907–921 DOI: 10.18514/MMN.2018.2558 UNIQUENESS OF SOLUTION AND FULLY DISCRETE SCHEME TO NONLINEAR INTEGRO-DIFFERENTIAL AVERAGED MODEL WITH SOURCE TERMS T

Vol. 19 (2018), No. 2, pp. 907–921 DOI: 10.18514/MMN.2018.2558

UNIQUENESS OF SOLUTION AND FULLY DISCRETE SCHEME TO NONLINEAR INTEGRO-DIFFERENTIAL AVERAGED MODEL

WITH SOURCE TERMS

T. JANGVELADZE, Z. KIGURADZE, AND M. KRATSASHVILI Received 14 March, 2018

Abstract. Uniqueness of solution and finite difference scheme of corresponding initial-boundary value problem for one nonlinear partial integro-differential averaged model with source terms are studied. Mentioned model is based on Maxwell system which describes electromagnetic field penetration into a substance. Mixed boundary condition is considered. Large time behavior of solution is fixed too. Convergence of the fully discrete scheme is proved. Wider class of nonlinearity is studied than one has been investigated before.

2010Mathematics Subject Classification: 65N06; 45K05; 35K55

Keywords: Maxwell equations, system of nonlinear integro-differential equations, uniqueness, asymptotic behavior, finite difference scheme

1. INTRODUCTION

Investigating many applied problems in nature we are facing to nonlinear integro- differential models which contain derivatives of several variables. Numerous public- ations deal with the study of integro-differential equations of various kinds (see, for example, the bibliography in [6], [8]). Integro-differential models arise for example at the mathematical simulation of process of a magnetic field penetration into a medium whose electro-conductivity depends on temperature. Numerous works are dedicated to the investigation of Maxwell equations describing above-mentioned process (see, for example, [12], [19], [20] and references therein). In a quasi-stationary case the corresponding system of Maxwell equations [12] can be rewritten in the following form [5]:

@H

@t D rot 2 4a

0

@

t

Z

0

jrotHj²d 1 ArotH

3

5; (1.1)

The third author thanks Shota Rustaveli National Science Foundation (project PhDF2016 19) for the financial support.

c 2018 Miskolc University Press

whereH D.H1; H2; H3/is a vector of the magnetic field and functionaDa.S /is defined forS2Œ0;1/.

If vector of magnetic field has the formH D.0; U; V /, whereU DU.x; t /,V D V .x; t /, then from (1.1) we get the following system of nonlinear parabolic integro- differential equations:

@U

@t D @

@x

a.S /@U

@x

; @V

@t D @

@x

a.S /@V

@x

; (1.2)

where

S.x; t /D

t

Z

0

"

@U

@x 2

C @V

@x 2#

d : (1.3)

Study of the models of type (1.1) have begun in [5]. In that work, in particular, based on Galerkin modified method and compactness arguments as in [17] for nonlin- ear parabolic equations the theorems of existence of solution of the initial-boundary value problem with first kind boundary conditions for scalar and one-dimensional space case whena.S /D1CS and uniqueness for more general cases are proven.

One-dimensional scalar variant for the casea.S /D.1CS /^p, 0 < p1is studied in [3]. Investigations for multi-dimensional space cases at first was carried out in [4].

Multidimensional space cases are also discussed in [1], [13]. Asymptotic behavior ast ! 1of solutions of initial-boundary value problems for (1.1) type models are studied in [8], [9], [10], [15], [23] and in a number of other works as well. In those works main attention is paid to one-dimensional cases. Finite element analogues and Galerkin method algorithm as well as settling of semi-discrete and finite difference schemes for (1.1) type one-dimensional integro-differential models are studied in [2], [7], [8], [9], [15], [18], [22], [23], [24] and in the other works as well for the linear case of diffusion coefficient, i.e.a.S /D1CS.

