution Vx,t to the exact solution vx,t of (1) is acquired by
(4)Vx,t=∑j=−1N+1CjtBj4x,η.Here,Cjt,j=−1,⋯N+1 is time-dependent unknowns that must be evaluated and Bj4x,η are extended cubic B-spline (ECuBS) basis functions provided by [35].
(5)Bj4x,η=124h44h1−ηx−xj3+3ηx−xj4,x∈xj,xj+1,4−ηh4+12h3x−xj+1+6h22+ηx−xj+12−12hx−xj+13−3ηx−xj+14,x∈xj+1,xj+2,4−ηh4+12h3xj+3−x+6h22+ηxj+3−x2−12hxj+3−x3−3ηxj+1−x4,x∈xj+2,xj+3,4h1−ηxj+4−x3+3ηxj+4−x4,x∈xj+3,xj+4,0,otherwise,where η∈−8,1. Because of the local support characteristic of ECuBS, only Bj−14x,η,Bj4x,η and Bj+14x,η are preserved at the grid point xj. Consequently, the approximation Vq of vs,t at qth time level is given as
(6)Vx,tq=Vq=∑j=−1N+1CjqtBj4x,η.The unknowns,Cjqt,j=−1,⋯,N+1, are found by using collocation conditions on Bj4x,η and the given initial and boundary conditions. As a consequence, the approximations Vq and its essential derivatives are obtained as
(7)Vq=α1Cj−1q+α2Cjq+α1Cj+1q,Vxq=−β1Cj−1q+β2Cjq+β1Cj+1q,Vxxq=λ1Cj−1q+λ2Cjq+λ1Cj+1q,where α1=4−η/24,α2=8+η/12,β1=1/2h,β2=0,λ1=2+η/2h2, and λ2=−2+η/h2.Discretizing the time derivative in (1) by using forward difference scheme, we obtain
(8)∂vx,t∂t≈Vq+1x−VqxΔt.The term on the RHS of (1) can be written as
(9)∫0tt−z−γvxxx,zdz+gx,t=∫0tq+1tq+1−z−γvxxx,zdz+gx,tq+1.The first expression on the RHS of (9) is time discretized as
(10)∫0tq+1tq+1−z−γvxxx,zdz=∫0tq+1z−γvxxz,tq+1−zdz,=∑r=0q∫trtr+1z−γvxxx,tq+1−zdz,=∑r=0qvxxz,tq−r+1∫trtr+1z−γdz,=Δt1−γ1−γ∑r=0qvxxx,tq−r+1r+11−γ−r1−γ,=Δt1−γ1−γ∑r=0qlrvxxx,tq−r+1,where lr=r+11−γ−r1−γ. Equation (1) becomes
(11)vq+1x−vqxΔt=Δt1−γ1−γ∑r=0qlrvxxq−r+1x+gx,tq+1.LetW=Δt2−γ/1−γ, so that the last equation becomes
(12)vq+1x−Wl0vxxq+1x=vqx+W∑r=1qlrvxxq−r+1x+Δtgx,tq+1.The discretization of the space derivative is performed by (8) so that (12) reduces to
(13)Vq+1x,η−WVxxq+1x,η=Vqx,η+W∑r=1qlrVxxq−r+1x,η+Δtgq+1x.Forx=xj, where j=0,1,2,⋯,N, we have
(14)Vq+1xj,η−WVxxq+1xj,η=Vqxj,η+W∑r=1qlrVxxq−r+1xj,η+Δtgq+1xj.Setting(15)Gj=Vqxj,η+W∑r=1qlrVxxq−r+1xj,η+Δtgq+1xj,we can write, for j=0,1,⋯,N,
(16)Vq+1xj,η−Wl0Vxxq+1xj,η=GjThis implies that(17)∑j=−1N+1Cjq+1tBj4xj,η−W∑j=−1N+1Cjq+1tBj4′′xj,η=Gp,⇒∑j=−1N+1Bj4xj,η−WBj4′′xj,ηCjq+1t=Gj.Equation (17) forms a matrix system of order N+1×N+3. Approximating the boundary conditions (3) using (7), we obtain two equations from where we remove the unknowns, C−1 and CN+1. Consequently, a matrix system of order N+1×N+1 is generated to acquire the unique solution for this system. The matrix equation for this system is given by
(18)SC=G,where the matrices S, C, and G are
(19)S=ω000⋯0ρ1ρ2ρ10⋯00ρ1ρ2ρ1⋱⋮⋮⋱⋱⋱⋱00⋯0ρ1ρ2ρ10⋯000ω,(20)C=C0q+1C0q+1⋯CNq+1CNq+1T,(21)G=ϑ1tG0G1⋯GNGN+1ϑ2tT,where ρ1=α1−Wλ1 and ρ2=α2−Wλ2, ω=Wλ1/α1.Initial vector is as follows: the initial condition,(22)vxp,0=φxp,p=0,1,2,3,⋯,Ncan be used to find the initial vector,
(23)C0=C00C10⋯CN−10CN0T.Equation (22) produces a matrix system of N+1×N+1 order given as
(24)AC0=B,where
(25)A=α1α2α10⋯00α1α2α1⋯00⋱α1α2α1⋮⋮⋱⋱⋱⋱⋮0⋯α1α2α100⋯0α1α2α1,C0=c−10c00⋯cN0cN+10T,B=φx0φx1⋯φxN−1φxNT.Once the initial vectorC0 is obtained, the recurrence relation (17) gives the time evolution of vectors Cq, and thus the approximate solution can be calculated.


