# Efficient Numerical Algorithm for the Solution of Nonlinear Two-Dimensional Volterra Integral Equation Arising from Torsion Problem

Authors: A. M. Al-Bugami
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

In this article, an effective method is given to solve nonlinear two-dimensional Volterra integral equations of the second kind, which is arising from torsion problem for a long bar that consists of the nonlinear viscoelastic material type with a fixed elliptical cross section. First, the existence of a unique solution of this problem is discussed, and then, we find the solution of a nonlinear two-dimensional Volterra integral equation (NT-DVIE) using block-by-block method (B-by-BM) and degenerate kernel method (DKM). Numerical examples are presented, and their results are compared with the analytical solution to demonstrate the validity and applicability of the method.


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## Body

## 1. Introduction

The equations of the torsion problem were derived in detail with analytical solutions, by Muskhelishvili [1], Frank and Mises [2], Nowinski [3], and Sneddon and Berry [4]. The problem can be formulated as a boundary value problem of the Laplace equation. In [5], boundary element method was developed for the nonuniform torsion of simply or multiply connected cylindrical bars of arbitrary cross section, where the bar is subjected to an arbitrary distributed twisting moment while its edges are restrained by the most general linear torsional. In [6], nonlinear inelastic uniform torsion of bars by BEM was studied. Sapountzakis and Tsipiras in [7] used the boundary element method solution to the nonlinear inelastic uniform torsion problem of composite bars. El-Kalla and AL-Bugami in [8] discussed the nonlinear Volterra-Fredholm integral equation and torsion problems. Sheshtawy and Ghaleb in [9], discussed approximate solution to the problem of torsion by a boundary integral method. Assari, in [10], discussed the numerical solution of T-DFIE of the second kind on nonrectangular domains. Fattahzadeh, in [11], solved two-dimensional linear and nonlinear Fredholm integral equations of the first kind based on Haar wavelet. Authors, in [12], solved two-dimensional integral equation of the first kind by a multistep method. Alturk, in [13], solved two-dimensional Fredholm integral equations of the first kind using regularization-homotopy method. In this work, effective numerical methods are proposed to obtain the solution of nonlinear two-dimensional Volterra integral equations of the second kind and study the values of absolute errors.


## 2. Basic Formulas

While one end of the bar, of lengthb, is prevented from rotating, the other end is rotated about the z-axis. So that a section at distancezfrom the fixed end turns through angleθ, the variation of angleθwithz,z∈0,b, is taken as
(1)θ=zαx,t.α is a twist angle. The displacement uθ of a particle in a tangential direction is given by
(2)uθ=rθ,where r is the radius of the particle. Then, we get
(3)ux=−yruθ,uy=xruθ.From (1), (2), and (3), we obtained
(4)ux=−yzαx,t,uy=xzαx,t,uz=αx,tψx,y.Hence, we get(5)ex=∂ux∂x=0,ey=∂uy∂y=0,ez=∂uz∂z=0,exy=12∂ux∂y+∂uy∂x=0,ezx=12∂ψ∂x−yα,ezy=12∂ψ∂y+xα,(6)σxy=0,σzx=∂ψ∂x−yGα,σzy=∂ψ∂y+xGα.The stress equilibrium equations are now examined:(7)∂σx∂x+∂σxy∂y+∂σxz∂z=0,∂σy∂y+∂σyz∂z+∂σyx∂x=0,∂σz∂z+∂σzx∂x+∂σzy∂y=0,(8)σzx=∂ϕ∂yGα,σzy=∂ϕ∂xGα.A comparison of (6) and (8) now gives
(9)∂ϕ∂y=∂ψ∂x−y,−∂ϕ∂x=∂ψ∂y+x.The functionsθ and ψ must satisfy the relations
(10)∇2ϕ=−2,∇2ψ=0.Stresses are derived from scalarϕ in (6) in such a way that rectangular axes with any orientation may be used.Let the origin of coordinate axes (n,s) be situated on the boundary of the section, direction n being normal to the boundary and directions being tangential to it. Local values of stresses are now given by
(11)σzn=∂ϕ∂sGα,σzn=∂ϕ∂nGα.The forcepx acting on a vertical strip of width δx is given by
(12)px=Gαδx∫AB∂ϕ∂ydy.It can be seen that the torque on the section is given by(13)T=∬Rxσzy−yσzxdxdy,where T is the moment of torque.


