Applied Mathematics and Computation 313 (2017) 442–452
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Numerical simulation of three-dimensional telegraphic equation using cubic B-spline differential quadrature method R.C. Mittal, Sumita Dahiya∗ Indian Institute of Technology, Roorkee, India
a r t i c l e
i n f o
Keywords: Telegraph equation Cubic B-spline functions Modified cubic B-spline differential quadrature method Thomas algorithm SSP–RK43 method
a b s t r a c t This paper employs a differential quadrature scheme that can be used for solving linear and nonlinear partial differential equations in higher dimensions. Differential quadrature method with modified cubic B-spline basis functions is implemented to solve threedimensional hyperbolic equations. B-spline functions are employed to discretize the space variable and their derivatives. The weighting coefficients are obtained by semi-explicit algorithm. The partial differential equation results into a system of first-order ordinary differential equations (ODEs). The obtained system of ODEs has been solved by employing a fourth stage Runge–Kutta method. Efficiency and reliability of the method has been established with five linear test problems and one nonlinear test problem. Obtained numerical solutions are found to be better as compared to those available in the literature. Simple implementation, less complexity and computational inexpensiveness are some of the main advantages of the scheme. Further, the scheme gives approximations not only at the knots but also at all the interior points in the domain under consideration. The scheme is found to be providing convergent solutions and handles different cases. High order time discretization using SSP–RK methods guarantee stability with respect to a given norm and a proper constraint on time step. Matrix method has been used for stability analysis in space and it is found to be unconditionally stable. The scheme can be used effectively to handle higher dimensional PDEs. © 2017 Elsevier Inc. All rights reserved.
1. Introduction In this paper, we consider the three-dimensional hyperbolic equations that are being widely used in many different fields of science and mathematical engineering such as vibration of structures [1], random walk theory [2] and transmission and propagation of electric signal [3]. In the recent scenario, the communication system plays a significant part in social and industrial periphery. In the industrial arena, a major role is played by the radio frequency (RF) and microwave frequency (MW) systems that generate high frequency. These system employ the transmission media for point to point transfer of information carrying signals. We can possibly divide the transmission media in two types, namely as unguided and guided. For unguided medium, the signals get carried by the electro-magnetic waves over parts of the entire path of communication by MW and RF channels. Antenna is used to transmit and receive these electromagnetic waves. The guided media are able to transport the current waves and high frequency voltage, as in this medium the signals are transferred through the coaxial cable or transmission line. ∗
Corresponding author. E-mail addresses:
[email protected] (R.C. Mittal),
[email protected] (S. Dahiya).
http://dx.doi.org/10.1016/j.amc.2017.06.015 0 096-30 03/© 2017 Elsevier Inc. All rights reserved.
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In the guided transmission media, the problem of efficient telegraph transmission is addressed by investigating cable transmission medium. A cable transmission medium classified as a guided transmission medium acts as a physical system that directly propagates the transmission between to or more different locations. For optimizing the guided communication system, the determination or projection of power and signal losses are to be ensured in the system, because every system has these types of losses. The necessity to minimize these losses and for the maximization of the output, formulation of some equation is required. Consider the hyperbolic equation in three dimensions.
∂ 2v ∂ 2v ∂ 2v ∂ 2v ∂v + 2α + β 2 v = A 2 + B 2 + C 2 + f (x, y, z, t ) 2 ∂t ∂t ∂x ∂y ∂z
(1)
in the domain = {(x, y, z, t)|0 < x, y, z < 1, t > 0}, where A, B, C, α , β are positive functions of x, y, and z. For A = B = C = 1, the given equation is known as telegraph equation. Initially
∂v (x, y, z, 0 ) = v2 (x, y, z ) ∂t
v(x, y, z, 0 ) = v1 (x, y, z );
(2)
conditions on the boundary are given as
v(0, y, z, t ) = a1 (y, z, t ) v(x, 0, z, t ) = b1 (x, z, t ) v(x, y, 0, t ) = c1 (x, y, t )
v(1, y, z, t ) = a2 (y, z, t ) v(x, 1, z, t ) = b2 (x, z, t ) v(x, y, 1, t ) = c2 (x, y, t ).