## 3. Stability Analysis

In this section, the stability of the proposed method is presented. The proposed scheme is proved stable by using the Von-Neumann stability method. For this purpose, putgq+1x=0 in (13) so that
(26)Vq+1x,η−WVxxq+1x,η=Vqx,η+W∑r=1qlrVxxq−r+1x,η.Using (8) in (26), we obtain
(27)ρ1Cj−1q+1+ρ2Cjq+1+ρ1Cj+1q+1=α1Cj−1q+α2Cjq+α1Cj+1q+W∑r=1qlrλ1Cj−1q−r+1+λ2Cjq−r+1+λ1Cj+1q−r+1.Substituting the Fourier mode,Cjq=ξqeιjϕh in (17), where ϕ is the mode number, h is the step size, ξ is the growth factor, ι=−1, and we obtain
(28)Kξq+1eιjϕh=Lξqeιjϕh+Wh2∑r=1qMξq−r+1eιjϕh,where
(29)K=ρ1e−ιϕh+eιϕh+ρ2,L=α1e−ιϕh+eιϕh+α2,M=δ1e−ιϕh+eιϕh+δ2,and δ1=h2λ1, δ2=h2λ2. Now, substituting the values of ρ1, ρ2, α1, α2, δ1, and δ2 in (29), we get
(30)K=4−η24−W2+η2h22cosϕh+8+η12−W2+ηh2,L=4−η242cosϕh+8+η12,M=2+η22cosϕh−2+η.Rearranging the terms of (28) to obtain
(31)ξq−LK+WMKh2ξq−1−WMKh2∑r=2qlrξq−r=0.Lettingb1=−L/K−WM/Kh2 and br=−WM/Kh2lr, and r=2,⋯,q in (31) to obtain
(32)ξq+b1ξq−1+b2ξq−2+⋯+bq−1ξ+bq=0.It is quite clear from (30) that K and L are positive and M≤0. Thus, the coefficients, b1,b2,⋯bq are positive. Here, it is necessary to mention the following theorem for further procedure.Theorem 1 (see [36]).
For all rootsξj of an arbitrary polynomial pξ=a0ξn+⋯+an with a0≠0, we have ξj≤max1,∑j=1naj/a0.It is necessary to prove that all the roots,ξj, of (32) satisfy ξj≤1 for stability. Since from Theorem 1, a0=1 and aj>0, j=1,⋯,q, we have
(33)∑j=1qaja0=−L+WM/h2∑r=1qlrK,where∑r=1qlr=∑r=1qr+11−γ−r1−γ=q+11−γ−1.()Let us assume thatDγ=q+11−γ−1, then (33) becomes
(34)∑j=1qaj=−L+WMDγ/h2K.From the definition ofM in (29), if we let M=0, then h=0. Consequently, (31) becomes
(35)ξq−LKξq−1=0⇒ξ=0orξ=1.Thus, the required condition for stability is fulfilled that isξj≤1. Next, if ≠0M<0, then from (34), we have
(36)−L−WMDγh2<K,so that the stability condition, i.e., ξj≤1 is satisfied. Now, using the values of K, L, and M in above inequality,
(37)−4−ηcosηh−8+η+62+ηh2WDγ−11−cosηh<,0.cosηh>−8+η/6+2+ηWDγ−1/h24−η/6+2+ηWDγ−1/h2.This inequality implies the unconditional stability of introduced scheme.