## 3. Solution in the Form of NTVIE

The deviator strain of nonlinear elastic material is as follows:(14)Eyz=eyz,Exz=exz.Using (5) in (14), we get
(15)Eyz=12∂ψ∂y+xαx,t,Exz=12∂ψ∂x−yαx,t.The strain deviator tensor is defined as(16)Eij=eij‐13ekkδij,ekk=exx+eyy+ezz,i,j=x,y,z,Sij=σij‐13σkkδij,σkk=σxx+σyy+σzz,i,j=x,y,z.The second invariant of strain and stress tensor is as follows:(17)E2=eijeij‐13ekkekk.Using (5) in (17), we get
(18)E2=2eyz2+ezx2.Using (17) again, we get
(19)E2=12∂ψx,y∂y+x2+∂ψx,y∂x−y2α2x,t.Therefore, for the stress components, we find thatσxz,σyz are the only nonvanishing components of stress; thus, we find
(20)σxx=σyy=σzz=σxy=0.Also,(21)Sii=0,Sxz=σxz,Syz=σyz.In addition, the principal cubic theory is given by(22)Sij=2GEij+∫0t∫0xJt−τ,x−yEijdydτ+∫0t∫0xkt−τ,x−yE2Eijdydτ,where G is the shear modulus of the material and Jt−τ,x−y, kt−τ,x−y are the kernel functions.From (21) and (22),
(23)σxz=Sxz=2GExz+∫0t∫0xJt−τ,x−yExzdydτ+∫0t∫0xkt−τ,x−yE2Exzdydτ,(24)σyz=Syz=2GEyz+∫0t∫0xJt−τ,x−yEyzdydτ+∫0t∫0xkt−τ,x−yE2Eyzdydτ.Using (15) and (19) in (22), (23), and (24), we obtain
(25)σxz=G∂ψ∂x−yαx,t+12∂ψ∂x−y∫0t∫0xJt−τ,x−yαy,τdydτ+14∂ψ∂x−y3+∂ψ∂y+x2∂ψ∂x−y.∫0t∫0xkt−τ,x−yα3y,τdydτ.Also,(26)σyz=∂ψ∂y+xGαx,t+12∫0t∫0xJt−τ,x−yαy,τdydτ+14∂ψ∂x−y2+∂ψ∂y+x2∫0t∫0xkt−τ,x−yα3y,τdydτ.Using (25) and (26) in (34), we have
(27)T=∬Rx∂ψ∂y−y∂ψ∂x+x2+y2dxdyGαx,t+12∫0t∫0xJt−τ,x−yαdydτ+14∬Rx∂ψ∂y−y∂ψ∂x+x2+y2∂ψ∂y+x2+∂ψ∂x−y2dxdy∫0t∫0xkt−τ,x−yα3y,τdydτ.Let(28)A1=∬Rx∂ψ∂y−y∂ψ∂x+x2+y2dxdy,(29)A1=∬Rx∂ψ∂y−y∂ψ∂x+x2+y2∂ψ∂y+x2+∂ψ∂x−y2dxdy.Then, equation (27) becomes
(30)T=A1Gαx,t+12∫0t∫0xJt−τ,x−yαy,τdydτ+14A2∫0t∫0xkt−τ,x−yα3dydτ.Then, we get(31)ψx,y=xyb2−a2a2+b2,where a and b are the semimajor axis of the ellipse.By calculating∂ψ/∂x and ∂ψ/∂yfrom equation (31) and introducing the result in (29), we find
(32)A1=1a2+b2∬Ra2y2+b2x2dxdy=πa3b3a2+b2.Also,(33)A2=2a2+b2∬Rb2x2+a2y24b4x2a2+b22+4a4y2a2+b22dxdy.Then, we have(34)A2=4πa5b53a2+b22.Here,A1 is the torsional rigidity and A2 is the polar moment of inertia of the cross section of the bar.Write formula (27) in the form
(35)Gαx,t+12∫0t∫0xJt−τ,x−yαy,τdydτ+A24A1∫0t∫0xkt−τ,x−yα3dydτ=TA1,λ=2Gυ1−2υ,K=2G1+υ31−2υ.Here,υ is the Poison ratio, and then, we obtain
(36)αx,t+λ1∫0t∫0xJt−τ,x−yαy,τdydτ+λ2∫0t∫0xkt−τ,x−yα3dydτ=fx,t,where λ1λ/2G,λ2A2K/4A1G, and fx,t=T/A1G.Formula (36) represents NTVIE of the second kind, Jt−τ,x−y is the kernel of linear term, and kt−τ,x−y is the kernel nonlinear term.If the bar is a linear and viscoelastic materialkt−τ,x−y, then we get
(37)αx,t+λ1∫0t∫0xJt−τ,x−yαy,τdydτ=fx,t.If the bar is a nonlinear and viscoelastic material, we getJt−τ,x−y=0; then,
(38)αx,t+λ2∫0t∫0xkt−τ,x−yα3dydτ=fx,t.The general form of formula (36) is
(39)μαx,t+λ1∫0t∫0xJt−τ,x−yαy,τdydτ+λ2∫0t∫0xkt−τ,x−yγτ,y,αy,τdydτ=fx,t,whereλ1=λ/2μ,λ2=A2K/4A1μ,andfx,t=T/A1μ⋅γy,τ,αy,τ,fx,t∈L20,b×C0,T, are given continuous functions and αx,t is an unknown function. λ1,λ2 known constants, which have many physical meaning, may be complex. Jt−τ,x−y and kt−τ,x−y are continuous.


