# A boundary value approach for solving three-dimensional elliptic and hyperbolic partial differential equations

- T. A. Biala
^{1}Email author and - S. N. Jator
^{2}

**Received: **18 July 2015

**Accepted: **16 September 2015

**Published: **9 October 2015

## Abstract

In this article, the boundary value method is applied to solve three dimensional elliptic and hyperbolic partial differential equations. The partial derivatives with respect to two of the spatial variables (*y*, *z*) are discretized using finite difference approximations to obtain a large system of ordinary differential equations (ODEs) in the third spatial variable (*x*). Using interpolation and collocation techniques, a continuous scheme is developed and used to obtain discrete methods which are applied via the Block unification approach to obtain approximations to the resulting large system of ODEs. Several test problems are investigated to elucidate the solution process.

### Keywords

Boundary value methods Method of lines Hyperbolic and elliptic PDEs Systems of ordinary differential equations### Mathematics Subject Classification

65N06 65N40## Background

*u*, the variable coefficients

*a*,

*b*,

*c*,

*d*,

*e*,

*f*and the forcing term G are assumed to be continuously differentiable and have the required partial derivatives on \(\Omega\).

The numerical approximation of (1) has received great attention from researchers in the past few decades. This is because most modeled physical and engineering processes results in PDEs of the form (1). The standard convection diffusion equation is obtained if \(a=b=c\ne 0\) and \(G\equiv G(x,y,z)\) for all \(x, y, z \in \Omega\) (see Ge and Zhang 2002). Many transport processes including fluid flows and heat transfer can be modeled by a convective diffusive equation which describes the convection and diffusion of various physical quantities such as heat and momentum (see Roache 1976). Several numerical schemes have been developed for (1) which include the compact difference scheme (Ge and Zhang 2002; Spotz 1995; Spotz and Carey 1996; Zhang 1998a, b, c, d; Zhang et al. 2000), compact alternating direct implicit scheme (Cui 2010; Liao and Sun 2010), finite difference methods (Jain et al. 1992; Mohanty and Singh 2006), method of lines (Brastos 2007; Dehghan and Shokri 2008) , Jacobi elliptic function method (Bhrawy and Abdelkawy 2013), the use of collocation and radial basis functions (Shakeri and Dehghan 2008), Riccati–Bernoulli sub ODE method (Yang et al. 2015), the use of the expansion methods (He 1998; Roshid and Rahman 2014; Alam et al. 2014; Roshid et al. 2013, 2014a, b, c) among others.

*y*and

*z*variable with mesh spacings

*k*and whose

*k*additional conditions are not only imposed at the beginning of the integration process but also at the end so that they form a discrete analog of the continuous boundary value problems. Thus, they are used for the numerical approximation of both initial and boundary value problems. They have been used for the solution of first order 1D initial and boundary value problems and their convergence and linear stability properties have been fully discussed in Brugnano and Trigiante (1998). Recently, Biala and Jator (2015) developed BVMs for the direct solution of systems of the general second order ODEs. One main feature of the BVMs is that they can be used in the same way for solving both initial and boundary value problems. Therefore, such methods are the best candidate for solving the semididscretized PDEs in (3). The BVMs simultaneously generates approximate solutions \((u_{1,m,n}, u_{2,m,n}, \ldots , u_{l,m,n})^T\) to the exact solution \((u_{m,n}(x_1), u_{m,n}(x_2), \ldots , u_{m,n}(x_l))^T\) on the entire interval of integration. This approach has the advantage of producing smaller global errors than those produced by the step-by-step methods due to the presence of accumulated errors at each step in the step-by-step method.

The paper is organized as follows: in "Description of the BVM", we derive a continuous scheme which is used to formulate the BVM as well as investigate the properties of the BVM. The computational complexities associated with the method is addressed in "Computational procedures". Several numerical test examples are given in "Numerical examples" to show the accuracy of the method. We give some concluding remarks in "Conclusion".

## Description of the BVM

### Development of the continuous BVM

In this section, we discuss the construction of the continuous scheme, via the interpolation and collocation approach (Jator and Li 2012), from which (6) is derived.