## 4. Convergence Analysis

The suggested scheme’s convergence analysis for spatial and temporal directions is provided separately in this section. The convergence is evaluated independently for spatial and temporal directions for this purpose. The following theorem must be proven for spatial convergence.Theorem 2.
Ifv^x is the exact solution of equations (1)–(3) and b^x is the B-spline collocation approximation tov^x, the technique is then second order convergent, and
(38)∥v^x−b^x∥∞≤σh2,where σ=κ0Łh2+R is a finite constant.Proof.
Assume thatv^x is the exact solution of equations (1)–(3), then the approximation, b^x to v^x, is given by
(39)b^x=∑j=−1N+1C^jtBj4x,η,where C^=C^−1,C^0,⋯,C^N+1. Further, suppose that b~x is the evaluated extended cubic B-spline collocation approximation to b^x, namely,
(40)b~x=∑j=−1N+1C~jtBj4x,η,where C~=C~−1,C~0,⋯,C~N+1. To approximate the error, ∥v^x−b^x∥∞, we have to determine the errors, ∥v^x−b~x∥∞ and ∥b~x−b^x∥∞ separately. To compute b~x and b^x, the values of vectors C^ and C~ must be computed from two linear equations,
(41)SC^=G^,(42)SC~=G~.
Now, by subtracting (42) from (41), we obtain
(43)SC~−C^=G~−G^.
The specification of matrixS in equation (19) makes S strictly diagonally dominant making it nonsingular. Thus,
(44)C~−C^=S−1G~−G^.
Taking infinity norm of above equation, we obtain(45)∥C~−C^∥∞≤∥S−1∥∞∥G~−G^∥∞.
Let the sum ofith row of matrix S=xijN+1×N+1 be τi0≤i≤N. Then, we have
(46)τ0=∑j=0Na0j=ω,τi=∑j=0Naij=ρ2+2ρ1i=1,⋯,N−1,τN=∑j=0NaNj=ω.
It is well known in the theory of matrices that(47)∑j=0Nxij−1τj=1,i=0,1,⋯,N.
Here,xij−1 represents the entries of S−1. Then,
(48)∥S−1∥∞=∑j=0Nxij−1≤1τ,where τ=min0≤i≤Nτi=minω,ρ2+2ρ1=minW2+η/2h224/4−η,1. Using (48) in (45) to acquire
(49)∥C~−C^∥∞≤1τ∥G~−G^∥∞.
Using (15), the upper bound of ∥G~−G^∥∞ is computed as
(50)G~i−G^i≤v~i−v^i+Δtg~iq+1−g^iq+1+Wh2∑r=1qlrδ1C~j−1q−r+1−C^j−1q−r+1+δ2C~jq−r+1−C^jq−r+1+δ1C~j+1q−r+1−C^j+1q−r+1.
To simplify the RHS of (50), we present the following theorem.Theorem 3 (see [37, 38]).
IfPx∈c4a,b,P4x≤Ł,∀s∈a,b, the interval a,b is partitioned by Δ=a=x0<x1<⋯<xN=b into subintervals of length h. If bx is the unique spline function interpolates Px at knots x0,x1,⋯,xN, then there exists a constant κj such that,
(51)∥Pl−bl∥≤κlŁh4−ll=0,1,2,3.
Using the aforementioned theorem,(52)v~i−v^i=b~ix−b^ix≤κ0Łh4.
Furthermore,lrr=1q is a decreasing sequence of positive terms and lr≤1 for 1≤r≤n. Using (52) and letting g~iq+1=g^iq+1, we can thus write (50) as
(53)G~i−G^i≤κ0Łh4+Wh2∑r=1qdr.where
(54)δ1C~i−1q−r+1−C^i−1q−r+1+δ2C~iq−r+1−C^iq−r+1+δ1C~i+1q−r+1−C^i+1q−r+1=dr.