Some generalization of the system of type (1.1) is proposed by Prof. G. I. Laptev [14]. In particular, in some physical assumptions, the process of penetration of the magnetic field into a material is modeled by so-called averaged integro-differential model, the (1.2), (1.3) type analog of which have the following form:

@U

@t Da.S /@²U

@x²; @V

@t Da.S /@²V

@x²; (1.4)

where

S.t /D

t

Z

0 1

Z

0

"

@U

@x 2

C @V

@x 2#

dxd : (1.5)

The literature on the questions of existence, uniqueness, regularity, asymptotic be- havior of the solutions and numerical resolution of the initial-boundary value prob- lems to the (1.2), (1.3) and (1.4), (1.5) type models and models like them is very rich (see, for example, [5], [6], [7], [8], [14], [15], [16], [18] and references therein). The

asymptotic behavior and numerical solution of two-dimensional case for the (1.4), (1.5) type averaged integro-differential system is considered for example in [11], [24].

Our aim is to investigate system of nonlinear integro-differential equations (1.4), (1.5) with source terms. Uniqueness and large time behavior of solution of initial- boundary value problem with mixed boundary condition as well as convergence of corresponding fully discrete scheme is studied. In the present note we consider new class of nonlinearity considering more general cases of the diffusion coefficientaD a.S /.

2. UNIQUENESS AND ASYMPTOTIC BEHAVIOR OF SOLUTION ASt! 1 In the cylinderŒ0; 1Œ0;1/let us consider the following initial-boundary value problem:

@U

@t a 0

@

t

Z

0 1

Z

0

"

@U

@x 2

C @V

@x 2#

dxd 1 A

@²U

@x² Cg.U /Df1.x; t /;

@V

@t a 0

@

t

Z

0 1

Z

0

"

@U

@x 2

C @V

@x 2#

dxd 1 A

@²V

@x² Cg.V /Df2.x; t /;

(2.1)

U.0; t /DV .0; t /D @U.x; t /

@x ˇ ˇ ˇ ˇ_x

D1

D @V .x; t /

@x ˇ ˇ ˇ ˇ_x

D1

D0; (2.2)

U.x; 0/DU0.x/; V .x; 0/DV0.x/; (2.3) whereaDa.S /,g,f1,f2,U0andV0are given functions of their arguments.

The following statement takes place.

Theorem 1. IfaDa.S /a0DC onst > 0,a⁰.S /0,a⁰⁰.S /0, gis mono- tonically increased function, U0; V02H¹.0; 1/, U0.0/DV0.0/ D ^{d U}_dx^0.x/

ˇ ˇ ˇ_x

D1D

d V₀.x/

dx

ˇ ˇ ˇ_x

D1D0,f1; f2;^@f_@x¹;^@f_@x² 2L2.Q/and problem (2.1) - (2.3) has a solution then it is unique and exponential stabilization of solution ast! 1takes place.

Here we use usualL2and SobolevH¹spaces.

To prove the uniqueness of solution we assume that there exist two different.U ; V / and .U ; V / solutions of problem (2.1) - (2.3) and introduce the differences ZD U U andW DV V. To show thatZDW 0the methodology of proving the convergence theorem, which is given in the next section, monotone growth feature of functiongand the following identity is mainly used:

8

<

: a

0

@

t

Z

0 1

Z

0

2 4

@U

@x

!² C @V

@x

!²3 5dxd

1 A

@U

@x

a 0

@

t

Z

0 1

Z

0

2 4

@U

@x

!2

C @V

@x

!23 5dxd

1 A

@U

@x 9

=

;

@U

@x

@U

@x

!

C 8

<

: a

0

@

t

Z

0 1

Z

0

2 4

@U

@x

!² C @V

@x

!²3 5dxd

1 A

@V

@x

a 0

@

t

Z

0 1

Z

0

2 4

@U

@x

!2

C @V

@x

!23 5dxd

1 A

@V

@x 9

=

;

@V

@x

@V

@x

!

