a new numerical approach for the solutions of partial differential … · 2019. 5. 21. ·...

Appl. Math. Inf. Sci.10, No. 5, 1663-1672 (2016) 1663

Applied Mathematics & Information SciencesAn International Journal

http://dx.doi.org/10.18576/amis/100504

A New Numerical Approach for the Solutions of PartialDifferential Equations in Three-Dimensional Space

Brajesh Kumar Singh1 and Carlo Bianca2,∗

1 Department of Applied Mathematics, School of Allied Sciences, Babasaheb Bhimrao Ambedkar University, Lucknow, UttarPradesh226025, India

2 Laboratoire de Physique Statistique, Ecole Normale Superieure, PSL Research University; Universite Paris DiderotSorbonne Paris-Cite; Sorbonne Universites UPMC Univ Paris 06; CNRS; 24 rue Lhomond, 75005 Paris, France

Received: 10 Jan. 2016, Revised: 18 Apr. 2016, Accepted: 27 Apr. 2016Published online: 1 Sep. 2016

Abstract: This paper deals with the numerical computation of the solutions of nonlinear partial differential equations in three-dimensional space subjected to boundary and initial conditions. Specifically, the modified cubic B-spline differential quadrature methodis proposed where the cubic B-splines are employed as a set ofbasis functions in the differential quadrature method. Themethodtransforms the three-dimensional nonlinear partial differential equation into a system of ordinary differential equations which is solvedby considering an optimal five stage and fourth-order strongstability preserving Runge-Kutta scheme. The stability region of thenumerical method is investigated and the accuracy and efficiency of the method are shown by means of three test problems: the three-dimensional space telegraph equation, the Van der Pol nonlinear wave equation and the dissipative wave equation. The results showthat the numerical solution is in good agreement with the exact solution. Finally the comparison with the numerical solution obtainedwith some numerical methods proposed in the pertinent literature is performed.

Keywords: 3D nonlinear wave equation; modified cubic B-spline differential quadrature method; SSP-RK54 scheme, Thomasalgorithm

1 Introduction

The development of numerical methods for the simulationof mathematical models has gained much attentionconsidering that recently the power of the computerssciences has been increased. Various numerical methodshave been proposed for obtaining numerical solutions ofpartial differential equations, see, among others, [1,2,3,4,5,6]. A highly accurate non-polynomial tension splinescheme for one-dimensional wave equation has beendeveloped in [7] and applied to the one-dimensional waveequation. The numerical solution of the one-dimensionalhyperbolic telegraph equation by using cubic B-splinecollocation method has been obtained in [8]; numericalsolutions of the multi-dimensional telegraphic has beeninvestigated in [4]. Singh and Lin [9] have proposed ahigh order variable mesh off-step discretization schemefor the one-dimensional nonlinear hyperbolic equation;the reader interested to numerical methods for linear andnonlinear hyperbolic partial differential equations in

three-dimensional space is referred to papers [4,10,11,12,13,14,15] and references cited therein. Recently in [5]an element-free Galerkin scheme has been proposed forthe solution of the three-dimensional wave equation, andin [16] a element-free Galerkin method and a meshlesslocal Petrov-Galerkin method have been proposed for thethree-space-dimensional nonlinear wave equation.

The differential quadrature method (DQM) dates backto Bellman et al. [17,18]. In DQM the derivative of afunction is approximated by introducing the weightedsum of the function values at certain discrete points. Afterthe seminal paper of Bellman, various test functions havebeen proposed, among others, spline functions, sincfunction, Lagrange interpolation polynomials, radial basisfunctions, see [19,20,21,22,23,24] and the referencescited therein. In particular Shu and Richards [25] havedeveloped one of the most generalized approach to solvethe incompressible Navier-Stokes equation.

Recently, Arora and Singh [26] have proposed amodified cubic B-spline differential quadrature method

∗ Corresponding author e-mail:[email protected]

c© 2016 NSPNatural Sciences Publishing Cor.

http://dx.doi.org/10.18576/amis/100504

1664 B. K. Singh, C. Bianca: A new numerical approach for the solutions of...

(MCB-DQM) for the numerical computation of thesolution of the one-dimensional Burger equation. TheMCB-DQM has been further generalized for thecomputational modeling of partial differential equationsin two-dimensional space [27] (the reader is referred alsoto papers [28,29]).