Let∑r=1qdr=Dq and κ0Łh4+W/h2Dq=Rq, then, (53) becomes
(55)G~i−G^i≤Rq.
Using (55) in (49) to get
(56)∥C~−C^∥∞≤1τRq=Rh2,where Rh2=1/τRq=max1/W2h2/2+η4−η/24,1.
To proceed further, we have to follow the next theorem.Theorem 4 (see [39]).
TheB-splines B−1,B0,B1,⋯,BN−1,BN,BN+1 satisfy the following inequality
(57)∑i=−1N+1Bix≤1,0≤s≤1.
Now, by subtracting (40) from (39), we have
(58)b~x−b^x=∑j=−1N+1C~j−C^jBj4x.
Taking the infinity norm on both sides, we obtain(59)∥b~x−b^x∥∞=∥∑j=−1N+1C~j−C^jBj4x∥∞,≤∑j=−1N+1Bj4x∥C~j−C^jBj4x∥∞,≤Rh2,that is,
(60)∥b~x−b^x∥∞≤Rh2.
From Theorem (3) and equation (52), we have
(61)∥v^x−b~x∥∞≤κ0Łh4.Thus, from (60) and (61), we have
(62)∥v^x−b^x∥∞≤∥v^x−b~x∥∞+∥b~x−b^x∥∞,≤κ0Łh4+Rh2=σh2,where σ=κ0Łh2+R.Now, for temporal convergence, applying Taylor expansion on (16), we have
(63)vqx+Δtvtqx+Δt22vttqx+⋯−Tvxxqx+Δtvxxtqx+Δtvxxtqx+Δt22!vxxttqx+⋯=vqx+W∑r=1qlrvxxq−r+1x+Δgqx+Δgtqx.Rearranging terms in above equation, we obtain(64)Δtvtq−gq−Wl0vxxq+∑r=1qlrvxxq−r+1+Δvxxtq+Δt22!vxxttq+Δt22!vxxq+gtq=OΔt.Assumingvx,t to be the exact and vqx,t the approximate solutions of the equations (1)-(3), we have from (62) and (64) that
(65)∥vx,t−vqx,t∥≤ρk+h2,where ρ is a finite constant.


## 5. Numerical Results

The efficiency and the validity of the suggested methodology are confirmed in this part using various test problems by utilizing theL2 and L∞ error norms defined as
(66)L2=∥V−Vq∥2=h∑j=0NVxj,tq−Vjq2,L∞=∥V−Vq∥∞=maxjVxj,tq−Vjq.All numerical calculations have been performed using Mathematica 12.Example 5 (see [23]).
Consider equation (1) with
(67)gx,t=2t1/2ππ52sinπx−4t3/2πsin2πx−2π52t2sin2πx,subject to the BCs,
(68)v0,t=v1,t=0,0≤t<1,and the IC,
(69)vx,0=sinπx,x∈0,1.The analytical solution for this problem isvx,t=sinπx−4t5/2/πsin2πx with γ=1/2. The suggested approach is implemented on the aforementioned problem to get numerical results. The estimated errors are compared to those provided in [23, 32–34] at various time stages in Table 1. Figure 1 presents an efficient comparison of approximate and exact solutions at various time levels. 3D comparison between approximate and exact solution is depicted in Figure 2. Figure 3 exhibits the 2D and 3D error profiles. The comparison reveals that the proposed algorithm has far better accuracy. The numerical solution when h=0.05, t=1, and Δt=0.01 is given as
(70)Vx,1=1.43183×10−18−1.53174x+2.05977×10−14x2+25.8542x3−6.29505x4,x∈0,120,−0.0000243493−1.52879x−0.118318x2+27.831x3−18.1748x4,x∈120,110,−0.000218991−1.51313x−0.471333x2+30.9718x3−27.9956x4,x∈110,320,⋮⋮2.92425−32.3551x+104.229x2−115.32x3+40.5233x4,x∈1720,910,−6.19315+8.75716x+34.725x2−63.10583+25.817x4,x∈910,1920,−19.686+65.9016x−56.0282x2+0.949023x3+8.86352x4,x∈1920,1.Example 6 (see [32]).
Consider the equation (1) with vx,t=t+12sinπs subject to initial and boundary conditions vs,0=sinπs, v0,t=v1,t=0, respectively. gx,t is to be chosen with γ=0.5.Table 1
Error comparison for Example5 when Δt=10−5,and N=10.
