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European Journal of Applied Sciences – Vol. 12, No. 6

Publication Date: December 25, 2024

DOI:10.14738/aivp.126.17966.

Umurdin, D., & Dilfuza, X. (2024). Comparative Analysis of Schemes with Movable Nodes for a Parabolic Equation. European

Journal of Applied Sciences, Vol - 12(6). 344-352.

Services for Science and Education – United Kingdom

Comparative Analysis of Schemes with Movable Nodes for a

Parabolic Equation

Dalabaev Umurdin

Department of System Analysis and Mathematical Modeling,

University of World Economy and Diplomacy, Tashkent, Uzbekistan

Xasanova Dilfuza

Department of System Analysis and Mathematical Modeling,

University of World Economy and Diplomacy, Tashkent, Uzbekistan

ABSTRACT

The article considers an approximate analytical solution of a linear parabolic

equation with initial and boundary conditions. Many problems in engineering

applications are reduced to solving an initial-boundary value problem of parabolic

type. There are various analytical, approximate-analytical and numerical methods

for solving such problems. The most popular difference methods for solving an

initial-boundary value problem of a parabolic equation are explicit, implicit and

Crank-Nicolson schemes. Here, we consider methods for obtaining an

approximate-analytical solution based on the movable node method and their

comparative analysis of these schemes for specific test problems. A comparison of

the exact and approximate solutions is made using specific examples.

Keywords: parabolic equation, approximate-analytical solution, moving nodes.

INTRODUCTION

Processes in hydrodynamics, heat transfer, boundary layer flow, elasticity, quantum

mechanics and electromagnetic theory are modeled by differential equations. Only some of

these equations can be solved by an analytical method. But the search for exact solutions,

when they exist, is always necessary to better explain the modeled phenomenon. The search

for an analytical solution gives an advantage for analyzing processes [1,2,3].

Analytical methods have a relatively low degree of universality for solving such problems.

More universal are approximate analytical methods (projection, variational methods, the

small parameter method, operational methods, various iterative methods) [4,5,6,7].

Comparative analysis for solving shifted boundary value problems is carried out based on the

method of moving nodes [8,9,10]. The method combines the approximation of derivatives

appearing in the equation, difference relations and obtaining an approximate analytical

expression for the solution of the problem. In this case, we can obtain an approximate

analytical solution to the problem, which is a hybrid of known methods. Note that obtaining

an approximate analytical solution to differential equations is based on numerical methods.

The nature of numerical methods also allows obtaining an approximate analytical expression

for the solution of differential equations. For this purpose, the so-called "movable node" is

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Umurdin, D., & Dilfuza, X. (2024). Comparative Analysis of Schemes with Movable Nodes for a Parabolic Equation. European Journal of Applied

Sciences, Vol - 12(6). 344-352.

URL: http://dx.doi.org/10.14738/aivp.126.17966

introduced [8]. The aim of the study is a comparative analysis of various difference schemes

for applying the method of moving nodes for a parabolic type of equation and is a

continuation of the work [10]. Compare explicit, implicit and Crank-Nicholson for a mixed

problem of a parabolic equation and provide test examples.

STATEMENT OF THE PROBLEM

Let us consider a one-dimensional differential equation of parabolic type in the domain Ω:

(0<t<T, W<x<E)

( , ); u u A f x t

t x

 

= +

 

(1)

with initial

u x t u x ( , 0, ) ( ). = =

(2)

and boundary conditions

( , ) ( ) ( , ) ( ). W E u x W t u t u x E t u t = = = =

(3)

We assume that the solution to problem (1)-(3) exists and is unique.

For the numerical solution of problem (1)--(3) there are various difference schemes [11]. Let

us consider various variants of difference approximation of a linear one-dimensional equation

in space by a moving node.

(а) (b) (с)

Fig 1: Template of a moving node

Fig. 1 shows templates with one moving node. Fig. 1 (a) corresponds to the template by the

explicit scheme, Fig. 1 (b) ̶the implicit scheme, and Fig. 1 (c) ̶the Crank Nicholson scheme. In

Fig. 1, the point (t, x) ε Ω refers to the moving node. Points (0, x), (t, W) and (t, E) are also

movable nodes: point (0, x) moves only along the x-axis, point (t, W) moves along the left, and

point (t, E) moves along the right boundary of the region.

SOLUTION BY MOVING NODES METHOD

Using one movable node, we can obtain a rough analytical representation of the solution to

problem (1)-(3).

(W,0)

(x,0) (E,0)

( x,t)

(t,E)

(x,0)

(W,t) ( x,t) (E,t) (W ,t) (E,t)

(x,0)

(x,t)

(E,0)

(W,0)

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European Journal of Applied Sciences (EJAS) Vol. 12, Issue 6, December-2024

Let us denote by

U x t ( , ).

an approximate analytical solution to the problem obtained using the

movable node method and using the boundary and initial conditions. Let

( , ) x t 

be an

arbitrary moving point. We approximate equation (1) with an explicit scheme

0 0 0 0 0 ( , ) ( ) 2 ( ) ( ) ( ) ( ) ( , ), U x t U x U x U W U E U x E A f x t

t E W E x x W

− −   −

= − +  

− − −  

(4)

In (4) is an approximate analytical solution to the problem. When the point runs through, we

obtain a solution in the region under consideration. From (3) we obtain

0 0 0 0

0 2 ( ) ( ) ( ) ( ) ( , ) ( ) ( , ), U E U x E

t U x U W U x t U x A tf x t

E W E x x W

  − −

= + − +  

− − −  

(5)

If we perform approximation using the implicit scheme, we have

( , ) ( ) 2 ( ) ( , ) ( , ) ( ) ( , ), U x t U x U t U x t E U x t U t W A f x t

t E W E x x W

−   − −

= − +  

− − −  

(6)

By solving this equation, we obtain an approximate analytical solution to the problem in the

case of an implicit scheme

( )( ) 0

2 ( )( ) ( )( )  

( , ) ( )

2 ( )( ) 2 ( )( )

( )( ) ( , ).

2 ( )( )

E x x W At U t x W U t E x E W U x t U x

At E x x W At E x x W

E x x W t f x t

At E x x W

− − − + −

= + +

+ − − + − −

− −

+ − −

(7)

The Crank-Nicholson scheme has the form:

0 0 0 0

( , ) ( ) 2 ( ) ( , ) ( , ) ( )

2 ( ) ( ) ( ) ( ) (1 ) ( , ),

U x t U x U t U x t E U x t U t W A

t E W E x x W

U E U x U x U W A f x t

E W E x x W



−   − −

= − +  

− − −  

  − −

− − +  

− − −  

(8)

From here we determine the approximate analytical solution using the Crank-Nicholson

scheme:

0 0 0 0

( )( ) ( ) ( ) ( ) ( ) ( , ) ( )

( )( ( ) ( )) ( )( ( ) ( )) ( )( ) ( , ),

E W E x x W x W U t E x U t U x t U x B

D D

x W U E U x E x U x U W E x x W t C f x t

D D

− − − − −

= + +

− − − − − − −

(9)

In expression (9), the notation is introduced