Boundary-value problems with non-local condition for degenerateparabolic equations

7252019 Boundary-value problems with non-local condition for degenerateparabolic equations

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Contemporary Analysis and Applied Mathematics

Vol1 No1 42-48 2013

Boundary-value problems with non-local condition for degenerate

parabolic equations

John M Rassias1 and Erkinjon T Karimov2

1 Pedagogical department Mathematics and Informatics section National and Capodistrian

University of Athens Athens Greece

2 Institute of Mathematics National University of Uzbekistan named after Mirzo Ulugbek

Tashkent Uzbekistan

e-mail jrassiasprimeduuoagr erkinjongmailcom

Abstract In this work we deal with degenerate parabolic equations with three lines

of degeneration Using rdquoa-b-crdquo method we prove the uniqueness theorems defining

conditions to parameters We show nontrivial solutions for considered problems

when uniqueness conditions to parameters participating in the equations are not

fulfilled

Key words Degenerate parabolic equation boundary-value problems with non-

local initial conditions classical rdquoa-b-crdquo method eigenvalues and eigenfunctions

1 Introduction

Degenerate partial differential equations have many applications in practical problems Even

finding some particular solutions for these kinds of equations is interesting for specialists in

numerical analysis

We note works by Friedman Gorkov [12] on finding solutions for degenerate parabolic

equations Regarding the usage of nonlocal conditions used in the present work for parabolic

equations we cite the work by Shopolov [4] There are some papers [36-12] containing interesting

results on investigation various type of degenerate partial differential equations For detailed

information on degenerate partial differential equations one can find in the monograph [5]

This work is continuation of the work [13] and only for convenience of the reader we did not

omit some details of proofs

42



John M Rassias Erkinjon T Karimov

2 Boundary problem with nonlocal condition for parabolic

equations with three lines of degeneration

Let Ω be a simple-connected bounded domain in R3

with boundaries S

i (i = 1

6) Here

S 1 = (x y t) t = 0 0 lt x lt 1 0 lt y lt 1

S 2 = (x y t) x = 1 0 lt y lt 1 0 lt t lt 1

S 3 = (x y t) y = 0 0 lt x lt 1 0 lt t lt 1




We consider a degenerate parabolic equation

xnymut = tkymuxx + tkxnuyy minusλtkxnymu (21)

in the domain Ω Here m gt 0 n gt 0 k gt 0 λ = λ1 + iλ2 λ1 λ2 isin R

Problem 21 To find a function u (x y t) satisfying the following conditions

1 u (x y t) isin C 852008

Ω852009 cap C 2

21xyt (Ω)

2 u (x y t) satisfies the equation (21) in Ω

3 u (x y t) satisfies boundary conditions

u (x y t) |S 2cupS 3cupS 4cupS 5 = 0 (22)

4 u (x y t) satisfies a non-local condition

u (xy 0) = αu (xy 1) (23)

Here α = α1 + iα2 α1 α2 are real numbers and moreover α21 + α2

2 = 0

Theorem 21 If α21 + α2

2 lt 1 λ1 ge 0 and a solution of Problem 21 exists then it is unique

Proof Let us suppose that Problem 21 has two solutions u1 u2 Denoting u = u1 minus u2 we

claim that u equiv 0 in Ω

43



Boundary-value problems with non-local condition for degenerate parabolic equations

First we multiply Equation (21) to the function u (x y t) which is complex conjugate func-

tion of u (x y t) Then integrate it along the domain Ωε with boundaries

S 1ε =

(x y t) t = ε ε lt x lt 1

minusε ε lt y lt 1

minusε

S 2ε = (x y t) x = 1 minus ε ε lt y lt 1 minus ε ε lt t lt 1 minus ε

S 3ε = (x y t) y = ε ε lt x lt 1 minus ε ε lt t lt 1 minus ε

S 4ε = (x y t) x = ε ε lt y lt 1 minus ε ε lt t lt 1 minus ε

S 5ε = (x y t) y = 1 minus ε ε lt x lt 1 minus ε ε lt t lt 1 minus ε

S 6ε = (x y t) t = 1 minus ε ε lt x lt 1 minus ε ε lt y lt 1 minus ε

Then taking real part of the obtained equality and considering

Re852008

tkymuuxx

852009 = Re

852008tkymuux

852009x minus tkym |ux|2

Re 852008tkxnuuyy852009 = Re 852008tkxnuuy

852009y minus tkxn |uy|2

Re (xnymuut) =

10486161

2xnym |u|2

1048617t

after using Greenrsquos formula we pass to the limit at ε rarr 0 Then we get

intint part Ω

Re1048667

tkymuux cos (ν x) + tkxnuuy cos (ν y) minus 12

xnym |u|2 cos(ν t)1048669

dτ

=intintint Ω

983080tkym |ux|2 + tkxn |uy|2 + λ1tkxnym |u|2

983081dσ

where ν is outer normal Considering Re [uux] = Re [uux] Re [uuy] = Re [uuy] we obtain