D

1

Z

0

d da

0

@

t

Z

0 1

Z

0

8

<

:

"

@U

@x C @U

@x

@U

@x

!#² C

"

@V

@x C @V

@x

@U

@x

!#²9

=

; dxd

1 A

"

@U

@x C @U

@x

@U

@x

!#

d @U

@x

@U

@x

!

C

1

Z

0

d da

0

@

t

Z

0 1

Z

0

8

<

:

"

@U

@x C @U

@x

@U

@x

!#² C

"

@V

@x C @V

@x

@U

@x

!#²9

=

; dxd

1 A

"

@V

@x C @V

@x

@V

@x

!#

d @V

@x

@V

@x

! :

For obtaining stabilization of solution stated in the Theorem 1the method of a- priori estimates based on analogical methodology given in [7] is used and large time behavior of solution is obtained.

3. CONVERGENCE OF THE FULLY DISCRETE SCHEME

In the rectangleQT DŒ0; 1Œ0; T , whereT is a positive constant, let us consider again problem (2.1) - (2.3). OnQT let us introduce a net with mesh points denoted by .xi; tj/D.ih; j /;wherei D0; 1; :::; MIj D0; 1; :::; N withhD1=M, DT =N. The initial line is denoted byjD0. The discrete approximation at.xi; tj/is designed by.u^j_i; v_i^j/and the exact solution to the problem (2.1) - (2.3) by.U_i^j; V_i^j/. We will use the following known notations [21] of forward and backward derivatives:

r_x;i^j Dr_i^j_C1 r_i^j

h ; r_x;i^j_N Dr_i^j r_{i 1}^j

h ; r_t;i^j Dr_i^jC1 r_i^j and inner products and norms:

.r^j; y^j/Dh

M 1

X

iD1

r_i^jy_i^j; .r^j; y^jDh

M

X

iD1

r_i^jy_i^j; kr^jk D.r^j; r^j/¹⁼²; kr^jj D.r^j; r^j¹⁼²:

For problem (2.1) - (2.3) let us consider the following finite difference scheme:

u^j_i^C1 u^j_i

a

0

@ h

jC1

X

kD1 M

X

`D1

h

.u^k_{x;`N</sub> /<sup>2</sup>C.v<sub>x;`}^k_N /² i

1

Au^j_xx;iN^C1Cg.u^j_i^C1/Df_1;i^j ;

v_i^jC1 v_i^j

a

0

@ h

jC1

X

kD1 M

X

`D1

h

.u^k_{x;`N</sub> /<sup>2</sup>C.v<sub>x;`}^k_N /²i 1

Av_xx;i^j_N^C1Cg.v^j_i^C1/Df_2;i^j ; iD1; 2; :::; M 1I j D0; 1; :::; N 1;

(3.1)

u^j₀Dv₀^j Du^j_x;MN Dv_x;M^j_N D0; j D0; 1; :::; N; (3.2)

u⁰_i DU0;i; v⁰_i DV0;i; iD0; 1; :::; M: (3.3) Multiplying equations in (3.1) scalarly byu^j_i^C1 andv_i^jC1 respectively, it is not difficult to get the inequalities:

kuⁿk²C

n

X

jD1

ku^j_xNj² < C; kvⁿk²C

n

X

jD1

kv_x^j_Nj² < C; nD1; 2; :::; N; (3.4)

where here and belowC is a positive constant independent fromandh.

The a priori estimates (3.4) guarantee the stability of the scheme (3.1) - (3.3). Note, that applying the technique as we prove convergence theorem blow, it is not difficult to prove the uniqueness of the solution of the scheme (3.1) - (3.3) too.

The main statement of the present section can be stated as follows.