(3)
Telegraph equation is found to be much more appropriate than ordinary diffusion equations for modeling of reaction– diffusion phenomenon. The vibration of structures, for e.g. buildings and machines, is modeled through hyperbolic PDE, which are the basis of atomic physics. Problems of wave propagation, random walk theory and signal analysis are significantly modeled through telegraph equation [2,3]. In the recent years, many numerical methods e.g. finite element, finite difference and collocation methods have been developed to solve the linear and non-linear hyperbolic equations. Onedimensional telegraph and fractional telegraph equations of order two has been dealt with by Dehghan et al. [4–6] using DRBIE method, Chebyshev cardinal functions and He’s variational method. Abbas and Dehghan [7] used Chebyshev tau’s method, whereas Mohebbi et al. [8] used meshless method of radial basis functions for numerical solution of telegraph and fractional telegraph equations respectively. Mohanty and Jain [9] developed an ADI scheme for two-dimensional linear hyperbolic equation. Again Mohanty et al. [10] have given an ADI scheme for three-dimensional linear hyperbolic equation. Dehghan and Shokri [11,12] and Dehghan and Nikpour [13] introduced a computational scheme to solve the hyperbolic systems in one and two dimensions by applying collocation method. The solutions were approximated taking the thin-plate spline radial basis functions. Dehghan et al. [14,15] developed two mesh-less methods, MLPG and MLWS to solve the telegraph equation in two dimensions. MLS approximation technique was employed to construct the shape functions for each of the methods. To deal with time derivatives, a time stepping scheme is used. For approximation of two-dimensional telegraph equation of order two, explicit group relaxation techniques of unconditional stability built upon the consolidation of centered and rotated five-point finite difference approximation were proposed by Ali and Kew [16] on different grid spacing. The scheme is of second order accuracy and was proven to require the least computing effort and execution timings compared to the other explicit group methods belonging to same class [17]. They have extended the same idea for three-dimensional telegraph equations [18]. Mohanty [19] also proposed operator splitting method for this class of equation. In this paper, we extend the modified cubic B-spline differential quadrature method to formulate the solution of three-dimensional linear and non linear hyperbolic equations. The paper is coordinated in six sections. Section 2 elaborates the use of the modified cubic B-spline differential quadrature method. In Section 3, the method is applied to the considered equation in three dimensions with the treatment of boundary conditions. Stability analysis is carried out in Section 4. Section 5 presents some test examples of the hyperbolic equations. A summary is given at the end of the paper in Section 6. 2. Differential quadrature method Differential quadrature method [20] is a numerical approach to approximate numerically the solutions of ordinary and partial differential equations. DQM approximates the derivatives of function at any given discrete points as a linear sum of functional values at points in the whole domain. Ease of application, accuracy and computational inexpensiveness make this method preferable. DQM has been widely experimented and its versatility has been established in variety of physical problems. By employing differential quadrature method, we are capable of approximating the derivatives of functions in space at any of the given knots with the weighted linear sum of function values at all knots in the domain under consideration. To apply DQM in three dimensions, firstly we discretize the domain = {(x, y, z ) : 0 ≤ x, y, z ≤ 1} as D1 = {(xi , y j , zr ), i = 1, 2, . . . , N; j = 1, 2, . . . , M; r = 1, 2, . . . L} by taking step length x = xi − xi−1 in the direction of x-axis, y = y j − y j−1 in y-axis direction and z = zr − zr−1 in z-axis direction. Approximation of the partial derivative of order one of the dependent function u(x, y, z, t) in the direction of x-axis, at point xi , keeping yj and zr fixed, is given as follows
ux ( xi , y j , zr , t ) =
N (1 ) aik u ( xk , y j , zr , t ), k=1
i = 1, 2, . . . , N
(4)
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R.C. Mittal, S. Dahiya / Applied Mathematics and Computation 313 (2017) 442–452 Table 1 Values of ϕ m (x) and its derivatives at the nodal points.