This paper is devoted to the development of a newMCB-DQM for the numerical simulation of the followingpartial differential equation in three-dimensional space:

∂ 2u∂ t2 +α

∂u∂ t

+βu= 2u+ δg(u)

∂u∂ t

+ f (x,y,z, t), (1)

subject to the following initial condition (ICs):

u(x,y,z,0) = ψ1(x,y,z), (x,y,z) ∈ Ω

∂u∂ t (x,y,z,0) = ψ2(x,y,z), (x,y,z) ∈ Ω

(2)

and to the Dirichlet boundary condition (BCs):

u(x,y,z, t) = ξ (x,y,z), (x,y,z) ∈ ∂Ω , t > 0, (3)

where2 ≡ ∂ 2

∂x2 +∂ 2

∂y2 +∂ 2

∂z2 , Ω = (x,y,z) : 0≤ x,y,z≤ 1

is the computational domain and∂Ω is the boundary ofΩ .The functionu(x,y,z, t) is the unknown function whereasf ,ψ1,ψ2 andξ are known functions.

The MCB-DQM is used for computing the spatialderivatives. Accordingly the partial differential equationis transformed into a system of first-order ordinarydifferential equations which is then solved by using theSSP-RK54 scheme [28,29]. The stability region of thenumerical method is investigated within the paper and theaccuracy and efficiency of the method are studied bymeans of three test problems: the three-dimensional spacetelegraph equation, the Van der Pol nonlinear waveequation and the dissipative wave equation. The resultsshow that the numerical solution is in good agreementwith the exact solution. Finally the comparison with thenumerical solution obtained with some numericalmethods proposed in the pertinent literature is performed.Specifically the Root Mean Square (RMS) error norm inthe MCB-DQM solutions is compared with the errorobtained with the MLPG [16] and the EFP [16].

The paper is organized into five more sections, whichfollow this introduction. Specifically Section2 deals withthe description of the modified cubic B-spline differentialquadrature method. Section3 is devoted to the procedurefor the implementation of method for the problem (1)with the initial conditions (2) and boundary conditions(3). The stability analysis of the MCB-DQM is discussedin Section 4. Section 5 is concerned with three testproblems with the main aim to establish the accuracy ofthe proposed method in terms of the RMS error norm.Finally Section6 concludes the paper with reference tocritical analysis and research perspectives.

2 The modified cubic B-spline differentialquadrature method

This section deals with the description of the MCB-DQM[26,27,30,31] for the partial differential equation in three-dimensional space (1). LetD be the following domain:

D= (x,y,z) ∈R3 : a≤ x≤ b, c≤ y≤ d, ℓ≤ z≤ m

which is uniformly partitioned in each direction with thefollowing knots:

a= x1 < x2 < .. . < xi < .. . < xNx−1 < xNx = b,

c= y1 < y2 < .. . < y j < .. . < yNy−1 < yNy = d,

ℓ= z1 < z2 < .. . < zk < .. . < zNz−1 < zNz = m,

where

hx =b−aNx−1

, hy =d− cNy−1

, hz =m− ℓ

Nz−1,

is the discretization step in thex, y and z directions,respectively. Let(xi ,y j ,zk) be the generic grid point and

ui jk ≡ ui jk(t)≡ u(xi,y j ,zk, t),

for i ∈ ∆x = 1,2, . . . ,Nx, j ∈ ∆y = 1,2, . . . ,Ny andk∈∆z = 1,2, . . . ,Nz.The rth-order partial derivatives ofu(x,y,z, t), forr ∈ 1,2, with respect tox, y, z and evaluated in the gridpoint(xi ,y j ,zk) are approximated as follows:

∂ ru∂xr (xi ,y j ,zk) =

Nx

∑p=1

a(r)ip up jk, i ∈ ∆x,

∂ ru∂yr (xi ,y j ,zk) =

Ny

∑p=1

b(r)jp uipk, j ∈ ∆y,

∂ ru∂zr (xi ,y j ,zk) =

Nz

∑p=1

c(r)kpui jp, k∈ ∆z,

(4)

wherea(r)ip , b(r)jp andc(r)kp, called the weighting functions ofthe rth-order partial derivative, are the unknown timedependent quantities to be determined.The cubic B-splines functionϕi = ϕi(x), in thex directionand at the knots, reads:

ϕi =1h3

x

(x− xi−2)3 x∈ [xi−2,xi−1)

(x− xi−2)3−4(x− xi−1)

3 x∈ [xi−1,xi)(xi+2− x)3−4(xi+1− x)3 x∈ [xi ,xi+1)(xi+2− x)3 x∈ [xi+1,xi+2)0 otherwise

(5)

The setϕ0,ϕ1,ϕ2, . . . ,ϕNx,ϕNx+1 is a basis over the set[a,b]. The values ofϕi and its first and second derivatives