## 4. The Existence of a Unique Solution of T-DVIE

To discuss the existence and uniqueness solution of equation (39), we write it in the integral operator form
(40)Q¯αx,t=1μfx,t+1μQαx,tμ≠0,where
(41)Qαx,t=Q1αx,t+Q2αx,tμ≠0,(42)Q1αx,t=λ1∫0t∫0xJt−τ,x−yαy,τdydτ,Q2αx,t=λ2∫0t∫0xkt−τ,x−yγτ,y,αy,τdydτIn addition, we assume the following conditions:(1)
Jt−τ,x−yandkt−τ,x−y∈L20,b×C0,Tand satisfiesJt−τ,x−y≤M1andkt−τ,x−y≤M2,M1,M2 constants, M>M1,M>M2∀t,τ∈0,T,x,y∈0,b(2)
fx,t, with its partial derivatives with respect to x and t, is continuous in L20,b×C0,T, and its norm is defined as
(43)fx,t=max0≤t≤T∫0t∫0xf2x,τdx1/2dτ=GGisaconstant(3)
The known continuous functionγy,τ,αy,τ satisfies, for the constants A>A1,A>P, the following conditions:
(44)a−γx,t,αx,t≤A1αx,t,b−γx,t,αx,t−γx,t,α2x,t≤Nt,xα1x,t−α2x,t,Nt,xL20,b×C0,T=max0≤t≤T∫0t∫0xN2τ,xdx1/2dτ=P<∞(4)
The unknown functionαx,t satisfies the Lipchitz condition for the first argument of position and Ho¨lder condition for the second argument of time, where
(45)αx,t=max0≤t≤T∫0t∫0xαx,t1/2dx1/2dτ(5)
The kernels satisfies the Lipchitz condition with respect to position and Ho¨lder condition with respect to time, where
(46)Jt−τ,x1−y1−Jt−τ,x2−y2≤L1x1−y1−x2−y2,kt−τ,x1−y1−kt−τ,x2−y2≤L1x1−y1−x2−y2,Jt1−τ1,x−y−Jt2−τ2,x−y≤L2t1−τ1−t2−τ2,kt1−τ1,x−y−kt2−τ2,x−y≤L2t1−τ1−t2−τ2.Theorem 1.
If conditions (1)–(3) are satisfied, then an equation has a unique solution in the Banach space L20,b×C0,T.Lemma 2.
Under the conditions (1)–((3)-a), the operator Q¯ maps the space L20,b×C0,T into itself.Proof.
In the light of the two formulas (40) and (42), we have
(47)Q¯αx,t≤1μfx,t+λ1μ∫0t∫0xJt−τ,x−yαy,τdydτ+λ2μ∫0t∫0xkt−τ,x−yγτ,y,αy,τdydτ.☐Using conditions (1) and (2), then applying Cauchy-Schwarz inequality, we get
(48)Q¯αx,t≤Gμ+λ1μM1∫0t∫0xαy,τdydτ+λ2μM2∫0t∫0xγτ,y,αy,τdydτ.In view of condition (3-a), the above inequality takes the form
(49)Q¯αx,t≤Gμ+λ1μM1Tαx,t+λ2μM2ATαx,t,Q¯αx,t≤Gμ+σαx,t,σ=λ1μM1T+λ2μM2AT,T=max0≤t≤Tt.Inequality (49) shows that the operator Q¯ maps the ball Sρ into itself, where
(50)ρ=Gμ−λ1M1T+λ2M2AT.Sinceρ>0 and G>0, therefore we have σ<1. Moreover, the inequality (49) involves the boundedness of the operator Q of equation (42), where
(51)Qαx,t≤σαx,t.In addition, the inequalities (49) and (51), define the boundedness of the operator Q.Lemma 3.
Assume that the conditions (1) and (3-b) are verified, and then, Q¯ is a contraction operator in the space L20,b×C0,T.Proof.
Forα1x,t and α2x,t in the space L20,b×C0,T and from equations (40) and (42), we find
(52)Q¯α1x,t−Q¯α2x,t≤λ1μ∫0t∫0xJt−τ,x−yα1y,τ−α2y,τdydτ+λ2μ∫0t∫0xkt−τ,x−yγτ,y,α1y,τ−γτ,y,α2y,τdydτ.☐With the aid of conditions (1) and (3-b), the above inequality becomes
(53)Q¯α1x,t−Q¯α2x,t≤λ1μM1∫0t∫0xα1y,τ−α2y,τdydτ+λ2μM2∫0t∫0xNτ,yα1y,τ−α2y,τdydτ.Then, we get(54)Q¯α1x,t−Q¯α2x,t≤σα1x,t−α2x,t.From inequality (54), we see that Q¯ is continuous in the space L20,b×C0,T, and then, Q¯ is a contraction operator under the condition σ<1.


## 5. Solution of NT-DVIE

We consider the bar in the nonlinear case; then, the integral equation (39) with continuous kernel reduced to
(55)μαx,t=fx,t−λ∫0t∫0xkt−τ,x−yγτ,y,αy,τdydτ,where γy,τ,αy,τand fx,t∈L20,b×C0,T are given continuous functions. λ, which have many physical meaning, may be complex. The kernel kt−τ,x−y is continuous.