Theorem 2. IfaDa.S /a0DC onst > 0,a⁰.S /0,a⁰⁰.S /0,gis monoton- ically increased function and problem (2.1) - (2.3) has a sufficiently smooth solution .U.x; t /; V .x; t //, then the solutionu^j D.u^j₁; u^j₂; : : : ; u_{M 1}^j /,v^j D.v^j₁; v₂^j; : : : ; v_{M 1}^j /, j D1; 2; : : : ; N of the difference scheme (3.1) - (3.3) tends to the solution of continu- ous problem (2.1) - (2.3) U^j D.U₁^j; U₂^j; : : : ; U_{M 1}^j /, V^j D.V₁^j; V₂^j; : : : ; V_{M 1}^j /, j D1; 2; : : : ; N as!0; h!0and the following estimates are true:

ku^j U^jk C.Ch/; kv^j V^jk C.Ch/: (3.5)

Proof. Introducing the differences´^j_i Du^j_i U_i^j andw^j_i Dv_i^j V_i^j we get the following relations:

´^j_t;i^C1 8

<

: a

0

@ h

jC1

X

kD1 M

X

`D1

h

.u^k_{x;`N</sub> /<sup>2</sup>C.v<sub>x;`}^k_N /²i 1 Au^j_x;iN^C1

a 0

@ h

jC1

X

kD1 M

X

`D1

h

.U_{x;`</sub><sup>k</sup><sub>N</sub> /<sup>2</sup>C.V<sub>x;`N}^k /²i 1 AU_x;iN^jC1

9

=

;

x

Cg.u^j_i^C1/ g.U_i^jC1/D 1;i^j ;

w^j_t;i^C1 8

<

: a

0

@ h

jC1

X

kD1 M

X

`D1

h

.u^k_{x;`N</sub> /<sup>2</sup>C.v<sub>x;`}^k_N /² i

1 Av_x;i^j_N^C1

a 0

@ h

jC1

X

kD1 M

X

`D1

h

.U_{x;`N</sub><sup>k</sup> /<sup>2</sup>C.V<sub>x;`N}^k /²i 1 AV_x;iN^jC1

9

=

;

x

Cg.v^j_i^C1/ g.V_i^jC1/D 2;i^j ;

(3.6)

´^j₀Dw^j₀ D´^j_x;MN Dw^j_x;MN D0; (3.7)

´⁰_i Dw_i⁰D0; (3.8)

where _1;i^j and _2;i^j are approximation errors of scheme (3.1) and

j

k;i DO.Ch/; kD1; 2:

Multiplying the first equation of system (3.6) scalarly by the grid function ´^jC1D .´^j₁^C1; ´^j₂^C1; : : : ; ´^j_{M 1}^C1/and using the boundary conditions (3.7) we get

k´^jC1k² .´^jC1; ´^j/C h

M

X

iD1

8

<

: a

0

@ h

jC1

X

kD1 M

X

`D1

h

.u^k_{x;`N</sub> /<sup>2</sup>C.v<sub>x;`}^k_N /² i

1 Au^j_x;iN^C1

a 0

@ h

jC1

X

kD1 M

X

`D1

h

.U_{x;`</sub><sup>k</sup><sub>N</sub> /<sup>2</sup>C.V<sub>x;`N}^k /²i 1 AU_x;iN^jC1

9

=

;

´^j_x;iN^C1

C

g.u^j_i^C1/ g.U_i^jC1/; u^jC1 U^jC1

D . ₁^j; ´^jC1/:

Analogously,

kw^jC1k² .w^jC1; w^j/C h

M

X

iD1

8

<

: a

0

@ h

jC1

X

kD1 M

X

`D1

h

.u^k_{x;`N</sub> /<sup>2</sup>C.v<sub>x;`}^k_N /²i 1 Av_x;i^j_N^C1

a 0

@ h

jC1

X

kD1 M

X

`D1

h

.U_{x;`</sub><sup>k</sup><sub>N</sub> /<sup>2</sup>C.V<sub>x;`N}^k /²i 1 AV_x;iN^jC1

9

=

; w_x;i^j_N^C1

C

g.v_i^jC1/ g.V_i^jC1/; v^jC1 V^jC1

D . ₂^j; w^jC1/:

Adding these two equalities and taking into account monotonicity of the function g, from these two equalities we have

k´^jC1k² .´^jC1; ´^j/C kw^jC1k² .w^jC1; w^j/

C h

M

X

iD1

8

<

: a

0

@ h

jC1

X

kD1 M

X

`D1

h

.u^k_{x;`N</sub> /<sup>2</sup>C.v<sub>x;`}^k_N /²i 1 Au^j_x;iN^C1

a 0

@ h

jC1

X

kD1 M

X

`D1

h

.U_{x;`</sub><sup>k</sup><sub>N</sub> /<sup>2</sup>C.V<sub>x;`N}^k /²i 1 AU_x;iN^jC1

9

=

;

´^j_x;iN^C1

C h

M

X

iD1

8

<

: a

0

@ h

jC1

X

kD1 M

X

`D1

h

.u^k_{x;`N</sub> /<sup>2</sup>C.v<sup>k</sup><sub>x;`N} /²i 1 Av^j_x;iN^C1

a 0

@ h

jC1

X

kD1 M

X

`D1

h

.U_{x;`</sub><sup>k</sup><sub>N</sub> /<sup>2</sup>C.V<sub>x;`N}^k /²i 1 AV_x;iN^jC1

9

=

; w_x;i^j_N^C1

. ₁^j; ´^jC1/ . ₂^j; w^jC1/:

(3.9)

Note that, using the Hadamard formula '.y/ '.´/D

1

Z

0

d

d'Œ´C.y ´/d;

below we prove one of the main inequality to estimate terms with nonlinear diffusion coefficienta.S /

( a h

jC1

X

kD1 M

X

`D1

h.u^k_{x;`N</sub> /<sup>2</sup>C.v<sup>k</sup><sub>x;`N} /²i

! u^j_x;iN^C1

a h

jC1

X

kD1 M

X

`D1

h.U_{x;`</sub><sup>k</sup><sub>N</sub> /<sup>2</sup>C.V<sub>x;`N}^k /²i

! U_x;iN^jC1

)

u^j_x;iN^C1 U_x;i^j_N^C1

C (

a h

jC1

X

kD1 M

X

`D1

h

.u^k_{x;`N</sub> /<sup>2</sup>C.v<sub>x;`}^k_N /²i

! v^j_x;iN^C1

a h

jC1

X

kD1 M

X

`D1

h.U_{x;`</sub><sup>k</sup><sub>N</sub> /<sup>2</sup>C.V<sub>x;`N}^k /²i

! V_x;iN^jC1

)

v_x;i^j_N^C1 V_x;iN^jC1

D

1

Z

0

( d da h

jC1

X

kD1 M

X

`D1

h

U_{x;`</sub><sup>k</sup><sub>N</sub> C.u<sup>k</sup><sub>x;`N} U_x;`N^k /i2

Ch

V_{x;`N</sub><sup>k</sup> C.v<sup>k</sup><sub>x;`N} V_x;`N^k /i2!

h

U_x;i^j_N^C1C.u^j_x;iN^C1 U_x;iN^jC1/io d

u^j_x;iN^C1 U_x;iN^jC1

C

1

Z

0

(d da h

jC1

X

kD1 M

X

`D1

h

U_{x;`</sub><sup>k</sup><sub>N</sub> C.u<sup>k</sup><sub>x;`N} U_x;`^k_N /i2

Ch

V_{x;`N</sub><sup>k</sup> C.v<sup>k</sup><sub>x;`N} V_x;`N^k /i2!

h

V_x;iN^jC1C.v^j_x;iN^C1 V_x;iN^jC1/io d

v_x;i^j_N^C1 V_x;iN^jC1

D2

1

Z

0

a⁰ h

jC1

X

kD1 M

X

`D1

h

U_{x;`N</sub><sup>k</sup> C.u<sup>k</sup><sub>x;`N} U_x;`N^k /i2

Ch

V_{x;`N</sub><sup>k</sup> C.v<sup>k</sup><sub>x;`N} V_x;`N^k /i2!