Appl. Math. Inf. Sci.10, No. 5, 1663-1672 (2016) /www.naturalspublishing.com/Journals.asp 1665

in the grid pointx j , denoted byϕi j := ϕi(x j), ϕ ′i j := ϕ ′

i (x j)

andϕ ′′i j := ϕ ′′

i (x j), respectively, read:

ϕi j =

4, if i − j = 01, if i − j =±10, otherwise

(6)

ϕ ′i j =

3/hx, if i − j = 1−3/hx, if i − j =−10, otherwise

(7)

ϕ ′′i j =

−12/h2x, if i − j = 0

6/h2x, if i − j =±1

0 otherwise(8)

The modified cubic B-splines basis functions areobtained by modifying the cubic B-spline basis functions(5) as follows [26]:

φ1(x) = ϕ1(x)+2ϕ0(x)

φ2(x) = ϕ2(x)−ϕ0(x)

...

φ j(x) = ϕ j(x), for j = 3,4, . . . ,Nx−2

...

φNx−1(x) = ϕNx−1(x)−ϕNx+1(x)

φNx(x) = ϕNx(x)+2ϕNx+1(x)

(9)

The setφ1,φ2, . . . ,φNx is a basis over the set[a,b].Analogously procedure is followed for they and zdirections.

2.1 Computation of the weighting coefficients

In order to compute the weighting coefficientsa(1)ip of Eq.(4), we use the modified cubic B-splineφp(x), p∈ ∆x. Letφ ′

pi := φ ′p(xi) and φpℓ := φp(xℓ). Accordingly the

approximation of the first-order derivative is obtained asfollows:

φ ′pi =

Nx

∑ℓ=1

a(1)iℓ φpℓ, p, i ∈ ∆x. (10)

Setting Φ = [φpℓ], A = [a(1)iℓ ] (the unknown weightingcoefficient matrix), andΦ ′ = [φ ′

pi], then Eq. (10) can bere-written as the following system of linear equations:

ΦAT = Φ ′. (11)

The coefficient matrixΦ of orderNx can be obtained from(6) and (9):

Φ =

6 10 4 1

1 4 1...

..... .

1 4 11 4 0

1 6

and in particular the columns of the matrixΦ ′ read:

Φ ′[1] =

−6/hx6/hx

0...

00

,Φ ′[2] =

−3/hx0

3/hx0...

0

, . . . ,

Φ ′[Nx−1] =

0...

0−3/hx

03/hx

, andΦ ′[Nx] =

0

...

0−6/hx6/hx

.

It is worth stressing that the cubic B-splines are modified inorder to have a diagonally dominant coefficient matrixΦ,see Eq. (11). The system (11) is thus solved by employingthe Thomas Algorithm [32].

Similarly, the weighting coefficientsb(1)ip andc(1)ip canbe computed considering the grid in they andzdirections.

The weighting coefficientsa(r)ip , b(r)ip andc(r)ip , for r ≥ 2,can be computed by using the following Shu’s recursiveformulae [21]:

a(r)i j = r

a(1)i j a(r−1)ii −

a(r−1)i j

xi − x j

, i 6= j : i, j ∈ ∆x,

a(r)ii =−Nx

∑i=1,i 6= j

a(r)i j , i = j : i, j ∈ ∆x.

b(r)i j = r

b(1)i j b(r−1)ii −

b(r−1)i j

yi − y j

, i 6= j : i, j ∈ ∆y

b(r)ii =−Ny

∑i=1,i 6= j

b(r)i j , i = j : i, j ∈ ∆y.

c(r)i j = r

c(1)i j c(r−1)ii −

c(r−1)i j

zi − zj

, i 6= j : i, j ∈ ∆z

c(r)ii =−Nz

∑i=1,i 6= j

c(r)i j , i = j : i, j ∈ ∆z.

(12)

3 The numerical scheme of MCB-DQM

Setting ∂u∂ t = v and thus ∂ 2u

∂ t2= ∂v

∂ t , andf (xi ,y j ,zk, t) = fi jk , the numerical scheme transforms


www.naturalspublishing.com/Journals.asp


Eqs. (1)-(2) into the following problem:

dui jk

dt= vi jk

dvi jk

dt=

Nx

∑p=1

a(2)ip up jk+Ny

∑p=1

b(2)jp uipk+Nz

∑p=1

c(2)kp ui jp +Ki jk

ui jk(t = 0) = ψ1(xi ,y j ,zk),

vi jk(t = 0) = ψ2(xi ,y j ,zk),(13)

wherei ∈ ∆x, j ∈ ∆y,k∈ ∆z and

Ki jk = (δg(ui jk)−α)vi jk −βui jk + fi jk .