ϕ m (x) ϕm (x ) ϕm ( x )
xm−2
xm−1
xm
xm+1
xm+2
0 0 0
1 3/h 6/h2
4 0 −12/h2
1 3/h 6/h2
0 0 0
(1 ) where aik represent the the weighting coefficients to be determined to approximate the partial derivatives of order one in the direction of x. Various approaches are available in literature that can compute these weighting coefficients for e.g. Quan and Chang’s approach [21,22], Shu’s approach [23] etc. Korkmaz and Dagˆ [24,25] used DQM based on cosine expansion and Sinc DQM for many nonlinear PDEs. Lagrange interpolation and sine–cosine expansion based quadrature methods has been very frequently used in literature. A differential quadrature method based on polynomials has been suggested by Mittal et al. [26] for numerical approximation of the the solutions of PDEs both linear and nonlinear in nature. The idea of iterative DQM has been introduced by Tomasiello [27]. In her work, iterative DQM and the stability of DQ solutions have been discussed. In the present work, weighting coefficients of DQM are determined using a methodology that employs modified cubic B-spline functions. In this work, we have used cubic B-splines as basis functions and weighting coefficients are computed accordingly.
2.1. Modified cubic B-spline functions Computation of weighting coefficients is a key procedure which is of paramount importance. In this work, the authors (1 ) have used computed the weighting coefficients aik and a¯ (jk1 ) by adopting modified cubic B-spline functions. These functions can be written as
⎧ (x − xm−2 )3 ⎪ ⎪ ⎨(x − xm−2 )3 − 4(x − xm−1 )3 1 ϕm (x ) = 3 (xm+2 − x )3 − 4(xm+1 − x )3 h ⎪ 3 ⎪ ⎩(xm+2 − x ) 0
x ∈ [xm−2 , xm−1 ) x ∈ [xm−1 , xm ) x ∈ [xm , xm+1 ) x ∈ [xm+1 , xm+2 ) otherwise
m = 0, 1, . . . , N + 1
(5)
Here, {ϕ0 (x ), ϕ1 (x ), . . . , ϕN (x )} defines a basis over the computational domain [a, b]. Table 1 comprises the values of original cubic B-splines and their derivatives at the knots. The B-spline functions are modified in such a manner with which the diagonal dominance of the resultant matrix can be preserved. With this modification, number of B-spline functions equates to the number of knot points in the considered domain. The boundary conditions can be implemented effectively. Mittal and Jain [28] define the modified cubic B-spline basis functions at the knot points as
φ1 ( x ) = ϕ 1 ( x ) + 2 ϕ 0 ( x ) , φ2 ( x ) = ϕ 2 ( x ) − ϕ 0 ( x ) , φl ( x ) = ϕ l ( x ) , l = 3 , 4 , . . . , N − 2 , φN−1 = ϕN−1 (x ) − ϕN+1 (x ) φN (x ) = ϕN (x ) + 2ϕN+1 (x )
(6)
The function φl (x ), l = 1, 2, . . . , N again form a basis over the interval [a, b]. 2.2. Determination of weighting coefficients To determine the weighting coefficients aik and a¯ jk , using modified B-spline functions φm (xi ), m = 1, 2, . . . , N in Eq. (4), gives
φm (xi ) =
N
aik φm (xk ),
i = 1, 2, . . . , N
k=1
(1 ) To compute the weighting coefficients aik , we fix y-axis in Eq. (4)[29]. “Thomas algorithm” is adopted to solve the resultant
(1 ) (1 ) tri-diagonal system and the weighting coefficients a11 , a12 , . . . , a1(1N) are computed. Similarly, the weighting coefficients for each knot i = 2, 3, . . . , N can be computed. With these coefficients, we can approximate partial derivatives of order one in the direction of x-axis. The following recurrence relation [23] is used to compute derivatives of higher order. (r )
(1 ) (r−1 )
ai j = r ai j aii
i, j = 1, 2, . . . , N;
−
ai(jr−1) xi − x j
, f or i = j
r = 2, 3, . . . , N − 1
(7)
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1 N=5 M=5 L=5
0.8 0.6 0.4
Imaginary
0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −36,000
−30,000
−24,000
−18,000 Real
−12,000
−6,0000
0
1 N = 10 M = 10 L = 10
0.8 0.6 0.4
Imaginary
0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1,05,000
−90,000
−75,000
−60,000
−45,000 Real
−30,000
−15,000
0
Fig. 1. Eigen values of matrix D using different grid points.