### 5.1. The B-by-BM

In this section, we use the B-by-BM for solving the NT-DVIE of the second kind.The interval0,b is divided into steps of width h,xj=jh,j=0,1,⋯,n, and h=b−a/n. The approximate solution of α1x will be defined at mesh points xj and denoted by αij,j=0,1,⋯,n, such as αij is an approximation to αixj.To solve the NT-DVIE,(56)Ut=Ft−λ∫0t∫0xGt,τ,x,y,Uy,τdydτ,where
(57)Ut=α1t,⋯,αltT,Ut=α1τ,⋯,αlτ,Ft=f1t,⋯,fltT,Gt,τ,x,y,Uy,τ=Gt,τ,x,yγτ,y,Uy,τ,Gt,τ,Uτ=g1,1t,τ,Uτ⋯g1,st,τ,Uτ⋮gs,1t,τ,Uτ⋯gs,st,τ,Uτ.Then, we get(58)αn+λ∫0t∫0xGt,τ,x,y,Uy,τdydτ.Rewrite equation (58) as follows:
(59)αntk=fitk−λ∫0tpm∫0xpmgi,stk,τ,xk,y,Uy1,τdydτ−λ∫tpmtn∫xpmxngi,stk,τ,xk,y,Uy1,τdydτ.Here,p is some integer and m is (∣k/p∣), s=1,2,⋯. If the values αi0,αi1,⋯αi,nm, are known, then the first integral is obtained by a quadrature rule using values of the integrand at τ=tpm;tpm+1,⋯;tpm+1;y=xpm;xpm+1,⋯;andxpm+1.Then,(60)αik=fixk−λh∑j=0mpwkjgi,stk,τj,xk,yj,u1j,⋯,ulj−λh∑j=mpm+1pwkj∗gi,stk,τmp+j,xk,ymp+j,u1,mp+jj,⋯,ul,mp+j,for n=mp+1,mp+2,⋯,m+1p,m=0,1,⋯,N−1, where wkj,w∗kj depend on the quadrature rule used.


### 5.2. Modified Method of Two Blocks

For this method, we takep=2; the integration over a,t2m can be accomplished by Simpson’s rule, and the integral overt2l,tncan be accomplished by using a quadratic interpolation of the integrand at the point t2m,t2m+1,t2m+2; then, equation (58) becomes
(61)αi,2m+1=fit2m+1−λ∫a2m+1h∫b2m+1hgi,st2m+1,τ,x2m+1,y,Uy,τdydτ,(62)αi,2m+2=fit2m+2−λ∫a2m+2h∫b2m+2hgi,st2m+2,τ,x2m+2,y,Uy,τdydτ,where i=1,2,⋯,l,m=0,1,⋯.On the other hand, from (59), equations (61) and (62) can be written as
(63)αi,2m+1=fit2m+1−λ∫a2mh∫b2mhgi,st2m+1,τ,x2m+1,y,Uy,τdydτ−λ∫2mh2m+1h∫2mh2m+1hgi,st2m+1,τ,x2m+1,y,Uy,τdydτ,αi,2m+2=fit2m+2−λ∫a2mh∫b2mhgi,st2m+2,τ,x2m+2,y,Uy,τdydτ−λ∫2mh2m+1h∫2mh2m+1hgi,st2m+2,τ,x2m+2,y,Uy,τdydτ.Therefore, by equation (60), the approximate solution is computed by
(64)αi,2m+1=fit2m+1−λh3∑j=02mwjgi,st2m+1,τj,x2m+1,yj,u1j,⋯,ulj+λh125gi,st2m+1,τ2m,x2m+1,y2m,u1,2m,⋯,ul,2m+8gi,st2m+1,τ2m+1,x2m+1,y2m+1,u1,2m+1,⋯,ul,2m+1−gi,st2m+1,τ2m+2,x2m+1,y2m+2,u1,2m+1,⋯,ul,2m+2,(65)αi,2m+2=fit2m+2−λh3∑j=02m+2wj′gi,st2m+2,τj,x2m+2,yj,u1j,⋯,ulj,where α1,0=fit0.Thus, replace the second term in equation (64) by formulas (62) and (64), and then, we get
(66)αi,2m+1=fit2m+1−λh3∑j=02mwjgi,st2m+1,τj,x2m+1,yj,α1j,⋯,αlj++λh6gi,st2m+1,τ2m,x2m+1,y2m,α1,2m,⋯,αl,2m+4gi,st2m+1,τ2m+1/2,x2m+1,y2m+1/2,38α1,2m+34αi,2m+1−18αl,2m+2,⋯,38αl,2m+34αl,2m+1‐18αl,2m+2+gi,st2m+1,τ2m+1,x2m+1,y2m+1,α1,2m+1,⋯,αl,2m+1,(67)αi,2m+2=fit2m+2−λh3∑j=02m+2wj′gi,st2m+2,τj,x2m+2,yj,α1j,⋯,αlj,or
(68)αi,2m+2=fit2m+2−λh3gi,st2m+2,τ0,x2m+2,y0,α10,⋯,αl0+4gi,st2m+2,τ0,x2m+2,y0,α11,⋯,αl1+⋯+gi,st2m+2,τ2m+2,x2m+2,y2m+2,α1,2m+2,⋯,αl,2m+2,where
(69)w0=w2m=1,wj=3−−1j,j=1,2,⋯,2m−1,w0′=w2m+2′=1,wj′=3−−1j,j=1,2,⋯,2m+1.Finally, we construct2l linear equations from (67) and (68) to find the unknown functionsui,2m+1,ui,2m+2.