Bearing the boundary condition (3) in mind, Eq. (13) isrewritten as follows:

dui jk

dt= vi jk

dvi jk

dt=

Nx−1

∑p=2

a(2)ip up jk+Ny−1

∑p=2

b(2)jp uipk+Nz−1

∑p=2

c(2)kp ui jp +Fi jk

ui jk(t = 0) = ψ1(xi ,y j ,zk),

vi jk(t = 0) = ψ2(xi ,y j ,zk),(14)

where 2≤ i ≤ Nx−1,2≤ j ≤ Ny−1,2≤ k≤ Nz−1 and

Fi jk = Ki jk +a(2)i1 u1 jk +a(2)iNxuNx jk +b(2)j1 ui1k

+b(2)jNyuiNyk+ c(2)k1 ui j 1+ c(2)k,Nz

ui jNz.(15)

Various numerical schemes have been proposed to solveinitial value problems, among others, the SSP-RK schemeallows low storage and large region of absolute property[29,28]. In particular in what follows we consider thefollowing SSP-RK54 scheme which is strongly stable fornonlinear hyperbolic differential equations:

u(1) = um+0.391752226571890tL(um)

u(2) = 0.444370493651235vm+0.555629506348765u(1)

+0.368410593050371tL(u(1))

u(3) = 0.620101851488403um+0.379898148511597u(2)

+0.251891774271694tL(u(2))

u(4) = 0.178079954393132um+0.821920045606868u(3)

+0.544974750228521tL(u(3))

um+1 = 0.517231671970585u(2)+0.096059710526147u(3)

+0.063692468666290tL(u(3))+0.386708617503269u(4)

+0.226007483236906tL(u(4))

4 Stability Analysis

In what follows, we assumeα > δg. The system (14) canbe rewritten in compact form as follows:

dUdt

= AU+G,

or

ddt

[

uv

]

=

[

O IB (δg−α)I

][

uv

]

+

[

O1F

]

(16)

wherea)O andO1 are null matrices;b)I is the identity matrix of order(Nx−2)(Ny−2)(Nz−

2);c)U = (u,v)T the vector solution at the grid points:

u= (u222,u223, . . . ,u22(Nz−1),u232,u233, . . . ,u23(Nz−1),

. . . ,u(Nx−1)(Ny−1)3, . . . ,uNx−1)(Ny−1)(Nz−1)).v= (v222,v223, . . . ,v22(Nz−1),v232,v233, . . . ,v23(Nz−1),. . . ,v(Nx−1)(Ny−1)3, . . . ,vNx−1)(Ny−1)(Nz−1)).

d)F = (F222,F223, . . . ,F22(Nz−1),F232,F233, . . . ,F23(Nz−1),. . . ,F(Nx−1)(Ny−1)3, . . . , FNx−1)(Ny−1)(Nz−1)), whereFi jk

is defined in Eq. (15).e)B= −β I +Bx+By+Bz, whereBx, By andBz are the

following matrices (of order(Nx−2), (Ny−2), (Nz−

2), respectively) of the weighting coefficientsa(2)i j , b(2)i j

andc(2)i j :

Bx =

a(2)22 Ix a(2)23 Ix . . . a(2)2(Nx−1)Ix

a(2)32 Ix a(2)33 Ix . . . a(2)3(Nx−1)Ix...

.... . .

...

a(2)(Nx−1)2Ix a(2)

(Nx−2)3Ix . . . a(2)(Nx−1)(Nx−1)Ix

(17)

By =

My Oy . . . OyOy My . . . Oy...

.... . .

...Oy Oy . . . My

Bz =

Mz Oz . . . OzOz Mz . . . Oz...

.... . .

...Oz Oz . . . Mz

(18)where

My =

b(2)22 Iz b(2)23 Iz . . . b(2)2(Ny−1)Iz

b(2)32 Iz b(2)33 Iz . . . b(2)3(M−1)Iz...

.... . .

...

b(2)(Ny−1)2Iz b(2)

(Ny−1)3Iz . . . b(2)(Ny−1)(Ny−1)Iz

and

Mz =

c(2)22 c(2)23 . . . c(2)2(Nz−1)

c(2)32 c(2)33 . . . c(2)3(Nz−1)...

.... . .

