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

Biala TA, Jator SN - Springerplus (2015)

Bottom Line: 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.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics and Computer Science, Jigawa State University, Kafin Hausa, P.M.B 048, Kafin Hausa, Nigeria.

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.

No MeSH data available.


Graphical evidence when  for  for test problem 5
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Fig5: Graphical evidence when for for test problem 5

Mentions: The singularly perturbed elliptic boundary value problem given in Mohanty and Singh (2006)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \epsilon \left( \dfrac{\partial ^2u}{\partial x^2} + \dfrac{\partial ^2u}{\partial y^2} + \dfrac{\partial ^2u}{\partial z^2} +\dfrac{\alpha }{x}\dfrac{\partial u}{\partial x}\right) = G(x,y,z) \end{aligned}$$\end{document}ϵ∂2u∂x2+∂2u∂y2+∂2u∂z2+αx∂u∂x=G(x,y,z)defined in the domain with boundary and subject to the Dirichlet boundary conditions and where the forcing term G is set to satisfy the exact solution given as . We have solved the problem using . The norms are given in Table 5 with different meshsizes and for different values of . Figure 5 also shows the plot of the exact, approximate and error function when Table 5


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

Biala TA, Jator SN - Springerplus (2015)

Graphical evidence when  for  for test problem 5
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4627984&req=5

Fig5: Graphical evidence when for for test problem 5
Mentions: The singularly perturbed elliptic boundary value problem given in Mohanty and Singh (2006)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \epsilon \left( \dfrac{\partial ^2u}{\partial x^2} + \dfrac{\partial ^2u}{\partial y^2} + \dfrac{\partial ^2u}{\partial z^2} +\dfrac{\alpha }{x}\dfrac{\partial u}{\partial x}\right) = G(x,y,z) \end{aligned}$$\end{document}ϵ∂2u∂x2+∂2u∂y2+∂2u∂z2+αx∂u∂x=G(x,y,z)defined in the domain with boundary and subject to the Dirichlet boundary conditions and where the forcing term G is set to satisfy the exact solution given as . We have solved the problem using . The norms are given in Table 5 with different meshsizes and for different values of . Figure 5 also shows the plot of the exact, approximate and error function when Table 5

Bottom Line: 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.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics and Computer Science, Jigawa State University, Kafin Hausa, P.M.B 048, Kafin Hausa, Nigeria.

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.

No MeSH data available.