### 5.3. The DKM

In this part, we replace the given kernelg∣t−τ∣,∣x−y∣ approximately by a degenerate kernelgi∣t−τ∣,∣x−y∣, that is,
(70)git−τ,x−y=∑j=1nB′jxCjy∑i=1nBitCiτ,such that
(71)∫0t∫0tg∣t−τ∣,∣x−y∣−gi∣t−τ∣,∣x−y∣2dtdτ1/2⟶0asl⟶∞.Hence, the solution of equation (55) associated with the kernel gi∣t−τ∣,∣x−y∣ takes the form
(72)μαx,t+λ∫0t∫0xg∣t−τ∣,∣x−y∣γτ,y,αy,τdydτ=fx,t.Using (70) in (72), we have
(73)μαx,t+λ∑i=0n∑j=0nAi,jBitB′jx=fx,t,where
(74)Ai,j=∫0t∫0xCiτCjyγτ,y,βy,τdydτ.Here,Ai,j are constants to be determined from the following formula:
(75)Am,m′=∫0t∫0xCmτCm′yγτ,y,1μfy,τ−λμ∑i=1n∑j=1nAi,jBiτB′jydydτ,where j=0,1,2,⋯n;m=1,2,⋯l. If we define
(76)Hm,mA1m′,A2m′,⋯,Alm′=∫0t∫0xCmτCm′yγτ,y,1μfy,τ−λμ∑i=1n∑j=1nAi,jBiτB′jydydτ,m=1,2,⋯l, then equation (75) can be solved numerically.


## 5.1. The B-by-BM

In this section, we use the B-by-BM for solving the NT-DVIE of the second kind.The interval0,b is divided into steps of width h,xj=jh,j=0,1,⋯,n, and h=b−a/n. The approximate solution of α1x will be defined at mesh points xj and denoted by αij,j=0,1,⋯,n, such as αij is an approximation to αixj.To solve the NT-DVIE,(56)Ut=Ft−λ∫0t∫0xGt,τ,x,y,Uy,τdydτ,where
(57)Ut=α1t,⋯,αltT,Ut=α1τ,⋯,αlτ,Ft=f1t,⋯,fltT,Gt,τ,x,y,Uy,τ=Gt,τ,x,yγτ,y,Uy,τ,Gt,τ,Uτ=g1,1t,τ,Uτ⋯g1,st,τ,Uτ⋮gs,1t,τ,Uτ⋯gs,st,τ,Uτ.Then, we get(58)αn+λ∫0t∫0xGt,τ,x,y,Uy,τdydτ.Rewrite equation (58) as follows:
(59)αntk=fitk−λ∫0tpm∫0xpmgi,stk,τ,xk,y,Uy1,τdydτ−λ∫tpmtn∫xpmxngi,stk,τ,xk,y,Uy1,τdydτ.Here,p is some integer and m is (∣k/p∣), s=1,2,⋯. If the values αi0,αi1,⋯αi,nm, are known, then the first integral is obtained by a quadrature rule using values of the integrand at τ=tpm;tpm+1,⋯;tpm+1;y=xpm;xpm+1,⋯;andxpm+1.Then,(60)αik=fixk−λh∑j=0mpwkjgi,stk,τj,xk,yj,u1j,⋯,ulj−λh∑j=mpm+1pwkj∗gi,stk,τmp+j,xk,ymp+j,u1,mp+jj,⋯,ul,mp+j,for n=mp+1,mp+2,⋯,m+1p,m=0,1,⋯,N−1, where wkj,w∗kj depend on the quadrature rule used.


## 5.2. Modified Method of Two Blocks

For this method, we takep=2; the integration over a,t2m can be accomplished by Simpson’s rule, and the integral overt2l,tncan be accomplished by using a quadratic interpolation of the integrand at the point t2m,t2m+1,t2m+2; then, equation (58) becomes
(61)αi,2m+1=fit2m+1−λ∫a2m+1h∫b2m+1hgi,st2m+1,τ,x2m+1,y,Uy,τdydτ,(62)αi,2m+2=fit2m+2−λ∫a2m+2h∫b2m+2hgi,st2m+2,τ,x2m+2,y,Uy,τdydτ,where i=1,2,⋯,l,m=0,1,⋯.On the other hand, from (59), equations (61) and (62) can be written as
(63)αi,2m+1=fit2m+1−λ∫a2mh∫b2mhgi,st2m+1,τ,x2m+1,y,Uy,τdydτ−λ∫2mh2m+1h∫2mh2m+1hgi,st2m+1,τ,x2m+1,y,Uy,τdydτ,αi,2m+2=fit2m+2−λ∫a2mh∫b2mhgi,st2m+2,τ,x2m+2,y,Uy,τdydτ−λ∫2mh2m+1h∫2mh2m+1hgi,st2m+2,τ,x2m+2,y,Uy,τdydτ.Therefore, by equation (60), the approximate solution is computed by
(64)αi,2m+1=fit2m+1−λh3∑j=02mwjgi,st2m+1,τj,x2m+1,yj,u1j,⋯,ulj+λh125gi,st2m+1,τ2m,x2m+1,y2m,u1,2m,⋯,ul,2m+8gi,st2m+1,τ2m+1,x2m+1,y2m+1,u1,2m+1,⋯,ul,2m+1−gi,st2m+1,τ2m+2,x2m+1,y2m+2,u1,2m+1,⋯,ul,2m+2,(65)αi,2m+2=fit2m+2−λh3∑j=02m+2wj′gi,st2m+2,τj,x2m+2,yj,u1j,⋯,ulj,where α1,0=fit0.Thus, replace the second term in equation (64) by formulas (62) and (64), and then, we get
(66)αi,2m+1=fit2m+1−λh3∑j=02mwjgi,st2m+1,τj,x2m+1,yj,α1j,⋯,αlj++λh6gi,st2m+1,τ2m,x2m+1,y2m,α1,2m,⋯,αl,2m+4gi,st2m+1,τ2m+1/2,x2m+1,y2m+1/2,38α1,2m+34αi,2m+1−18αl,2m+2,⋯,38αl,2m+34αl,2m+1‐18αl,2m+2+gi,st2m+1,τ2m+1,x2m+1,y2m+1,α1,2m+1,⋯,αl,2m+1,(67)αi,2m+2=fit2m+2−λh3∑j=02m+2wj′gi,st2m+2,τj,x2m+2,yj,α1j,⋯,αlj,or
(68)αi,2m+2=fit2m+2−λh3gi,st2m+2,τ0,x2m+2,y0,α10,⋯,αl0+4gi,st2m+2,τ0,x2m+2,y0,α11,⋯,αl1+⋯+gi,st2m+2,τ2m+2,x2m+2,y2m+2,α1,2m+2,⋯,αl,2m+2,where
(69)w0=w2m=1,wj=3−−1j,j=1,2,⋯,2m−1,w0′=w2m+2′=1,wj′=3−−1j,j=1,2,⋯,2m+1.Finally, we construct2l linear equations from (67) and (68) to find the unknown functionsui,2m+1,ui,2m+2.