QECuBSTCuBS [34]CuBS [32]QBCM [33]QWM [23]L2L∞L2L∞L∞L∞L∞508.40×10−81.13×10−75.96×10−68.93×10−61.24×10−61.18×10−41.58×10−31501.48×10−72.00×10−73.09×10−54.42×10−56.34×10−66.75×10−47.89×10−32501.90×10−72.57×10−76.65×10−59.45×10−51.36×10−51.40×10−31.61×10−23502.18×10−72.98×10−71.10×10−41.56×10−42.25×10−52.51×10−32.53×10−24502.38×10−73.26×10−71.60×10−42.28×10−43.28×10−53.70×10−33.46×10−2Figure 1
The exact and approximate (triangles, starts, circles) solutions for Example5 at various times when h=0.01.Figure 2
The approximate and exact solutions for Example5 when h=1/60,t=0.1,Δt=0.01.
(a)(b)Figure 3
2D and 3D error profiles for Example5 when h=1/60,t=0.1,Δt=0.01 .The suggested technique is applied on the above problem to acquire approximate solutions and absolute errors. Table2 reports the contrast between the computed errors of present scheme and those of [32, 34] for different time levels. For various time stages, a sharp contrast between exact and approximate solutions is presented in Figure 4. Figure 5 depicts a 3D comparison between exact and approximate solutions. 2D and 3D absolute errors are plotted in Figure 6. The numerical solution when h=0.05, t=1, and Δt=0.01 is given as
(71)Vx,1=7.69735×10−18+12.4842x+3.76073×10−13x2−20.4956x3+0.0307403x4,x∈0,120,−0.0000611671+12.4879x−0.0729451x2−20.0154x3+0.091464x4,x∈120,110,−0.00102794+12.5168x−0.361224x2−19.0661x3+0.149936x4,x∈110,320,⋮⋮−6.76161+44.8042x−56.66x2+18.4664x3+0.149936x4,x∈1720,910,−7.50901+47.3382x−59.5702x2+19.6495x3+0.091464x4,x∈910,1920,−7.98062+48.8796x−61.3023x2+20.3726x3+0.0307403x4,x∈1920,1.Example 7 (see [32]).
Consider an analytic solution of (1)
(72)vx,t=t+1cosπxx∈0,1.
The initial and boundary conditions are to be evaluated from (72). The function gx,t is to be chosen accordingly.Table 2
Error comparison for Example6 when N=100,and Δt=10−5.
QECuBSTCuBS [34]CuBS [32]L2L∞L2L∞L2L∞508.37×10−101.18×10−93.97×10−95.61×10−96.11×10−98.64×10−91005.28×10−97.47×10−91.41×10−81.99×10−82.01×10−82.84×10−81501.20×10−81.71×10−82.82×10−83.99×10−83.93×10−85.56×10−82002.08×10−82.95×10−84.56×10−86.45×10−86.26×10−88.86×10−82503.12×10−84.42×10−86.59×10−89.31×10−88.97×10−81.27×10−83004.31×10−86.10×10−88.86×10−81.25×10−71.20×10−71.70×10−7Figure 4
The exact and approximate (triangles, starts, circles) solutions for Example6 at various times when h=0.01.Figure 5
The approximate and exact solutions for Example6 when h=1/60,t=0.1,Δt=0.01.
(a)(b)Figure 6
2D and 3D error profiles for Example6 when h=1/60,t=0.1,Δt=0.01.The proposed methodology is utilized to acquire the numerical results for this problem. Table3 shows the efficiency of the presented scheme by comparing the errors with those presented in [32, 34]. Figures 7 and 8 illustrate the 2D and 3D comparison of the exact to approximate solutions. The graphs are in excellent affirmation. The 2D and 3D error functions for this example are shown in Figure 9. The numerical solution when h=0.05, t=1, and Δt=0.01 is given as
(73)Vx,1=2−0.0163534x−9.84508x2+0.613676x3+1.95785x4,x∈0,120,1.99985−0.00725949x−10.0272x2+1.83263x3+1.9096x4,x∈120,110,1.99867+0.0280521x−10.3822x2+3.03508x3+1.81434x4,x∈110,320,⋮⋮3.50607−4.37378x−9.60907x2+10.2924x3−1.81434x4,x∈1720,910,4.29238−6.92536x−6.92831x2+9.47104x3−1.9096x4,x∈910,1920,5.2899−10.0341x−3.74307x2+8.44509x3−1.95785x4,x∈1920,1.Example 8 (see [22]).
