Four Point Implicit Methods for the Second Derivatives of the Solution of First Type Boundary Value Problem for One Dimensional Heat Equation

Suzan Cival Buranay,Lawrence Adedayo Farinola,C Cattani,A Atangana,Z Hammouch,S Agoujil,H Bulut,H.M Baskonus,T Mekkaoui

doi:10.1051/itmconf/20182201011

Suzan Cival Buranay, Lawrence Adedayo Farinola + Show 7 more

Open Access

https://doi.org/10.1051/itmconf/20182201011

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Abstract

We construct four-point implicit difference boundary value problem for the first derivative of the solution u(x,t) of the first type boundary value problem for one dimensional heat equation with respect to the time variable t. Also, for the second derivatives of u(x,t) special four-point implicit difference boundary value problems are proposed. It is assumed that the initial function belongs to the Hölder space C8+α,0 < α < 1, the heat source function given in the heat equation is from the Hölder space [see formula in PDF], the boundary functions are from [see formula in PDF], and between the initial and the boundary functions the conjugation conditions of orders q = 0,1,2,3,4 are satisfied. We prove that the solution of the proposed difference schemes converge uniformly on the grids of the order O(h2+τ) (second order accurate in the spatial variable x and first order accurate in time t) where, h is the step size in x and τ is the step size in time. Theoretical results are justified by numerical examples.

Highlights

Many practical heat conduction questions lead to problems not conveniently solvable by classical methods such as separation of variables techniques or the use of Green's functions
The Hὄlder space Cx,t 2, the boundary functions are from C 2 and between the initial and the boundary functions the conjugation conditions of orders q 0,1, 2,3, 4 are satisfied
We prove that the solution of the proposed difference schemes converge uniformly on the grids of the order O(h2 τ) where, h is the step size in x and τ is the step size in time

Summary

Introduction

Many practical heat conduction questions lead to problems not conveniently solvable by classical methods such as separation of variables techniques or the use of Green's functions. The Hὄlder space Cx,t 2 , the boundary functions are from C 2 and between the initial and the boundary functions the conjugation conditions of orders q 0,1, 2,3, 4 are satisfied. Besides the solution of the heat conduction problem it is very important to provide information about some physical phenomena related with the derivatives. In this study we consider the first type boundary value problem for one dimensional heat equation of which the initial function belongs to C8 0 the heat source function is from Cx,t 2

Results

Conclusion