aii(r ) = −
N
ai(jr ) ,
f or i = j.
(8)
j=1, j=i
Here ai(jr−1 ) represents the weighting coefficients of partial derivatives of order (r − 1 ) w.r.t. x and ai(jr ) represents the weighting coefficients of derivatives of order (r) w.r.t. x. Following the same methodology, we can determine the weighting coefficients a¯ (jk1 ) of partial derivatives of order one in the direction of y-axis and z-axis using cubic B-spline functions in modified form. Recurrence formula can be successfully applied to calculate the derivatives of order two and higher. 3. Numerical scheme For numerical treatment of hyperbolic equations, we substitute vt = w in Eq. . After this substitution, we get a system of equations given as
∂w ∂ 2v ∂ 2v ∂ 2v + 2α w + β 2 v = A 2 + B 2 + C 2 + f (x, y, z, t ) ∂t ∂x ∂y ∂z
(9a)
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∂v =w ∂t
(9b)
The initial conditions are given as
v(x, y, z, 0 ) = v1 (x, y, z );
w(x, y, z, 0 ) = v2 (x, y, z )
(10)
with boundary conditions
v(0, y, z, t ) = a1 (y, z, t ) v(x, 0, z, t ) = b1 (x, z, t ) v(x, y, 0, t ) = c1 (x, y, t )
v(1, y, z, t ) = a2 (y, z, t ) v(x, 1, z, t ) = b2 (x, z, t ) v(x, y, 1, t ) = c2 (x, y, t ).
Rewriting Eq. (1) in the first order system, the spatial derivatives are discretized using modified cubic B-spline differential quadrature method, we get the following system of non-linear ordinary differential equations
dwi jk = −2α wi jk − β 2 vi jk + dt dvi jk = wi jk , dt
N M l (2 ) air(2) v(xr , y j , zk , t ) + a(jr2) v(xi , yr , zk , t ) + akr v ( xi , y j , zr , t ) r=1
0 < xi , y j , zk < 1,
r=1
t>0
(11a)
r=1
(11b)
subject to initial conditions
v ( xi , y j , zk , 0 ) = v0 ( xi , y j , zk ), w ( xi , y j , zk , 0 ) = w0 ( xi , y j , zk ).
(12)
Whenever Dirichlet’s boundary conditions are given, they can be used directly at the boundary without any manipulation. In the case of Neumann or mixed boundary conditions, we can deal with them using modified cubic B-spline DQM. They are simplified into boundary point solutions. Dirichlet’s boundary conditions are given as
v(0, j, k ) = a1 (y j , zk , t ), v(1, j, k ) = a2 (y j , zk , t ) v(i, 0, k ) = b1 (xi , zk , t ), v(i, 1, k ) = b2 (xi , zk , t ) v(i, j, 0 ) = c1 (xi , y j , t ), v(i, j, 1 ) = c2 (xi , y j , t ). 4. Why SSP–RK method SSP methods [30,31] also sometimes termed as TVD (time variation diminishing)time discretization are used for solving a system of ODEs. In [32], a general Runge–Kutta method of order m is formulate as
v ( 0 ) = vn , v (i ) =
i−1
(αi,k v(k) + t βi,k L(v(k) )), αi,k ≥ 0, i = 1, . . . , m
k=1
v(n+1) = v(m) . For the sake of consistent system
(13) i−1
k=0
αi,k = 1. The hyperbolic conservation law,
∂u = − f ( u )x , ∂t where the spatial derivative, f(u)x is discretized using TVD finite difference with method of lines approximation results into a system of ODEs
du = L ( u ), dt We invariably assume that the discretization in space −L(u ) posses the attribute that when it combines with forward first-order Euler time discretization,
un+1 = un + tL(un ), then, for a small enough time step governed by the CFL condition,
t ≤ ctF E , the total variation in discrete solution does not increase in time, where c = mini,k βα i,k , provided β i, k L is replaced by βi,k L¯ i,k whenever β i, k is negative. The prime objectives multistep time discretization or high-order SSP Runge–Kutta is to achieve temporal accuracy of higher order and to maintain strong stability property.