h

jC1

X

kD1 M

X

`D1

nh

U_{x;`N</sub><sup>k</sup> C.u<sup>k</sup><sub>x;`N} U_x;`N^k /i

u^k_{x;`N</sub> U<sub>x;`}^k_N

Ch

V_{x;`N</sub><sup>k</sup> C.v<sub>x;`}^k_N V_x;`N^k /i

v_{x;`</sub><sup>k</sup><sub>N</sub> V<sub>x;`N}^k o

h

U_x;iN^jC1C.u^j_x;iN^C1 U_x;iN^jC1/i d

u^j_x;iN^C1 U_x;i^j_N^C1

C

1

Z

0

a h

jC1

X

kD1 M

X

`D1

h

U_{x;`</sub><sup>k</sup><sub>N</sub> C.u<sup>k</sup><sub>x;`N} U_x;iN^k /i2

Ch

V_{x;`N</sub><sup>k</sup> C.v<sup>k</sup><sub>x;`N} V_x;`N^k /i2!

u^j_x;iN^C1 U_x;i^j_N^C1 d

u^j_x;iN^C1 U_x;i^j_N^C1

C2

1

Z

0

a⁰ h

jC1

X

kD1 M

X

`D1

h

U_{x;`</sub><sup>k</sup><sub>N</sub> C.u<sup>k</sup><sub>x;`N} U_x;`^k_N /i2

Ch

V_{x;`N</sub><sup>k</sup> C.v<sup>k</sup><sub>x;`N} V_x;`N^k /i2!

h

jC1

X

kD1 M

X

`D1

nh

U_{x;`N</sub><sup>k</sup> C.u<sup>k</sup><sub>x;`N} U_x;`N^k /i

u^k_{x;`N</sub> U<sub>x;`}^k_N

Ch

V_{x;`N</sub><sup>k</sup> C.v<sub>x;`}^k_N V_x;`N^k /i

v_{x;`</sub><sup>k</sup><sub>N</sub> V<sub>x;`N}^k o

h

V_x;iN^jC1C.v_x;i^j_N^C1 V_x;iN^jC1/i d

v^j_x;iN^C1 V_x;iN^jC1

C

1

Z

0

a h

jC1

X

kD1 M

X

`D1

h

U_{x;`N</sub><sup>k</sup> C.u<sup>k</sup><sub>x;`N} U_x;`^k_N /i2

Ch

V_{x;`N</sub><sup>k</sup> C.v<sub>x;`}^k_N V_x;`N^k /i2!

v_x;`^j_N^C1 V_x;iN^jC1 d

v_x;i^j_N^C1 V_x;iN^jC1

D2

1

Z

0

a⁰ h

jC1

X

kD1 M

X

`D1

h

U_{x;`N</sub><sup>k</sup> C.u<sup>k</sup><sub>x;`N} U_x;`N^k /i2

Ch

V_{x;`N</sub><sup>k</sup> C.v<sup>k</sup><sub>x;`N} V_x;`N^k /i2!

h

jC1

X

kD1 M

X

`D1

nh

U_{x;`N</sub><sup>k</sup> C.u<sup>k</sup><sub>x;`N} U_x;`N^k /i

u^k_{x;`N</sub> U<sub>x;`}^k_N

Ch

V_{x;`N</sub><sup>k</sup> C.v<sub>x;`}^k_N V_x;`N^k /i

v_{x;`</sub><sup>k</sup><sub>N</sub> V<sub>x;`N}^k o

nh

U_x;i^j_N^C1C.u^j_x;iN^C1 U_x;iN^jC1/i

u^j_x;iN^C1 U_x;iN^jC1

Ch

V_x;iN^jC1C.v^j_x;iN^C1 V_x;iN^jC1/i

v^j_x;iN^C1 V_x;iN^jC1o d

C

1

Z

0

a h

jC1

X

kD1 M

X

`D1

h

U_{x;`</sub><sup>k</sup><sub>N</sub> C.u<sup>k</sup><sub>x;`N} U_x;iN^k /i2

Ch

V_{x;`N</sub><sup>k</sup> C.v<sup>k</sup><sub>x;`N} V_x;`N^k /i2!