###
**Theorem 1**

*Let the continuous representation*

*satisfy the following conditions*

*then the continuous representation*(7)

*is equivalent to*

*where we define the matrix V as*

*is obtained by replacing the*

*jth*

*column of*

*V*

*by*

*W*

*where*

*T*

*denotes the transpose,*\(P_j(x) = x^j, j= 0(1)6\)

*are basis functions and W is a vector given by*

###
*Proof*

*L*can be obtained and are given as

*jth*column of

*V*by

*W*. We rewrite (11) as (9) using the newly found elements of

*L*. \(\square\)

### BVM and its block extension

###
*Remark 1*

We note that the method (12) is locally obtained on \([x_l,x_{l+4}]\) and is applied to simultaneously obtain approximations to the semidiscretized system (3) over the whole 3D space \([L_1,L_2,L_3] \times [L_4,L_5,L_6];\) in which case \(l=0(4)(L-4)\), \(m=0(1)(M-1)\) and \(n=0(1)(N-1)\). Also, we note that the first three formulas in (12) are of \(O(h^8)\) while the derivative formulas are of \(O(h^7)\).

### Convergence analysis

We discuss the convergence of the BVMs in the following theorem

###
**Theorem 2**

*Let*
\(\mathbf{U }\)
* be an approximation of the solution vector*
\({\overline{\mathbf{U }}}\)
* for the system obtained on a partition*
\(\pi _{L}:=\{L_1=x_{0}<x_{1}<\ldots <x_{L}=L_4,~ x_{m}=x_{m-1}+h\}\)
* from the method* (12).* If*
\(e_{l}=|u_{m,n}(x_{l})-u_{l,m,n}|\), \(he^{\prime }_{l}=|hu^{\prime }_{m,n}(x_{l})-hu^{\prime }_{l,m,n}|\),* where the exact solution*
\(u_{m,n}(x)\)
* is several times differentiable on*
\([L_1,L_4]\)
* and if*
\(\Vert E\Vert =\Vert \mathbf{U }-{\overline{\mathbf{U }}}\Vert\), *then, the BVM is convergent of order 6, which implies that *
\(\Vert E\Vert =O(h^{6})\).

###
*Proof*

*L*(

*h*) is the local truncation error vector of the formulas in (12).

*J*is the Jacobian matrix and its entries \(J_{11}\), \(J_{12}\), \(J_{21}\) and \(J_{22}\) are defined as

*L*so that (15) becomes

*h*, \(A + M\) is a monotone matrix and thus nonsingular. Therefore

## Computational procedures

*M*and

*N*are positive integers and

*m*,

*n*are the grid index in the

*y*and

*z*direction respectively.

The resulting system of ODEs (3) is then solved on the partition \(\pi _L\). The block unification of (17) lead to a large system of finite difference equations which is then solved to provide all solutions of (3) on \(\Omega\).

The following algorithm summarizes the numerical integration of (3) for some set of points *x* on \(\pi _L\).

*Step 1* Use the block unification of (17) for \(\mu =1, n=0\) to obtain \(\mathbf{V }_{1}\) on the domian \([x_{0}, L_2, L_3]\times [x_4, L_5, L_6]\) and for \(\mu =2, n=4\), \(\mathbf{V }_{2}\) is obtained on the domain \([x_{4}, L_2, L_3]\times [x_8, L_5, L_6]\), and on the domains \([x_{8}, L_2, L_3]\times [x_{12}, L_5, L_6], \ldots , [x_{L-4}, L_2, L_3]\times [x_L, L_5, L_6]\), for \(\mu =3, \ldots ,\Gamma\), \(n=8, 12 \ldots , N-4\), we obtain \(\mathbf{V }_{3} \ldots , \mathbf{V }_{\Gamma }\).

*Step 2* The unified block given by the system \(\mathbf{V }_{1} \bigcup \mathbf{V }_{2} \bigcup \ldots \bigcup \mathbf{V }_{\Gamma -1}\bigcup \mathbf{V }_{\Gamma }\) obtained in Step 1 is a large system with dimensions \(2L(M-1)(N-1)\) with \(u_{m,n}(x_l) \approx u(x_l, y_m, z_n)\), \(l=1, \ldots , L\), \(m=1, \ldots , M\), \(n=1, \ldots , N.\)

*Step 3* The system obtained in Step 2 is solved using the feature NSolve in Mathematica 10.0 for linear problems and FindRoot (which incorporates the Newton’s method) for nonlinear problems.

*Step 4* The solution of (1) is approximated by the solution in Step 3 as \(\mathbf{U }(x_l) = [u(x_l, y_m, z_n), \ldots , u(x_L, y_m, z_n)]^T\), \(m=1,\ldots , M\)
\(n=1, \ldots , N.\)

## Numerical examples

In this section, some examples are investigated to show the reliability and efficiency of the proposed scheme in this paper.