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4.1. Stability analysis In classical FD methods, Von Neumann stability analysis could be done with much ease. Unlike classical FD methods, DQM discretized systems can not employ Von Neumann stability analysis. For this purpose, matrix stability or energy stability methods have been extensively studied in literature [33–35]. In the present manuscript, we have done a matrix stability analysis for the systems (11). In our computation we have used N = M = L. Equation representing a time dependent system is given as
∂f = l( f ) ∂t
(14)
with suitable initial-boundary conditions. Here, l represents a spatial nonlinear differential operator. After discretization with MCB–DQM and linearization of nonlinear term by assuming U(x) locally constant [36], Eq. (14) is transformed into system of ODEs in time
d{U } = [D]{U } + {b}. dt
(15)
Here, {U} is the unknown vector of the functional values at the intermediate points, vector {b} represents the nonhomogeneous part and the boundary conditions and D is the coefficient matrix [37]. We consider the system in Eq. (11), rewriting in compact form as
dU = DU + G. dt or
d w −2α = I dt u
(16)
A O
w O1 + u Gi (t )
where, O = null matrix, I = identity matrix, U represents a square matrix of order (N − 2 )(M − 2 )(L − 2 ) with weighting coefficients as its elements. U = [u, w]T is solution vector at interior knots, given by U = [u2,2,2 , . . . , u2,2,L−1 , u3,2,2 , . . . , u3,2,L−1 , . . . , uN−1,2,L−1 , . . . , uN−1,M−1,L−1 , . . . , w2,2,2 , . . . , w2,2,L−1 , w3,2,2 , . . . , w3,2,L−1 , . . . , wN−1,2,L−1 , . . . , wN−1,M−1,L−1 ]T , for N = M = L. G = [O1 , Gi (t)]T is a vector containing non-homogeneous part and boundary conditions, where O1 is null and Gi (t) is a column vector. The stability of the system in Eq. (16) directly relates to the stability of numerical method that is adopted to solve it. Any stable numerical scheme for temporal discretization may not achieve convergent solutions if the corresponding system of ODEs is unstable. The stability of system in Eq. (11) depend on the eigenvalues of coefficient matrix D, since its exact solution can be found using the eigenvalues. If all the eigenvalues of D are having real part the system in Eq. (11) will be stable. Let λ be any eigenvalue of D and X1 and X2 be two components (each of order (N−2)), of eigenvectors corresponding to eigenvalue λ. We have
−2α I
A O
X1 X =λ 1 X2 X2
(17)
or
−2α X1 + AX2 = λX1 IX1 = λX2 . From above system of equations, we get
−2αλX2 + AX2 = λ2 X2
(18)
AX2 = (λ2 + 2αλ )X2 .
(19)
⇒
( λ2
It shows that + 2αλ ) is an eigenvalue of A, and X2 is the corresponding eigenvector. We have computed the eigenvalues of the matrix A, by taking different grid points. For each eigenvalue μ of the matrix A, the quadratic equation given by
(λ2 + 2αλ ) = μ or
λ2 + 2αλ − μ = 0 is solved for λ for given values of α . For any α , the eigenvalue of the matrix D are real or imaginary depending upon whether α 2 + μ < 0 or α 2 + μ > 0. But the real part of the eigenvalues λ comes out to be negative. From the above set of equations it is clear that the real part of eigenvalues of D is always negative, i.e. the system in Eq. (11) is stable.
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R.C. Mittal, S. Dahiya / Applied Mathematics and Computation 313 (2017) 442–452 Table 2 Errors of example 1 with t = 0.001 and x = y = z = 1/20 for varying time. t
L2 error u
w
u
w
5 10 15
5.0130E−004 3.550 0E−0 06 1.2490E−008
2.840 0E−0 04 2.1020E−006 1.1279E−008
1.9019E−005 1.3430E−006 5.0740E−008
1.4710E−004 9.1918E−006 5.3220E−008
L∞ error
5. Numerical experiments This section elaborates the work of proposed method (MCB–DQM) on some test problems of telegraph equation . The illustration of the solutions has been done purposely for the same parametric values as used by other researchers for the sake of valid comparison. The computations has been done with the help of DEV C++. Existence of analytical solutions help to analyze the correctness and reliability of the proposed scheme. Error analysis is done by using L∞ , L2 and relative errors for the computed solutions defined below
L∞ (v ) = max |vi jk − Vi jk |, L2 ( v ) =
|vi jk − Vi jk |2
and
Relative error =
|vi jk − Vi jk |2 v2i jk
where vi jk = v(xi , y j , zk , t ) is the exact solution and Vi jk = V (xi , y j , zk , t ) is the approximate solution at the corresponding mesh points (xi , yj , zk ) at time t. 5.1. Linear hyperbolic equations In this section, we present numerical test for three-dimensional linear hyperbolic problems. We compared the calculated results with exact solutions whenever available or compared our results with the solutions already available in literature. Problem 1. (Constant coefficient problem (a)) [38]: Consider the 3-D linear telegraph equation
∂ 2v ∂v ∂ 2v ∂ 2v ∂ 2v + 2 + v = + + . ∂t ∂t2 ∂ x2 ∂ y2 ∂ z 2
(20)
The initial conditions are given as
v(x, y, z, 0 ) = sinh(x )sinh(y )sinh(z ),
(21)
∂v (x, y, z, 0 ) = −sinh(x )sinh(y )sinh(z ). ∂t
(22)
The exact solution is given by
v(x, y, z, t ) = e−t sinh(x )sinh(y )sinh(z ).
(23)
In this case we have substituted v + vt = w, so that we get the system of equations as
∂w ∂ 2v ∂ 2v ∂ 2v +w = + + + f (x, y, z, t ) ∂t ∂ x2 ∂ y2 ∂ z 2 ∂u = w − v. ∂t
(24a) (24b)
We have calculated the L2 and L∞ errors at time t = 5, 10, 15 with t = 0.001, x = y = z = 1/20 and that have been reported in Table 2. Problem 2. (Constant coefficient problem (b)) [18]: Consider the linear 3-D linear telegraph equation
∂ 2v ∂ 2v ∂ 2v ∂ 2v ∂v + 2α + β 2v = + + + f (x, y, z, t ) 2 ∂t ∂t ∂ x2 ∂ y2 ∂ z 2
(25)
where f (x, y, z, t ) = ((β 2 − 4 ) cos(t ) − 2 sin(t )) sinh(x ) sinh(y ) sinh(z ), α = 10 and β = 5. The initial condition are given as
v(x, y, z, 0 ) = sinh(x )sinh(y )sinh(z ),
(26)
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Table 3 Errors of example 2 with different step sizes and for varying time. Error