## 5.3. The DKM

In this part, we replace the given kernelg∣t−τ∣,∣x−y∣ approximately by a degenerate kernelgi∣t−τ∣,∣x−y∣, that is,
(70)git−τ,x−y=∑j=1nB′jxCjy∑i=1nBitCiτ,such that
(71)∫0t∫0tg∣t−τ∣,∣x−y∣−gi∣t−τ∣,∣x−y∣2dtdτ1/2⟶0asl⟶∞.Hence, the solution of equation (55) associated with the kernel gi∣t−τ∣,∣x−y∣ takes the form
(72)μαx,t+λ∫0t∫0xg∣t−τ∣,∣x−y∣γτ,y,αy,τdydτ=fx,t.Using (70) in (72), we have
(73)μαx,t+λ∑i=0n∑j=0nAi,jBitB′jx=fx,t,where
(74)Ai,j=∫0t∫0xCiτCjyγτ,y,βy,τdydτ.Here,Ai,j are constants to be determined from the following formula:
(75)Am,m′=∫0t∫0xCmτCm′yγτ,y,1μfy,τ−λμ∑i=1n∑j=1nAi,jBiτB′jydydτ,where j=0,1,2,⋯n;m=1,2,⋯l. If we define
(76)Hm,mA1m′,A2m′,⋯,Alm′=∫0t∫0xCmτCm′yγτ,y,1μfy,τ−λμ∑i=1n∑j=1nAi,jBiτB′jydydτ,m=1,2,⋯l, then equation (75) can be solved numerically.


## 6. Numerical Applications

Consider(77)αx,t=fx,t−λ∫0t∫0xkt−τ,x−yαy,τkdydτ,μ=1.We consider equation (77) in the linear and nonlinear case, where if k=1, we obtain the LTVIE, so in this case, we take λ=λ∗/2G and λ∗=2Gυ/1−2υ also when k≥2. We obtain the NTVIE, and in this case, we take λ=A2K/4GA1 and K=2G1+υ/31−2υ,and we find A1 and A2 from equations (32) and (34); a=4 and b=2 are the major and the minor axes of the ellipse, respectively. In addition, we study three materials: plutonium λ steel υ=0.27, and Copper υ=0.33, where υ is the Poisson ratio 0≤υ≤1/2.Consider(78)αx,t=fx,t−λ∫0t∫0x1+tτy3αy,τkdydτ,μ=1,where the exact solution αx,t=x4t3,; B-by-BM and DKM are used to obtain the approximate numerical solutions and corresponding errors for materials: plutonium, steel, and copper, respectively, N=10 and 20 and time t=0.2 and 0.6, 1, respectively. Table 1 shows the approximate and the absolute relative error values for linear case. Table 2 shows the approximate and the absolute relative error values for nonlinear case. The codes were written in Maple 10 program.Table 1
The approximate values and the absolute relative error values using B-by-BM for linear case (k=1).
Ntxυ=0.21υ=0.27υ=0.33Approx.BMErrorBMApprox.BMErrorBMApprox.BMErrorBM100.20.21.31402E-072.90025E-081.49416E-074.70166E-081.80146E-077.77464E-080.62.26330E-042.38155E-062.27809E-043.86090E-062.30333E-046.38472E-061.08.02121E-032.12185E-058.03440E-033.44078E-058.05692E-035.69251E-050.60.25.13803E-062.37323E-066.61209E-063.84729E-069.12666E-066.36186E-060.66.25662E-032.10002E-046.38706E-033.40451E-046.60961E-035.63002E-041.02.17971E-011.97126E-032.19196E-013.19676E-032.21289E-015.28929E-0310.23.14827E-051.86827E-054.30869E-053.02869E-056.28823E-055.00823E-050.62.98837E-021.89016E-033.10579E-023.06430E-033.30610E-025.06746E-031.01.0208E+002.08810E-021.0338E+003.38655E-021.0560E+005.60418E-02200.20.21.31402E-072.90026E-081.49416E-074.70166E-081.80146E-077.77464E-080.62.26322E-042.37323E-062.27796E-043.84729E-062.30310E-046.36186E-061.08.01883E-031.88375E-058.03053E-033.05385E-058.05049E-035.04999E-050.60.25.13803E-062.37323E-066.61209E-063.84729E-069.12666E-066.36186E-060.66.25636E-032.09746E-046.38664E-033.40023E-046.60887E-035.62261E-041.02.17893E-011.89313E-032.19069E-013.06904E-032.21075E-015.07511E-0310.23.14827E-051.86827E-054.30869E-053.02869E-056.28823E-055.00823E-050.62.98822E-021.88869E-033.10554E-023.06180E-033.30565E-025.06297E-031.01.0203E+002.03892E-021.0330E+003.30540E-021.0546E+005.46597E-02Table 2
The approximate values and the absolute relative error values using B-by-BM for nonlinear case (k=2).
Ntxυ=0.21υ=0.27υ=0.33Approx.BMErrorBMApprox.BMErrorBMApprox.BMErrorBM100.20.21.02402E-072.89760E-121.02403E-073.83470E-121.02405E-075.43320E-120.62.23988E-043.99838E-082.24001E-045.29166E-082.24023E-047.49807E-081.08.01059E-031.05942E-058.01402E-031.40251E-058.01988E-031.98833E-050.60.22.77120E-066.40799E-092.77328E-068.48029E-092.77681E-061.20153E-080.66.06303E-031.64133E-056.06834E-032.17224E-056.07739E-033.07804E-051.02.16985E-019.85292E-042.17304E-011.30451E-032.17849E-011.84975E-0310.21.30339E-052.33973E-071.31096E-053.09638E-071.32387E-054.38713E-070.62.86493E-026.55769E-042.88614E-028.67894E-042.92234E-021.22981E-031.01.0352E+003.52274E-021.0466E+004.66473E-021.0661E+006.61596E-02200.20.21.02402E-072.92220E-121.02403E-073.86720E-121.02405E-075.47920E-120.62.23955E-046.40800E-092.23957E-048.48030E-092.23960E-041.20154E-081.08.00086E-038.66158E-078.00114E-031.14629E-068.00162E-031.62419E-060.60.22.77120E-066.40866E-092.77328E-068.48118E-092.77681E-061.20166E-080.66.06202E-031.54093E-056.06701E-032.03926E-056.07551E-032.88934E-051.02.16668E-016.68128E-042.16884E-018.84218E-042.17252E-011.25286E-0310.21.30339E-052.33976E-071.74931E-063.09642E-071.32387E-054.38719E-070.62.86437E-026.50129E-052.88539E-028.60376E-042.92126E-021.21903E-031.01.0332E+003.32083E-021.0439E+004.39488E-021.0622E+006.22722E-02In Tables1 and 2, Approx.BM →approximate solution of B-by-BM and ErrorBM → the absolute error of B-by-BM. In Tables 3 and 4, Approx.DKM →approximate solution of DKM and ErrorDKM → the absolute error of DKM.Table 3
The approximate values and the absolute relative error values using DKM for linear case (k=1).
Ntxυ=0.21υ=0.27υ=0.33Approx.DKMErrorDKMApprox.DKMErrorDKMApprox.DKMErrorDKM100.20.21.31400E-072.90000E-081.49413E-074.70130E-081.80140E-077.77400E-080.62.26287E-042.33900E-062.27741E-043.79300E-062.30223E-046.27500E-061.08.01120E-031.12000E-058.01863E-031.86300E-058.03201E-033.20100E-050.60.25.13839E-062.37359E-066.61170E-063.84690E-069.12583E-066.36103E-060.66.25352E-032.06910E-046.38227E-033.35660E-046.60223E-035.55620E-041.02.17224E-011.22400E-032.18091E-012.09100E-032.19685E-013.68500E-0310.23.14808E-051.86808E-054.30829E-053.02829E-056.28735E-055.00735E-050.62.98581E-021.86450E-033.10193E-023.02570E-033.30035E-025.00990E-031.01.0141E+001.41000E-021.0244E+002.44000E-021.0431E+004.31000E-02200.20.21.31400E-072.90000E-081.49413E-074.70130E-081.80140E-077.77400E-080.62.26287E-042.33900E-062.27741E-043.79300E-062.30223E-046.27500E-061.08.01120E-031.12000E-058.01863E-031.86300E-058.03201E-033.20100E-050.60.25.13783E-062.37303E-066.61170E-063.84690E-069.12583E-066.36103E-060.66.25352E-032.06910E-046.38227E-033.35660E-046.60223E-035.55620E-041.02.17224E-011.22400E-032.18091E-012.09100E-032.19685E-013.68500E-0310.23.14808E-051.86808E-054.30829E-053.02829E-056.28735E-055.00735E-050.62.98581E-021.86450E-033.10193E-023.02570E-033.30035E-025.00990E-031.01.0416E+004.16000E-021.0244E+002.44000E-021.0431E+004.31000E-02Table 4
The approximate values and the absolute relative error values using DKM method for nonlinear case (k=2).
Ntxυ=0.21υ=0.27υ=0.33Approx.DKMErrorDKMApprox.DKMErrorDKMApprox.DKMErrorDKM100.20.21.02397E-073.00000E-121.02396E-074.00000E-121.02395E-075.00000E-120.62.23831E-041.17000E-072.23801E-041.47000E-072.23756E-041.92000E-071.07.97493E-032.50700E-057.96913E-033.08700E-057.06090E-039.39100E-040.60.22.77078E-065.98000E-092.77274E-067.94000E-092.77611E-061.13100E-080.66.05367E-037.06000E-066.05706E-031.04500E-056.06346E-031.68500E-051.02.14910E-011.09000E-032.14854E-011.14600E-032.14879E-011.12100E-0310.21.30309E-052.30900E-071.31058E-053.05800E-071.32339E-054.33900E-070.62.85827E-025.89100E-042.87835E-027.89900E-042.91304E-021.13680E-031.01.0195E+001.95000E-021.0286E+002.86000E-021.0450E+004.50000E-02200.20.21.02397E-073.00000E-121.02396E-074.00000E-121.02395E-075.00000E-120.62.23831E-041.17000E-072.23801E-041.47000E-072.23756E-041.92000E-071.07.97493E-032.50700E-057.96913E-033.08700E-057.96090E-033.91000E-050.60.22.77078E-065.98000E-092.77274E-067.94000E-092.77611E-061.13100E-080.66.05367E-037.06000E-066.05763E-031.10200E-056.06346E-031.68500E-051.02.14910E-011.09000E-032.14854E-011.14600E-032.14879E-011.12100E-0310.21.30309E-052.30900E-071.31058E-053.05800E-071.32339E-054.33900E-070.62.85827E-025.89100E-042.87835E-027.89900E-042.91304E-021.13680E-031.01.0952E+009.52000E-021.0286E+002.86000E-021.0450E+004.50000E-02


## 7. The Conclusions

This paper deals with a new computational method for approximate solution of NT-DVIE of the second kind with continues kernels. For this purpose, B-by-BM and DKM has been presented to solution the problem. These methods have proven to be effective in solving an equation NT-DVIE. Error analysis and some numerical examples are presented for different materials to illustrate the effectiveness and accuracy of the methods.From the previous results in Tables1 and 2 and Figures 1–12, we notice the following:
(1)
When the values ofυ and λ are fixed in the linear and nonlinear case, then the error value increases with the time x,t=0.2, 0.6, 1(2)
In the linear and nonlinear case, when the values of time are fixed, the error value increases with the increase ofυ and λ(3)
When the values ofυ, λ, and time t are fixed, the error value decreases with N which is increasing, for the linear and nonlinear case and for each material (plutonium, steel, and copper)(4)
Asx is increasing and t is fixed, the errors are also increasing for the linear and nonlinear case and for each material(5)
The approximate solutions calculated by B-by-BM and DKM are best methods for LT-DVIE and NT-DVIE(6)
In general, the maximum value of the errors by B-by-BM and DKM in the linear case is less than the maximum value of the errors in the nonlinear case, for all materials, and the minimum value of the errors in the linear case is larger than the minimum value of the errors in the nonlinear case(7)
The previous numerical experiments illustrate the accuracy of the proposed methods to solve the problemFigure 1
The values of errors by B-by-BM and DKM atT=0.2, N=10, and k=1 for υ=0.21,0.27,0.33.Figure 2
The values of errors by B-by-BM and DKM atT=0.6, N=10, and k=1 for υ=0.21,0.27,0.33.Figure 3
The values of errors by B-by-BM and DKM atT=1, N=10, andk=1 for υ=0.21,0.27,0.33.Figure 4
The values of errors by B-by-BM and DKM atT=0.2, N=20, and k=1 for υ=0.21,0.27,0.33.Figure 5
The values of errors by B-by-BM and DKM atT=0.6, N=20, and k=1 for υ=0.21,0.27,0.33.Figure 6
The values of errors by B-by- BM and DKM atT=1, N=20, and k=1 for υ=0.21,0.27,0.33.Figure 7
The values of errors by B-by-BM and DKM atT=0.2, N=10, and k=2 for υ=0.21,0.27,0.33.Figure 8
The values of errors by B-by-BM and DKM atT=0.6, N=10, and k=2 for υ=0.21,0.27,0.33.Figure 9
The values of errors by B-by-BM and DKM atT=1, N=10, andk=2 for υ=0.21,0.27,0.33.Figure 10
The values of errors by B-by-BM and DKM atT=0.2, N=20, and k=2 for υ=0.21,0.27,0.33.Figure 11
The values of errors by B-by-BM and DKM atT=0.6, N=20, and k=2 for υ=0.21,0.27,0.33.Figure 12
The values of errors by B-by-BM and DKM atT=1, N=20, and k=2 for υ=0.21,0.27,0.33.



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*Source: 6559694-2021-07-16.xml*