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The Finite Difference Methods for Parabolic Partial Differential Equations
ISSN: 2168-9679

Journal of Applied & Computational Mathematics
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The Finite Difference Methods for Parabolic Partial Differential Equations

Nigatie Y*
Department of Mathematics, College of Natural and Computational Sciences, University of Gondar, Gondar, Ethiopia
*Corresponding Author: Nigatie Y, Department of Mathematics, College of Natural and Computational Sciences, University of Gondar, Gondar, Ethiopia, Tel: +251913080434, Email: [email protected]

Received Date: Aug 18, 2018 / Accepted Date: Sep 12, 2018 / Published Date: Sep 27, 2018

Keywords: Parabolic; Heat equation; Finite difference; Bender- Schmidt; Crank-Nicolson

Introduction

Parabolic partial differential equations

The well-known parabolic partial differential equation is the one dimensional heat conduction equation [1]. The solution of this equation is a function u(x,t) which is defined for values of x from 0 to l and for values of t from 0 to ∞ [2-4]. The solution is not defined in a closed domain but advances in an open ended region from initial values, satisfying the prescribed boundary conditions (Figure 1). In general the study of pressure waves in a fluid, propagation of heat and unsteady state problems lead to parabolic type of equations [5].

applied-computational-mathematics-boundary

Figure 1: Conditions of prescribed boundary.

Solution of one-dimensional Heat conduction equation

Consider the heat conduction which is of the form [6]

cut=uxx (1)

where C is a constant Let the plane (x,t) be divided in to smaller rectangles by means of the sets of lines [7]

x=ih,i=0,1,2,.

t=jk,j=0,1,2,.

Using the finite difference approximations [8-11]

image (2)

and

image (3)

Hence the finite difference analogue in eqn. (1) is

image(4)

Which can be rewritten as

ui,j+1=ui,j+λ(ui−1,j −2ui,j+ui+1,j)

where image this formula expresses the unknown function value at the (i,j+1)th interior point in terms of the known function values and hence it is called the explicit formula. It can be shown that this formula is valid only [10] for image For image becomes in eqn. (4)

image (5)

The above formula in eqn. (5) is called Bender-Schmidt recurrence relation. In formula in eqn. (4), we have used the function values along the jth row only in the approximation of uxx is replaced by the [11] average of its finite difference approximations on the jth and (i,j+1)th rows. Thus

image

and hence in eqn. (1) is replaced by

image

This implies

image (6)

where image, on the left side in eqn. (6) we have three unknowns and on the right side all the three quantities are known. The above formula in eqn. (6) which is an implicit scheme is called Crank-Nicolson formula and it is convergent for all finite values of λ. If there are N internal mesh points on each row, then the formula in eqn. (6) gives N simultaneous equations for the N unknowns in terms of the given boundary values. Similarly the internal mesh points on all rows can be calculated [12].

Example 1: Use the Bender-Schmidt recurrence relation to solve the equation uxx=2ut. With the conditions u(x,0)=4x−x2,u(0,t)=u(4,t)=0 Solution Taking h=1 we obtain [13]

image

Also u(0,0)=0,u(1,0)=3,u(2,0)=4,u(3,0)=3 and u(4,0)=0. For this the first time step, k=1.

For this the first time step, k=1.

Using Bender-Schmidt recurrence relation, we obtain

image

For k=2, we have

image

For k=3 we have

image

For k=4 we have

image

Similarly with k=5 we obtain

image

The computation can be continued to any number of time steps.

Example 2: Solve the equation ut=uxx, subject to the conditions,

u(x,0)=sinπx,0 ≤ x ≤ 1,u(0,t)=u(1,t)=0.

Using

(a) Bender-Schmidt Method

(b) Crank-Nicolson Method

Carry out the computations for two levels, taking image

Solution

Here image and image so thatimage

Also image and all boundary values are zero in Figure 2.

applied-computational-mathematics-levels

Figure 2: Computations for two levels using Crank-Nicolson method.

(a) Using Bender-Schmidt formula

Using in eqn. (4) of this section

we have ui,j+1=λui−1,j+(1−2λ)ui,jui+1,j

This becomes

image

For i=1,2; j=0

image

(b) By using Crank-Nicolson method

From eqn. (6) above we have

image

For i=1,2; j=0

image

Solving these equations, we find

u1,1=u2,1=0.67

For i=1,2; j=1

−u0,2+10u1,2−u2,2=u0,1+ 6u1,1+ u2,1

⇒ 10u1,2−u2,2=4.69

−u1,2+10u2,2−u3,2=u1,1+6u2,1+ u3,1

⇒−u1,2+10u2,2=4.69

Again solving the above two equation we can obtain

u1,2=u2,2=0.52.

Solution of two dimensional heat equations

ut =c2(uxx+uyy).

The method employed for the solution of one dimensional heat equation can be readily extended to the solution two dimensional heat equations in eqn. (7).

Consider a square region 0 ≤ x ≤ y ≤ a and assume that u is known at all points within and on the boundary of this square.

If h be the step size then a mesh point

(x,y,t) =(ih,jh,nl) may be denoted as (i,j,n).

Replacing the derivatives in eqn. (7) by their finite difference approximations, we get

image (8)

Where image . This equation needs the five points available on the nth plane (Figure 3).

applied-computational-mathematics-explicit

Figure 3: Alternating direct explicit on (n+1)th plane.

The computation process consists point by point evaluation in the (n+1)th plane using the points on the nth plane. This method is known as ADE (Alternating Direct Explicit) method.

Example: Solve the equation ut=uxx+uyy.

Subject to the initial conditions u(x,y,0) =sin2πxsin2πy,0 ≤ x,y ≤ 1 and the conditions u(x,y,t)=0, t ≥ 0, on the boundaries using ADE method image and image (Calculate the results for one time level).

Solution: In eqn. (8) becomes

image

The mesh points and the computation model is given in Figure 4 below

applied-computational-mathematics-zeroth

Figure 4: Mesh points and the computation model in zeroth level.

At the zeroth level (n=0), the initial and boundary conditions are image

And

ui,0,0=u0,j,0+u3,j,0=ui,3,0=0;i,j=0,1,2,3

Now we can calculate the mesh values at the first level.

For n=0, from (∗) we obtain

image

(i) Putting i=j=1 in (∗∗) above

image

(ii) Putting i=2,j=1 in(∗∗) above

image

(iii) Putting i=1,j=2 in (∗∗) above

image

(iv) Putting i=j=2 in (∗∗) above

image

Similarly the mesh values at the second and higher levels can be calculated.

References

Citation: Nigatie Y (2018) The Finite Difference Methods for Parabolic Partial Differential Equations. J Appl Computat Math 7: 418. DOI: 10.4172/21689679.1000418

Copyright: © 2018 Nigatie Y. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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