...

c(2)(Nz−1)2 c(2)

(Nz−1)3 . . . c(2)(Nz−1)(Nz−1)



−140 −120 −100 −80 −60 −40 −20 0−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Real

Ima

gin

ary

Nx ×Ny ×Nz = 6 × 6 × 6

−700 −600 −500 −400 −300 −200 −100 0−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Real

Ima

gin

ary

Nx ×Ny ×Nz = 11 × 11 × 11

−1500 −1000 −500 0−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Real

Ima

gin

ary

Nx ×Ny ×Nz = 16 × 16 × 16

−140 −120 −100 −80 −60 −40 −20 0−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Real

Ima

gin

ary

Nx ×Ny ×Nz = 6 × 6 × 6

−700 −600 −500 −400 −300 −200 −100 0−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Real

Ima

gin

ary

Nx ×Ny ×Nz = 11 × 11 × 11

−1500 −1000 −500 0−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Real

Ima

gin

ary

Nx ×Ny ×Nz = 16 × 16 × 16

−140 −120 −100 −80 −60 −40 −20 0−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Real

Ima

gin

ary

Nx ×Ny ×Nz = 6 × 6 × 6

−700 −600 −500 −400 −300 −200 −100 0−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Real

Ima

gin

ary

Nx ×Ny ×Nz = 11 × 11 × 11

−1500 −1000 −500 0−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Real

Ima

gin

ary

Nx ×Ny ×Nz = 16 × 16 × 16

Fig. 1: Eigenvalues ofBx (first row),By (second row) andBz (third row) for different values of the grid points.

where Oy and Oz are null matrices of order(Ny −2)(Nz−2) and(Nz−2), respectively;Ix and Izare the identity matrices of order(Ny−2)(Nz−2) and(Nz−2), respectively.

The stability of the numerical scheme proposed for (1)depends on the stability of the system of ODEs defined in(16). If the system of ODEs (16) is unstable, then thenumerical scheme for temporal discretization may notconverge. Since the exact solution can be directlyobtained by means of the eigenvalues method, thestability of (16) depends on the eigenvalues of thecoefficient matrixA. Accordingly the system (16) is stableif the real part of each eigenvalue ofA is zero or negative.

Let λA be an eigenvalue ofA associated with theeigenvector(X1,X2)

T , where each component is a vectorof order(Nx−2)(Ny−2)(Nz−2). Then from Eq. (16) wehave

A

[

X1X2

]

=

[

O IB (δg−α)I

][

X1X2

]

= λA

[

X1X2

]

, (19)

which implies that

IX2 = λAX1, (20)

andBX1+(δg−α)X2 = λAX2. (21)

Simplifying Eq. (20) and Eq. (21), we get

BX1 = λA(λA+α − δg)X1. (22)

This shows that the eigenvalueλB of B is λB = λA(λA+α − δg). We now consider the matrix

B=−β I +Bx+By+Bz, (23)

and we compute the eigenvalues ofBx,By and Bz fordifferent grid points: 6× 6 × 6, 11× 11 × 11 and16× 16× 16. As Fig1 shows, for different values of thegrid points the computed eigenvalues ofBx,By andBz arereal negative numbers. Sinceβ > 0, from Eq.(23) wehave

Re(λB)≤ 0 and Im(λB) = 0, (24)

whereRe(z) andIm(z) denote the real and the imaginarypart ofz, respectively. LetλA = x+ ιy, then

λB = λA(λA+α − δg)

= x2− y2+(α − δg)x+ ι(2x+(α− δg))y.(25)

According to Eq. (24) and Eq. (25), we have

x2− y2+(α − δg)x< 0

(2x+(α − δg))y= 0(26)

The possible solutions of Eq.(26) are

1)If y 6= 0, thenx=−α−δg2 ,

2)If y= 0, then(

x+ (α−δg)2

)2<(

(α−δg)2

)2.

The proposed scheme is thus stable ifα > δg.

5 Numerical experiments

This section is devoted to the accuracy analysis of theproposed numerical method. Specifically three test casesof (1) are taken into account. The accuracy and




0.00050.0005

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y

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0.60.7

0.80.9

10

1

2

3

4

5

6

x 10−3

xy

Ab

so

lute

Err

or

Fig. 2: The contour plot (left panel) and surface plot (right panel)of the absolute error in the three-dimensional telegraph equation ofthe Problem1 for z= 0.5, t = 1, h= 0.044,t = 0.01.

0.01

0.01

0.01

0.0

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0.60.7

0.80.9

10

0.02

0.04

0.06

0.08

0.1

xy

Ap

pro

x. S

oln

Fig. 3: The contour plot (left panel) and surface plot (right panel)of the numerical solution of the three-dimensional telegraph equationof the Problem1 for z= 0.5, t = 1, h= 0.04,t = 0.01.

consistency of the method is performed by consideringthe following RMS error norm:

RMS=

Nx

∑i=1

Ny

∑j=1

Nz

∑k=1

∣

∣ui jk −u∗i jk∣

∣

2

NxNyNz

12

whereui jk andu∗i jk denote the numerical solution and theexact solution at(xi ,y j ,zk), respectively.