u^j_x;iN^C1 U_x;i^j_N^C12

C

v^j_x;iN^C1 V_x;iN^jC12 d

D2

1

Z

0

a⁰ h

jC1

X

kD1 M

X

`D1

h

U_{x;`</sub><sup>k</sup><sub>N</sub> C.u<sup>k</sup><sub>x;`N} U_x;`N^k /i2

Ch

V_{x;`N</sub><sup>k</sup> C.v<sup>k</sup><sub>x;`N} V_x;`N^k /i2!

^jC1./^j_t./ d

C

1

Z

0

a h

jC1

X

kD1 M

X

`D1

h

U_{x;`N</sub><sup>k</sup> C.u<sup>k</sup><sub>x;`N} U_x;`^k_N /i2

Ch

V_{x;`N</sub><sup>k</sup> C.v<sub>x;`}^k_N V_x;`N^k /i2!

u^j_x;iN^C1 U_x;i^j_N^C12

C

v^j_x;iN^C1 V_x;iN^jC12 d;

where

^jC1./D h

jC1

X

kD1 M

X

`D1

nh

U_{x;`N</sub><sup>k</sup> C.u<sup>k</sup><sub>x;`N} U_x;`N^k /i

u^k_{x;`N</sub> U<sub>x;`}^k_N

Ch

V_{x;`N</sub><sup>k</sup> C.v<sub>x;`}^k_N V_x;`N^k /i

v_{x;`</sub><sup>k</sup><sub>N</sub> V<sub>x;`N}^k o

; ⁰./D0;

and therefore,

^j_t./Dh

M

X

`D1

h

U_{x;`N</sub><sup>jC1</sup>C.u<sup>j</sup><sub>x;`N}^C1 U_x;`^j_N^C1/i

u^j_{x;`N</sub><sup>C1</sup> U<sub>x;`N}^jC1

Ch

V_{x;`N</sub><sup>jC1</sup>C.v<sup>j</sup><sub>x;`N}^C1 V_x;`N^jC1/i

v^j_{x;`N</sub><sup>C1</sup> V<sub>x;`N}^jC1 : Introducing the following notation

s^jC1./D h

jC1

X

kD1 M

X

`D1

h

U_{x;`N</sub><sup>k</sup> C.u<sup>k</sup><sub>x;`N} U_x;`^k_N /i2

Ch

V_{x;`N</sub><sup>k</sup> C.v<sup>k</sup><sub>x;`N} V_x;`N^k /i2

;

from the previous equality we have 8

<

: a

0

@ h

jC1

X

kD1 M

X

`D1

h

.u^k_{x;`N</sub> /<sup>2</sup>C.v<sub>x;`}^k_N /²i 1 Au^j_x;iN^C1

a 0

@ h

jC1

X

kD1 M

X

`D1

h

.U_{x;`</sub><sup>k</sup><sub>N</sub> /<sup>2</sup>C.V<sub>x;`N}^k /² i

1 AU_x;iN^jC1

9

=

;

u^j_x;iN^C1 U_x;i^j_N^C1

C 8

<

: a

0

@ h

jC1

X

kD1 M

X

`D1

h

.u^k_{x;`N</sub> /<sup>2</sup>C.v<sub>x;`}^k_N /²i 1 Av_x;i^j_N^C1

a 0

@ h

jC1

X

kD1 M

X

`D1

h

.U_{x;`</sub><sup>k</sup><sub>N</sub> /<sup>2</sup>C.V<sub>x;`N}^k /² i

1 AV_x;iN^jC1

9

=

;

v_x;i^j_N^C1 V_x;iN^jC1

D2

1

Z

0

a⁰

s^jC1./

^jC1j_t d

C

1

Z

0

a

s^jC1./

u^j_x;iN^C1 U_x;i^j_N^C12

C

v^j_x;iN^C1 V_x;iN^jC12 d: