Crank-Nicolson and Modified Crank-Nicolson Scheme for One Dimensional Parabolic Equation
Omowo Babajide Johnson,
Longe Idowu Oluwaseun
Issue:
Volume 6, Issue 3, September 2020
Pages:
35-40
Received:
28 May 2020
Accepted:
13 July 2020
Published:
13 August 2020
Abstract: Partial differential equations are very important tools for mathematical modeling in some field like; physics, engineering and applied Mathematics. It’s worth knowing that only few of this equation can be solved analytically and numerical method have been proven to perform exceedingly well in solving even difficult partial differential equations. Finite difference method is a popular numerical method which has been applied extensively to solve partial differential equations. A well known type of this method is the Classical Crank-Nicolson scheme which has been used by different researchers. In this work, we present a modified Crank-Nicolson scheme resulting from the modification of the classical Crank-Nicolson scheme to solve one dimensional parabolic equation. We apply both the Classical Crank-Nicolson scheme and the modified Crank-Nicolson scheme to solve one dimensional parabolic partial differential equation and investigate the results of the different schemes. The computation and results of the two schemes converges faster to the exact solution. It is shown that the modified Crank-Nicolson method is more efficient, reliable and better for solving Parabolic Partial differential equations since it requires less computational work. The method is stable and the convergence is fast when the results of the numerical examples where compared with the results from other existing classical scheme, we found that our method have better accuracy than those methods. Some numerical examples were considered to verify our results.
Abstract: Partial differential equations are very important tools for mathematical modeling in some field like; physics, engineering and applied Mathematics. It’s worth knowing that only few of this equation can be solved analytically and numerical method have been proven to perform exceedingly well in solving even difficult partial differential equations. F...
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Practical Insight of Ferroconvection in Heterogeneous Brinkman Porous Medium
Ravisha Mallappa,
Basavarajaiah Doddagangavadi Mariyappa,
Mamatha Arabhaghatta Lingaraju,
Prakash Revanna
Issue:
Volume 6, Issue 3, September 2020
Pages:
41-48
Received:
3 August 2019
Accepted:
23 December 2019
Published:
7 September 2020
Abstract: Ferromagnetic fluids are made up of magnetic particles, which are suspended in a carrier liquid such as water, hydrocarbon (mineral oil or kerosene) or fluorocarbon with a surfactant to avoid clumping. Worldwide many literature revealed that, the ferromagnetic fluids application has been diversified in nature and widely used in engineering, technology, agricultural, animal and biomedical sciences etc. (evidence based medicine for cancer patients, fertigation in agriculture). Now a days, the driven applications are using in developing countries. The ferromagnetic fluids analytical applications are very limited scope in Indian scenario due to paucity of literature and technological gap. In the essence of this research gap the present study undertaking to demonstrate the various applications of ferroconvection in a heterogeneous Brinkman porous medium on theoretical basis. The resulting eigenvalue problem is solved numerically using the Galerkin method. The effects of vertical heterogeneity of permeability, Darcy parameter, Magnetic Rayleigh number, nonlinearity of magnetization, and internal heat source on the onset of ferromagnetic convection is investigated.
Abstract: Ferromagnetic fluids are made up of magnetic particles, which are suspended in a carrier liquid such as water, hydrocarbon (mineral oil or kerosene) or fluorocarbon with a surfactant to avoid clumping. Worldwide many literature revealed that, the ferromagnetic fluids application has been diversified in nature and widely used in engineering, technol...
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Solution of Lagrange’s Linear Differential Equation Using Matlab
Abdel Radi Abdel Rahman Abdel Gadir,
Neama Yahia Mohammed,
Marwa Eltayb Abu Elgasim Msis
Issue:
Volume 6, Issue 3, September 2020
Pages:
49-53
Received:
7 July 2020
Accepted:
14 August 2020
Published:
16 September 2020
Abstract: MATLAB, which stands for Matrix Laboratory, is a software package developed by Math Works, Inc. to facilitate numerical computations as well as some symbolic manipulation. It strikes us as being slightly more difficult to begin working with it than such packages as Maple, Mathematica, and Macsyma, though once you get comfortable with it, it offers greater flexibility. The main point of using it is that it is currently the package you will most likely found yourself working with if you get a job in engineering or industrial mathematics. So we found that the Matlab method in differential equations is very important and useful mathematical tools which help us to solve and plot differential equations. The aims of this paper is to solve Lagrange’s Linear differential equations and compare between manual and Matlab solution such that the Matlab solution is one of the most famous mathematical programs in solving mathematical problems. We followed the applied mathematical method using Matlab and we compared between the two solutions whence accuracy and speed. Also we explained that the solution of Matlab is more accuracy and speed than the manual solution which proves the aptitude the usage of Matlab in different mathematical methods.
Abstract: MATLAB, which stands for Matrix Laboratory, is a software package developed by Math Works, Inc. to facilitate numerical computations as well as some symbolic manipulation. It strikes us as being slightly more difficult to begin working with it than such packages as Maple, Mathematica, and Macsyma, though once you get comfortable with it, it offers ...
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