Problem 1.The first test case deals with thethree-dimensionallinear telegraph hyperbolicequation[4,16], which corresponds toδ = 0, α = β = 2. Weconsider as exact solution of the equation (1)-(3) thefollowing function:

u(x,y,z, t)= sinh(x)sinh(y)sinh(z)e−2t ,(x,y,z)∈Ω , t ≥ 0,

with ψ1(x,y,z),ψ2(x,y,z), ξ (x,y,z, t), and f (x,y,z, t)obtained accordingly.

The numerical solution for the problem1 is obtainedfor t = 0.01 and grid size 11× 11× 11. Table 1summarizes the RMS error obtained with the

MCB-DQM, the MLPG [16] and the EFP [16]. The Fig.2shows the RMS for the MCB-DQM solution forz= 0.5,grid size 26× 26× 26 and at timet = 1.0; the Fig. 3shows the MCB-DQM solution forz = 0.5, grid size25×25×25 and at timet = 1.0. The numerical solutionis in good agreement with the exact solution.

Problem 2.The second test case deals with theVan der Polnonlinear wave equation [16,4], which corresponds toα =δ = κ ,β = 0 andg(u) = u2. We consider as exact solutionof the equation (1)-(3) the following function:

u(x,y,z, t) = sin(x)sin(y)sin(z)e−κt ,(x,y,z) ∈ Ω , t ≥ 0.(27)

with ψ1(x,y,z), ψ2(x,y,z), ξ (x,y,z, t) and f (x,y,z, t)defined accordingly.

The numerical solution of the Problem2 is obtainedfor κ = 3, t = 0.01 and grid size 11× 11× 11. Table2 summarizes the RMS error norm for the MCB-DQM,the MLPG [16] and the EFP [16]. The Fig.4 shows theabsolute error for the MCB-DQM solution forz= 1.0, gridsize 25×25×25 and at timet = 1.0. The contour plot andthe surface plots of the MCB-DQM solution are depictedin Fig.5 and in Fig.6, respectively. The numerical solutionis in good agreement with the exact solution.



x

y

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1.6

1.8

2

x 10−19

xy

Ab

so

lute

Err

or

Fig. 4: The contour plot (left panel) and surface plot (right panel)of the absolute error in the three-dimensional nonlinear Van der Polequation of the Problem2 for z= 1.0, t = 1, h= 0.04, andt = 0.01.

1e−018

1e

−0

18

1e−018

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018

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018

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−0

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018

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−0

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018

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018

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−0

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2e−0182e−018

3e−018

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018

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018

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018

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018

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18

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018

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x

y

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1

Fig. 5: The contour plot of the exact solution (left panel) and of theMCB-DQM solution (right panel) for the three-dimensionalnonlinear Van der Pol equation of the Problem2 for z= 1.0, t = 1, h= 0.04,t = 0.01.

00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

0

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0.6

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10

1

2

3

4

5

6

7

x 10−18

xy

Exa

ct S

oln

0

1

2

3

4

5

6x 10

−18

00.1

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0.40.5

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1

0

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0.4

0.6

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10

1

2

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4

5

6

7

x 10−18

xy

Ap

pro

x. S

oln

0

1

2

3

4

5

6

x 10−18

Fig. 6: The surface plot of the exact solution (left panel) and numerical solution (right panel) of the three-dimensional nonlinear Vander Pol equation of the Problem2 for z= 1, t = 1, h= 0.04,t = 0.01.

Problem 3.The third test case is devoted to thethree-dimensional nonlinear wave equation inthedissipative form, which corresponds toα = β = 0,δ = −2 andg(u) = u. We consider as exact solution ofthe equation (1)-(3) the following function:

u(x,y,z, t) = sin(t)∏x,y,z

sin(πx),(x,y,z) ∈ Ω , t ≥ 0, (28)

with ψ1(x,y,z), ψ2(x,y,z), ξ (x,y,z, t) and f (x,y,z, t)defined accordingly.

The numerical solution of the Problem3 is obtainedfor t = 0.01 and grid size 11× 11× 11. Table 3summarizes the RMS error norm obtained with theMCB-DQM, the MLPG[16] and the EFP[16]. The Fig.7depicts the absolute error for the MCB-DQM solution forz= 0.5, grid size 25× 25× 25 and at timet = 1.0. The

contour plot and the surface plot of the MCB-DQMsolution are shown in Fig.8 and 9, respectively. Thenumerical solution is in good agreement with the exactsolution.