L2 error L∞ error Relative-error
N = 10
N = 16
N = 20
t=1
t=2
t=1
t=2
t=1
t=2
5.70 08E−0 04 1.120 0E−0 03
5.0974E−004 8.6344E−004
1.4645E−004 2.1278E−004 2.440 0E−0 04
5.9825E−005 4.9567E−005 1.2936E−004
1.32044E−004 1.7305E−004
5.5135E−005 1.9493E−005
Table 4 L2 errors given by Mohanty [19] and Kew and Ali [18] for t = 2 and different step sizes. t
L2 error, t = 2 Operator splitting method
Explicit group iterative method
N = 16 N = 128
3.280E−003 1.654E−004
4.515E−003 4.668E−004
Table 5 Errors of example 3 with t = 0.001 and x = y = z = 1/10 for varying time. Error
L2 error L∞ error Relative-error
N = 10
N = 16
N = 20
t=1
t=2
t=1
t=2
t=1
t=2
7.6603E–005 1.440 0E–0 04
2.9109E–005 5.3010E–005
4.4157E–005 6.9372E–005 1.1980E–004
1.6678E–005 2.5526E–005 1.2380E–004
3.6073E–005 4.15395E–005
1.3744E–005 1.5269E–005
∂u (x, y, z, 0 ) = 0. ∂t
(27)
The exact solution is given by
v(x, y, z, t ) = cos(t )sinh(x )sinh(y )sinh(z ),
(28)
In this case, we have substituted vt = w, so that we get the system of equations as
∂w ∂ 2v ∂ 2v ∂ 2v + 2α w + β 2 v = + + + f (x, y, z, t ) ∂t ∂ x2 ∂ y2 ∂ z 2 ∂v = w. ∂t
(29a) (29b)
We have computed the L2 and L∞ errors at time t = 1 and t = 2 with t = 0.001, x = y = z = 1/10, 1/16, 1/20, respectively. The results have been reported in Table 3. For comparison purposes, results obtained by Mohanty [19] and Kew and Ali [18] at t = 2 are also reported in Table 4 for different values of N. Our solutions are found to be better than those available in literature. Problem 3. (Constant coefficient problem (c)) [18]: Consider the linear 3-D linear telegraph equation in Eq. (25), with α = 10 and β = 5 and f (x, y, z, t ) = 3e−t sinh(x ) sinh(y ) sinh(z ). The initial condition are given as
v(x, y, z, 0 ) = sinh(x ) sinh(y ) sinh(z ),
(30)
∂v (x, y, z, 0 ) = − sinh(x ) sinh(y ) sinh(z ). ∂t
(31)
The exact solution is given by
v(x, y, z, t ) = e−t sinh(x ) sinh(y ) sinh(z ),
(32)
We have computed the L2 and L∞ errors at time t = 1 and t = 2 with t = 0.001, x = y = z = 1/10, 1/16 and 1/20 and the results have been reported in Table 5. The corresponding results obtained by Mohanty [19] and Kew and Ali [18] at t = 2 for different values of N are given in Table 6. Problem 4. (Variable coefficient problem) [19]: Consider the equation
∂ 2v ∂v ∂ 2v ∂ 2v ∂ 2v 2 + 2 e x+y+z + sin (x + y + z )v = (1 + x2 ) 2 + (1 + y2 ) 2 + (1 + z2 ) 2 + f (x, y, z, t ) 2 ∂t ∂t ∂x ∂y ∂z
(33)
with f (x, y, z, t ) = e−t sinh(x ) sinh(y ) sinh(z )(1 − 2ex+y+z + sin2 (x + y + z ) − (3 + x2 + y2 + z2 )). The exact solution and initial and boundary conditions are same as for Eq. (20). For N = 1/16, the error estimation has been performed and compared with those reported by Mohanty [19] in Table 7.