Consider the equation (1) with IC
(74)vx,0=2sin2πx,and the BCs,
(75)v0,t=0,v1,t=0.
The exact solution of this problems is(76)vx,t=2t2+t+1sin2πx.
gx,t is to be chosen accordingly. The introduced algorithm is employed to the aforementioned problem to obtain the numerical results. A comparison of computed errors with those of [22, 24] is discussed in Table 4. In Figure 10, a closed contrast between exact and approximate solution is exhibited. 3D profiles of both exact and approximate solutions are compared in Figure 11. Figure 12 displays the 2D and 3D error portrayals between the approximate and exact solutions. The numerical solution when h=0.05, t=1, and Δt=0.01 is given as
(77)Vx,1=−0.0162085x+58.9241x2−18.0337x3−11.5028x4,x∈0,120,0.0043058−0.274697x+64.0995x2−52.6488x3−10.3781x4,x∈120,110,0.0337757−1.16093x+73.0047x2−82.7609x3−8.23736x4,x∈110,320,⋮⋮−19.1207+136.384x−224.702x2+115.71x3−8.23736x4,x∈1720,910,0.802238+71.5344x−156.115x2+94.1611x3−10.3781x4,x∈910,1920,29.3714−17.7196x−64.194x2+64.045x3−11.5028x4,x∈1920,1.Table 3
Error comparison for Example7 when N=50, and Δt=10−4.
QECuBSTCuBS [34]CuBS [32]L2L∞L2L∞L2L∞501.13×10−71.73×10−72.73×10−74.19×10−73.48×10−75.34×10−71003.49×10−75.21×10−78.48×10−71.26×10−61.08×10−61.61×10−61506.75×10−79.95×10−71.64×10−62.42×10−62.09×10−63.08×10−62001.07×10−61.57×10−62.61×10−63.82×10−63.33×10−64.87×10−62501.54×10−62.27×10−63.73×10−65.51×10−64.76×10−67.02×10−63002.05×10−63.02×10−64.99×10−67.33×10−66.36×10−69.35×10−6Figure 7
The exact and approximate (triangles, starts, circles) solutions for Example7 at various times when h=0.01.Figure 8
The approximate and exact solutions for Example7 when h=1/60,t=0.1,Δt=0.01.
(a)(b)Figure 9
2D and 3D error profiles for Example7 when h=1/60,t=0.1,Δt=0.01.Table 4
Error comparison for Example8 when γ=1/4, N=50, and Δt=10−4.
QECuBSTCuBS [24]CuBS [22]L2L∞L2L∞L2L∞501.48×10−62.88×10−62.29×10−64.26×10−62.46×10−64.53×10−61005.15×10−68.37×10−67.89×10−61.24×10−58.44×10−61.32×10−51501.06×10−51.69×10−51.61×10−52.50×10−51.72×10−52.67×10−52001.75×10−52.66×10−52.66×10−53.94×10−52.84×10−54.19×10−52502.58×10−53.89×10−53.90×10−55.74×10−54.17×10−56.12×10−53003.53×10−55.19×10−55.33×10−57.76×10−55.69×10−58.29×10−5Figure 10
The exact and approximate (triangles, starts, circles) solutions for Example8 at various times when h=0.01.Figure 11
The approximate and exact solutions for Example8 when h=1/60,t=0.1,Δt=0.01.
(a)(b)Figure 12
2D and 3D error profiles for Example8 when h=1/60,t=0.1,Δt=0.01.


## 6. Conclusion

In this study, a numerical technique based on the extended cubicB-spline collocation method for the numerical solution of a second order PIDE is presented. The standard finite difference approach is used to discretize the temporal derivatives, while extended cubic B-splines are used to approximate the spatial derivatives. The stability and convergence of the proposed technique are established to ensure that errors do not magnify. Moreover, experimental outcomes substantiate the validity of the proposed scheme. The scheme’s accuracy is confirmed by comparing the numerical results with those computed by some available numerical schemes.



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