6 Conclusions

The present paper is concerned with the definition of anew numerical method based on the MCB-DQM for thederivation of numerical solutions for partial differentialequations in three-dimensional space. The main aim is toimprove the accuracy of the numerical solutions, whichrelies on the strong and efficient implementation of themethod. The large computational cost is the maindrawback of almost all methods available in the literature




Table 1: The RMS error norm for the MCB-DQM, the MLPG [16] and the EFP [16] (t = 0.01 and grid size 11×11×11) for theProblem1.

t MCB-DQM MLPG[16] EFP[16] CPU time (seconds)

0.1 1.01409e-006 6.389040e-004 1.361376e-001 0.0770.2 1.66777e-006 1.621007e-003 1.108673e-001 0.1400.3 1.72678e-006 2.069397e-003 9.031794e-002 0.2070.4 1.49946e-006 1.851491e-003 7.555177e-002 0.2660.5 1.19699e-006 1.406413e-003 6.113317e-002 0.3260.6 9.06725e-007 1.120239e-003 5.076050e-002 0.3850.7 7.06686e-007 8.762877e-004 4.276296e-002 0.4440.8 5.57041e-007 5.762842e-004 3.416178e-002 0.5050.9 4.76172e-007 7.778958e-004 3.072394e-002 0.5651.0 4.42082e-007 8.638225e-004 2.562088e-002 0.624

Table 2: The RMS error norm for the MCB-DQM, the MLPG [16] and the EFP [16] (t = 0.01 and grid size 11×11×11) for theProblem2

t MCB-DQM MLPG[16] EFP[16] CPU time(seconds)

0.1 5.67101e-006 2.777931e-003 1.653265e+000 0.1400.2 9.70224e-006 8.477482e-003 1.005632e+000 0.2500.3 1.23148e-005 1.352534e-002 9.786343e-001 0.3700.4 1.51181e-005 1.583307e-002 7.456237e-001 0.4900.5 1.82388e-005 1.550351e-002 6.213675e-001 0.6100.6 2.22188e-005 1.367202e-002 4.354421e-001 0.7300.7 2.57046e-005 1.052578e-002 1.345213e-001 0.8510.8 2.86607e-005 6.216680e-003 9.973233e-002 0.9710.9 3.11707e-005 5.280951e-003 7.132423e-002 1.0911.0 3.32916e-005 2.276681e-003 6.124572e-002 1.211

Table 3: The RMS error norm for the MCB-DQM, the MLPG [16] and the EFP [16] (t = 0.01 and grid size 11×11×11) for theProblem3.

t MCB-DQM MLPG[16] EFP[16] CPU time(seconds)

0.1 2.90131e-007 8.903029e-005 1.435666e-003 0.1400.2 1.25781e-006 9.910264e-005 3.867576e-003 0.2400.3 2.94185e-006 1.590358e-004 5.033494e-003 0.3600.4 5.34148e-006 3.776687e-004 7.655177e-003 0.4800.5 8.77107e-006 4.781290e-004 9.119769e-003 0.6120.6 1.35793e-005 6.416380e-004 1.034540e-002 0.7320.7 2.02462e-005 8.809498e-004 3.279875e-002 0.8520.8 2.91101e-005 9.279331e-004 5.233178e-002 0.9720.9 4.03845e-005 1.059260e-004 6.072234e-002 1.0821.0 5.41878e-005 1.529316e-003 7.545088e-002 1.202

for the solution of three-dimensional partial differentialequations.

The analysis of the accuracy and effectiveness of themethod is performed by considering three test cases: thethree-dimensionallinear telegraphic equation, the Vander Pol type nonlinear wave equation and the dissipativenonlinear wave equation. The analysis of the root mean

square error shows that the MCB-DQM solutions aremore accurate of the numerical solutions obtained withthe existing methods of the pertinent literature [16].

Research perspectives include the possibility todevelop further refinements of the method proposed in thepresent paper for the derivation of numerical solutions forkinetic equations [33] and specifically for thermostatted



0.0

005

0.0

00

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xy

Ab

so

lute

Err

or

0

0.5

1

1.5

2

2.5

3

3.5

4

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5

x 10−3

Fig. 7: The contour plot (left panel) and surface plot (right panel)of the absolute error in the three-dimensional nonlinear wave equationin dissipative form of the Problem3 for z= 0.5, t = 1, h= 0.04, andt = 0.01.

0.1

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Fig. 8: The contour plot of the exact solution (left panel) and of theMCB-DQM solution (right panel) for the three-dimensionalnonlinear wave equation in dissipative form of the Problem3 for z= 0.5, t = 1, h= 0.04,t = 0.01.