### Test 1

*G*(

*x*,

*y*,

*z*) and with zero boundary values on the entire \(\partial \Omega\) given in Zhang (1998c) and whose solution is given as

Errors in the \(l^\infty\) norm for test problem 1

\(L_4\) | \(\Delta y = \Delta z =0.25\) | \(\Delta y = \Delta z =0.125\) | \(\Delta y = \Delta z =0.0625\) | \(\Delta y = \Delta z =0.03125\) |
---|---|---|---|---|

0.1 | 1.903e−04 | 4.905e−05 | 1.242e−05 | 3.109e−06 |

0.2 | 1.413e−03 | 3.641e−04 | 9.208e−05 | 2.305e−05 |

0.3 | 4.238e−03 | 1.091e−03 | 2.754e−04 | 6.896e−05 |

0.4 | 8.610e−03 | 2.213e−03 | 5.576e−04 | 1.396e−04 |

0.5 | 1.401e−02 | 3.586e−03 | 9.028e−04 | 2.260e−04 |

0.6 | 1.972e−02 | 5.010e−03 | 1.263e−03 | 3.158e−04 |

0.7 | 2.506e−02 | 6.291e−03 | 1.592e−03 | 3.980e−04 |

0.8 | 2.953e−02 | 7.340e−03 | 1.855e−03 | 4.636e−04 |

0.9 | 3.280e−02 | 8.142e−03 | 2.029e−03 | 5.092e−04 |

1.0 | 3.472e−02 | 8.616e−03 | 2.144e−03 | 5.356e−04 |

### Test 2

Errors in the \(l^\infty\) norm for test problem 2

\(L_4\) | \(\Delta y = \Delta z =0.5\) | \(\Delta y = \Delta z =0.25\) | \(\Delta y = \Delta z =0.125\) | \(\Delta y = \Delta z =0.0625\) |
---|---|---|---|---|

0.2 | 1.266e−01 | 3.343e−02 | 8.471e−03 | 2.125e−03 |

0.4 | 4.016e−01 | 1.035e−02 | 2.609e−02 | 26.535e−03 |

0.6 | 6.692e−01 | 1.637e−02 | 4.067e−02 | 1.015e−02 |

0.8 | 8.183e−01 | 1.811e−01 | 4.406e−02 | 1.112e−02 |

1.0 | 8.115e−01 | 1.815e−01 | 4.406e−02 | 1.013e−02 |

1.2 | 8.344e−01 | 1.814e−01 | 4.406e−02 | 1.015e−02 |

1.4 | 7.892e−01 | 1.788e−01 | 4.380e−02 | 1.187e−02 |

1.6 | 8.224e−01 | 1.810e−01 | 4.380e−02 | 1.055e−02 |

1.8 | 8.429e−01 | 1.764e−01 | 4.371e−02 | 1.034e−02 |

2.0 | 8.308e−01 | 1.815e−01 | 4.408e−02 | 1.154e−02 |

### Test 3

Errors in the \(l^\infty\) norm for test problem 3

\(L_4\) | \(\Delta y = \Delta z =0.5\) | \(\Delta y = \Delta z =0.25\) | \(\Delta y = \Delta z =0.125\) | \(\Delta y = \Delta z =0.0625\) |
---|---|---|---|---|

\(-0.8\) | 5.340e−04 | 1.493e−04 | 3.896e−05 | 9.775e−06 |

\(-0.6\) | 2.042e−03 | 5.911e−04 | 1.493e−03 | 3.763e−05 |

\(-0.4\) | 4.878e−03 | 1.257e−03 | 3.167e−04 | 7.991e−05 |

\(-0.2\) | 7.896e−03 | 2.007e−03 | 5.273e−04 | 1.320e−05 |

0.0 | 1.017e−02 | 3.005e−03 | 7.539e−04 | 1.895e−04 |

0.2 | 1.502e−02 | 3.966e−03 | 9.964e−04 | 2.494e−04 |

0.4 | 1.739e−02 | 4.742e−03 | 1.231e−03 | 3.097e−04 |

0.6 | 2.005e−02 | 5.389e−03 | 1.436e−03 | 3.648e−04 |

0.8 | 2.324e−02 | 5.975e−03 | 1.514e−03 | 3.793e−04 |

1.0 | 1.995e−02 | 5.048e−03 | 1.349e−03 | 3.449e−04 |

### Test 4

Errors in the \(l^\infty\) norm for test problem 4

\(\epsilon\) | \(\Delta y = \Delta z =0.25\) | \(\Delta y = \Delta z =0.125\) | \(\Delta y = \Delta z =0.0625\) | \(\Delta y = \Delta z =0.03125\) |
---|---|---|---|---|

