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.
Highlights
Partial differential equations are the basis of many mathematical models of Physics, Chemistry, and Biology even in Finance
Many researchers have worked on the well-known parabolic partial differential equation using various numerical methods but of all the numerical methods, finite difference methods are mostly used
In this work, modified Crank-Nicolson method has been successfully compared with the classical Crank-Nicolson method by applying it to solve problems on parabolic partial differential equations (Heat conduction problems)
Summary
Partial differential equations are the basis of many mathematical models of Physics, Chemistry, and Biology even in Finance. Problems involving time t as one independent variable result most usually to parabolic equation, which are derives from the theory of heat conduction. Solutions of such problems can be obtained by numerical methods. Many researchers have worked on the well-known parabolic partial differential equation (the one dimensional heat conduction equation) using various numerical methods but of all the numerical methods, finite difference methods are mostly used. Is a well known example of a parabolic partial differential equations The solution of these equation is a temperature function (x, t) which is defined for values of x from 0 to l and for values of t from 0 to ∞. The solution is not defined in a closed domain but advances in an open- ended region from initial values satisfying the prescribed boundary conditions
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: International Journal of Applied Mathematics and Theoretical Physics
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