00.1

0.20.3

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xy

Exa

ct

So

ln

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xy

Ap

pro

x. S

oln

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Fig. 9: The surface plot of the exact solution (left panel) and of thenumerical solution (right panel) of the three-dimensionalnonlinearwave equation in dissipative form of the Problem3 for z= 0.5, t = 1, h= 0.04,t = 0.01.

kinetic equations [34] that have been recently proposedfor the modeling of complex systems.

References

[1] C. Bianca, F. Pappalardo and S. Motta, Computers &Mathematics with Applications 58, 579-588 (2009).

[2] C. Bianca and S. Motta, Computers & Mathematics withApplications 57, 831-840 (2009).

[3] G. Arora, R.C. Mittal and B.K. Singh, J. Engg. Sci. Tech.9(3), 104-116 (2014) special issue on ICMTEA 2013conference.

[4] R. K. Mohanty and V. Gopal, Appl. Math. Model. 37, 2802-2815 (2013).

[5] Z. Zhang, D. Li, Y. Cheng and K. Liew, Acta Mech. Sin.28(3), 808-818 (2012).

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Society 22, 280-285 (2014).[8] R. C. Mittal and R. Bhatia, Appl. Math. Comput. 220, 496-

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(2014).[10] R. K. Mohanty, K. George and M. K. Jain, Int. J. Comput.

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[11] R. K. Mohanty, U. Arora and M. K. Jain, Appl. Math.Comput. 17, 277-289 (2001).

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[13] R. K. Mohanty, Appl. Math. Comput. 162, 549-557 (2005).[14] R. K. Mohanty, S. Singh and S. Singh, Appl. Math. Comput.

232, 529-541 (2014).[15] V. A. Titarev and E. F. Toro, J. Comput. Phy. 204, 715-736

(2005).[16] E. Shivanian, Engg. Analysis with Boundary Elements 50,

249-257 (2015).[17] R. Bellman, B.G. Kashef and J. Casti, J Comput Phy 10,

40-52 (1972).[18] R. Bellman, B. Kashef and E. S. Lee, R. Vasudevan,

Pergamon, Oxford 371–376 (1976).[19] J.R. Quan and C.T. Chang, Comput Chem Eng 13, 779-788

(1989).[20] J.R. Quan and C.T. Chang, Comput Chem Eng 13, 1017-

1024 (1989).[21] C. Shu, Differential Quadrature and its Application in

Engineering, Athenaeum Press Ltd., Great Britain, 2000.[22] A. Korkmaz andI. Dag, Eng. Comput. Int. J. Comput. Aided

Eng. Software 30(3), 320-344 (2013).[23] C. Shu C and Y. T. Chew, Commun. Numer. Methods Eng.

13(8), 643-653 (1997).[24] C. Shu and H. Xue, J. Sound Vib. 204(3), 549-555 (1997).[25] C. Shu and B. E. Richards, Int J Numer Meth Fluids 15,

791-798 (1992).[26] G. Arora and B. K. Singh, Appl. Math Comput 224, 166-177

(2013).[27] R. Jiwari and J. Yuan, J. Math. Chem. 52(6), 1535-1551

(2014).[28] J. R. Spiteri and S. J. Ruuth, SIAM Journal Numer Anal

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38, 251-289 (2009).[30] B. K. Singh, G. Arora and M. K.

Singh, J. Egyp. Math. Society (2016),http://dx.doi.org/10.1016/j.joems.2015.11.003.

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[32] W. Lee, Tridiagonal matrices: Thomas algorithm,http://www3.ul.ie/wlee/ms6021−thomas.pdf, ScientificComputation, University of Limerick.

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Brajesh K. Singhreceived the Masterof Science degree inMathematics in 2005 fromCSJM University Kanpur.He received the Ph. D. degreein Mathematics from IndianInstitute of TechnologyRoorkee in 2012. Currently,he is an Assistant Professor in

Babasaheb Bhimarao Ambedkar University, LucknowINDIA. His research interests are in the areas of appliedmathematics including numerical analysis, mathematicalsimulation and cryptology. He has published researchpapers in reputed international journals of mathematicaland engineering sciences. He is referee of variousmathematical journals.

Carlo Biancareceived the PhD degree inMathematics for EngineeringScience at PolytechnicUniversity of Turin. Hisresearch interests arein the areas of appliedmathematics and in particularin mathematical physicsincluding the mathematical

methods and models for complex systems, mathematicalbilliards, chaos, anomalous transport in microporousmedia and numerical methods for kinetic equations. Hehas published research articles in reputed internationaljournals of mathematical and engineering sciences. He isreferee and editor of mathematical journals. He is editorin chief of the Journal of Mathematics and Statistics.


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