0.1 | 1.492e−01 | 2.086e−02 | 1.801e−03 | 2.767e−04 |

0.2 | 2.830e−02 | 5.918e−03 | 1.108e−03 | 2.626e−04 |

0.34 | 1.890e−2 | 4.807e−03 | 1.113e−03 | 2.742e−04 |

0.4 | 1.655e−02 | 4.744e−03 | 1.146e−03 | 2.843e−04 |

0.5 | 1.680e−02 | 4.805e−03 | 1.174e−03 | 2.916e−04 |

0.6 | 1.737e−02 | 4.867e−03 | 1.195e−03 | 2.916e−04 |

0.7 | 1.780e−02 | 4.919e−03 | 1.211e−03 | 3.970e−04 |

0.8 | 1.813e−02 | 4.962e−03 | 1.224e−03 | 3.048e−04 |

0.9 | 1.839e−02 | 4.998e−03 | 1.235e−03 | 3.075e−04 |

1.0 | 1.860e−02 | 5.028e−03 | 1.243e−03 | 3.097e−04 |

### Test 5

Errors in the \(l^\infty\) norm for different values of \(\alpha\) for test problem 5

\(\alpha\) | \(\Delta y = \Delta z =0.25\) | \(\Delta y = \Delta z =0.125\) | \(\Delta y = \Delta z =0.0625\) | \(\Delta y = \Delta z =0.03125\) |
---|---|---|---|---|

0 | 1.359e−04 | 3.664e−05 | 9.521e−06 | 2.395e−06 |

1 | 1.383e−04 | 3.725e−05 | 9.768e−05 | 3.226e−06 |

2 | 1.380e−04 | 3.951e−05 | 9.858e−05 | 1.892e−06 |

3 | 1.370e−04 | 4.582e−05 | 1.649e−05 | 8.940e−05 |

4 | 1.361e−04 | 5.382e−05 | 2.623e−05 | 1.892e−05 |

5 | 1.359e−04 | 6.466e−05 | 3.936e−05 | 3.226e−05 |

## Conclusion

In this paper, we have developed a highly accurate 3D problem solver. This has been achieved by the discretization of two of the spatial variables and the construction of a continuous BVM via the interpolation and collocation approach for solving the resulting semidiscretized system. The results given in "Numerical examples" show that the approach is highly efficient and highly accurate.

## Declarations

### Authors' contributions

SJ proposed the algorithm. TB developed, analyzed and implement the methods. SJ also contributed to the implementation of the methods. Both authors read and approved the final manuscript.

### Acknowledgements

The authors are very grateful to the referees whose valuable suggestions tremendously improved the quality of the manuscript.