450
R.C. Mittal, S. Dahiya / Applied Mathematics and Computation 313 (2017) 442–452 Table 6 L2 errors given by Mohanty [19] and Kew and Ali [18] for t = 2 and different step sizes for Problem 3. t
L2 error, t = 2 Operator splitting method
Explicit group iterative method
N = 16 N = 128
1.3090E–004 6.0760E–005
6.5720E–004 7.6670E–005
Table 7 Errors of example 4 with t = 0.001 and x = y = z = 1/16 for varying time. t
1 2
L2 - error
L∞ - error
Relative-error
L2 - error, Mohanty [19]
u
w
u
w
u
u
2.0942E–004 7.5505E–005
2.3752E–004 7.5284E–005
2.4439E–004 8.8827E–005
2.5219E–004 8.8522E–005
5.1230E–004 5.0260E–004
1.1650E–003 4.2250E–004
Table 8 Errors of example 5 with t = 0.001 and x = y = z = 1/16 for varying time. t
1 2
L2 - error
Relative-error
L2 - error, Mohanty [19]
u
w
L∞ - error u
w
u
u
4.0781E–005 1.4897E–005
8.8476E–005 1.6122E–005
6.8497E–005 2.5177E–005
1.6397E–004 2.5973E–005
9.9775E–004 9.9174E–004
5.8970E–004 4.3540E–004
Problem 5. (Singular problem) [19]: Consider the equation
2 ∂ 2 v 2 ∂v 1 ∂ v ∂ 2v ∂ 2v 2 2 2 + + v = (1 + x + y + z ) + + + f (x, y, z, t ) ∂ t 2 x2 ∂ t x2 ∂ x2 ∂ y2 ∂ z 2
(34)
where f (x, y, z, t ) = (1 − x12 − 3(1 + x2 + y2 + z2 ))e−t sinh(x ) sinh(y ) sinh(z ). The exact solution and initial and boundary conditions are same as for Eq. (20). For N = 1/16, the error estimation has been performed and compared with those reported by Mohanty [19] in Table 8. 5.2. Non linear telegraph equation Problem 6.(Nonlinear problem)[38]: Consider the 3-D non linear telegraph equation
∂ 2v ∂v ∂ 2v ∂ 2v ∂ 2v +2 + v2 = + + + e(2(x+y+z−2t )) − e(x+y+z−2t ) ∂t ∂t2 ∂ x2 ∂ y2 ∂ z 2
(35)
The initial condition are given as
v(x, y, z, 0 ) = e(x+y+z) ,
(36)
∂v (x, y, z, 0 ) = −e(x+y+z) ∂t
(37)
The exact solution is given by
v(x, y, z, t ) = e(x+y+z−t ) ,
(38)
In this case, we have substituted vt = w, so that we get the system of equations as
∂w ∂ 2v ∂ 2v ∂ 2v + 2w + v2 = + + + e(2(x+y+z−2t )) − e(x+y+z−2t ) ∂t ∂ x2 ∂ y2 ∂ z 2 ∂v =w ∂t
(39a) (39b)
We have calculated the L2 and L∞ errors at time t = 5, 10, 15 with t = 0.001, x = y = z = 1/20 and the results have been reported in Table 9. 6. Conclusion In this work, differential quadrature method is employed that elaborates an approach to numerically approximate solutions for three-dimensional hyperbolic equations. The method efficiently provides accurate approximations for variable
R.C. Mittal, S. Dahiya / Applied Mathematics and Computation 313 (2017) 442–452
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Table 9 Errors of example 6 with t = 0.001 and x = y = z = 1/20 for varying time. t
L2 error u
w
u
w
5 10 15
2.0818E–002 1.0497E–004 4.3210E–007
3.4110E–003 3.930 0E–0 05 1.530 0E–0 07
1.1405E–002 6.250 0E–0 05 3.3690E–007
4.0710E–003 3.0510E–005 1.5320E–007
L∞ error
sets of parametric values. The efficiency and reliability of the approach have been evaluated for various test problems under consideration. Different error norms are calculated for varying times or the results present in literature are compared with the numerical solutions achieved by the used method. The approximated solutions and the corresponding error norms are presented in tables for varying time levels and for varying sets of parametric values. The results shows that error decreases with the increase in time. We see that the solutions becomes better with decreasing values of time step. The method gives convergent solutions and handles the equations efficiently even in nonlinear case. The obtained approximations are satisfactory and in compliance with the results available in the literature. Therefore, it is suggested that the method can be used as an alternative for the class of three-dimensional hyperbolic equations. 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