### Compliance with ethical guidelines

**Competing interests** The authors declare that they have no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

## References

- Alam MN, Akbar MA, Roshid HO (2014) Traveling wave solutions of the Boussinesq equation via the new approach of generalized (G′/G)-expansion method. SpringerPlus 3(1):43View ArticleGoogle Scholar
- Bhrawy AH, Abdelkawy MA (2013) Cnoidal and snoidal wave solutions to coupled nonlinear wave equations by the extended Jacobi’s elliptic function method. Commun Nonlinear Sci Numer Simul 18(4):915–925View ArticleGoogle Scholar
- Biala TA, Jator SN (2015) A family of boundary value methods for the numerical approximation of systems of second order boundary value problems (submitted)Google Scholar
- Brastos AG (2007) The solution of the two-dimensional sine-Gordon equation using the method of lines. J Comput Appl Math 206:251–277View ArticleGoogle Scholar
- Brugnano L, Trigiante D (1998) Solving differential problems by multistep initial and boundary value problems. Gordon and Breach Science Publishers, New YorkGoogle Scholar
- Cui M (2010) High order compact alternating direction implicit method for the generalized sine-Gordon equation. J Comput Appl Math 235:837–849View ArticleGoogle Scholar
- Dehghan M, Shokri A (2008) A numerical method for one-dimensional nonlinear sine-Gordon equation using collocation and radial basis functions. Numer Methods Partial Differ Equ 24:687–698View ArticleGoogle Scholar
- Ge L, Zhang J (2002) Symbolic computation of high order compact difference scheme for three dimensional linear elliptic partial differential equations with variable coefficients. J Comput Appl Math 143:9–27View ArticleGoogle Scholar
- He JH (1998) An approximate solution technique depending on an artificial parameter: a special example. Commun Nonlinear Sci Numer Simul 3:92–97View ArticleGoogle Scholar
- Jain MK, Jain RK, Mohanty RK (1992) Fourth order difference method for three dimensional elliptic equations with nonlinear first derivative terms. Numer Methods Partial Diffe Equ 8:575–591View ArticleGoogle Scholar
- Jator SN, Li J (2012) An algorithm for second order initial and boundary value problems with an automatic error estimate based on a third derivative method. Numer Algorithms 59:333–346View ArticleGoogle Scholar
- Liao HL, Sun ZZ (2010) Maximum norm error bounds of ADI and compact ADI methods for solving parabolic equations. Numer Methods Partial Differ Equ 26:37–60View ArticleGoogle Scholar
- Mohanty RK, Singh S (2006) A new highly accurate discretization for three dimensional singularly perturbed nonlinear elliptic partial differential equations. Numer Methods Partial Differ Equ 22:1379–1395View ArticleGoogle Scholar
- Roache PJ (1976) Computational fluid dynamics. Hermosa, AlbuquerqueGoogle Scholar
- Roshid HO, Rahman N, Akbar MA (2013) Traveling wave solutions of nonlinear Klein–Gordon equation by extended-expansion method. Ann Pure Appl Math 3(1):10–16Google Scholar
- Roshid HO, Kabir MR, Bhowmik RC, Datta BK (2014) Investigation of solitary wave solutions for Vakhnenko–Parkes equation via exp-function and Exp-expansion method. SpringerPlus 3(692):1–10Google Scholar
- Roshid HO, Akbar MA, Alam MN, Hoque MF, Rahman N (2014) New extended-expansion method to solve nonlinear evolution equation: the (3 + 1)-dimensional potential-YTSF equation. SpringerPlus 3:122. doi:https://doi.org/10.1186/2193-1801-3-122 View ArticleGoogle Scholar
- Roshid HO, Hoque MF, Akbar MA (2014) New extended-expansion method for traveling wave solutions of nonlinear partial differential equations (NPDEs) in mathematical physics. Italian J Pure Appl Math 33:175–190Google Scholar
- Roshid HO, Rahman MA (2014) The exp-expansion method with application in the (1+1)-dimensional classical Boussinesq equations. Results Phys 4:150–155View ArticleGoogle Scholar
- Shakeri F, Dehghan M (2008) The method of lines for solution of the one-dimensional wave equation subject to an integral conservation condition. Comput Math Appl 56:2175–2188View ArticleGoogle Scholar
- Spotz WF (1995) High-order compact finite difference scheme for computational mechanics. Ph.D. Thesis, University of Texas at Austin, Austin, TXGoogle Scholar
- Spotz WF, Carey GF (1996) A high-order compact formulation for the 3D Poisson equation. Numer Methods Partial Differ Equ 12:235–243View ArticleGoogle Scholar
- Yang XF, Deng ZC, Wei Y (2015) A Riccati–Bernoulli sub-ODE method for nonlinear partial differential equations and its application. Adv Differ Equ 1:117–133View ArticleGoogle Scholar
- Zhang J (1998a) On convergence and performance of iterative methods with fourth-order compact schemes. Numer Methods Partial Differ Equ 14:262–2832Google Scholar
- Zhang J (1998b) An explicit fourth-order compact finite difference scheme for three dimensional convection-diffusion equation. Commun Numer Methods Eng 14:209–218View ArticleGoogle Scholar
- Zhang J (1998c) Fast and high accuracy multigrid solution of the three dimensional Poisson equation. J Comput Phys 143:449–461View ArticleGoogle Scholar
- Zhang J (1998d) Fourth-order compact discretization and iterative solution of the 3D convection-diffusion equation. In: Wang J, Allen MB, Chen BM, Mathew T (eds) Iterative methods in scientific computation. IMACS, New Brunswick, pp 331–336Google Scholar
- Zhang J, Ge L, Gupta MM (2000) Fourth order compact difference scheme for 3D convection diffusion equation with boundary layers on nonuniform grids. Neural Parallel Sci Comput 8(3, 4):373–392Google Scholar