Re991787991787 S 1

1

2xnym |u|2 dτ 1 +

991787991787 S 2

tkymRe [uux] dτ 2 minus991787991787 S 3

tkxnRe [uuy] dτ 3

minus991787991787 S 4

tkymRe [uux] dτ 4 +

991787991787 S 5

tkxnRe [uuy] dτ 5 minusRe

991787991787 S 6

1

2xnym |u|2 dτ 6 (24)

=

991787991787991787 Ω


983081dσ

From (24) and by using conditions (22) (23) we find

1

2

8520591 minus 852008α2

1 + α22

852009852061 1991787 0

1991787 0

xnym |u (xy 1)|2 dxdy+

+

991787991787991787 Ω


983081dσ = 0 (25)

44




Setting α21 + α2

2 lt 1 λ1 ge 0 from (24) we have u (x y t) equiv 0 in Ω

Theorem 21 is proved

We find below non-trivial solutions of Problem 21 at some values of parameter λ for which

the uniqueness condition Reλ = λ1 ge 0 is not fulfilled

We search the solution of Problem 21 as follows

u (x y t) = X (x) middot Y (y) middot T (t) (26)

After some evaluations we obtain the following eigenvalue problems

X primeprime (x) + micro1xnX (x) = 0

X (0) = 0 X (1) = 0(27)

Y primeprime

(y) + micro2ym

Y (y) = 0

Y (0) = 0 Y (1) = 0(28)

T prime (t) + (λ + micro) tkT (t) = 0

T (0) = αT (1) (29)

Here micro = micro1 + micro2 is a Fourier constant

Solving eigenvalue problems (27) and (28) we find

micro1l =

1048616n + 2

2

micro1l

10486172

micro2 p =

1048616m + 2

2

micro2 p

10486172

(210)

X l (x) = Al

1048616 2

n + 2

1048617 1n+2

micro1

2(n+2)

1l x12 J 1

n+2

10486162radic

micro1l

n + 2 x

n+22

1048617 (211)

Y p (y) = B p

1048616 2

m + 2

1048617 1m+2

micro1

2(m+2)

2 p y12 J 1

m+2

10486162radic

micro2 p

m + 2xm+22

1048617 (212)

where l p = 1 2 micro1l and micro2 p are roots of equations J 1n+2

(x) = 0 and J 1m+2

(y) = 0 respec-

tively J p(middot) is the first kind Bessel function of p-th order

The eigenvalue problem (29) has non-trivial solution only when

α1

α21+α22 = eminusλ1+microlpk+1

cos λ2

k+1 α2

α21+α22= eminus

λ1+microlpk+1 sin λ2

k+1

Here λ = λ1 + iλ2 α = α1 + iα2 microlp = micro1l + micro2 p After elementary calculations we get

λ1 = minusmicrolp + k + 1

2 ln

852008α21 + α2

2

852009 λ2 = (k + 1)

983131arctan

α2

α1

+ sπ

983133 s isin Z + (213)

45




Corresponding eigenfunctions have the form

T s (t) = C se

983131minus ln

radic α21+α22minusi

983080arctan

α2α1

+sπ983081983133

tk+1

(214)

Considering (26) (211) (212) and (214) we can write non-trivial solutions of Problem

21 in the following form

ulps (x y t) = Dlps

1048616 2

n + 2

1048617 1n+2

1048616 2

m + 2

1048617 1m+2

micro1

2(n+2)

1l micro1

2(m+2)

2 p

radic xy

timesJ 1n+2

10486162radic

micro1l

n + 2 x

n+22

1048617middot J 1

m+2

10486162radic

micro2 p

m + 2ym+22

1048617e

983131minus ln


983080arctan

α2α1

+sπ983081983133

tk+1

where Dlps = Al middotB p middotC s is constant

Remark 22 One can easily see that λ1 lt 0 in (213) which contradicts to condition Reλ =

λ1 ge 0 of the Theorem 21

Remark 23 The following problems can be studied in similar way Instead of condition (22)

we put zero conditions on surfaces as follows

Problemrsquos name P 2 P 3 P 4 P 5 P 6 P 7 P 8 P 9

S 2 ux u u ux u u ux u

S 3 uy u uy u uy u u u

S 4 u ux u ux u ux u u

S 5 u uy uy u u u u uy

3 Boundary problem with two nonlocal conditions

We consider equation

tkymU xx minus tkxnU y minus xnymU t minusΛxnymtkU = 0 (31)

in the domain Ω Here m n k gt 0 Λ = Λ11 + Λ21 + i (Λ12 + Λ22) Λjj isin R

Problem 31 To find a function U (x y t) from the class of

W =983163

U (x y t) U isin C 852008

Ω852009 cap C 2

11xyt (Ω)

983165

satisfying Equation (31) in Ω and boundary condition

U (x y t) |S 2cupS 3 = 0 (32)

46




nonlocal conditions

U (x 0 t) = βU (x 1 t) (33)

U (xy 0) = γ U (xy 1)

Here γ = γ 1 + iγ 2 β = β 1 + iβ 2

Theorem 31 If β 21 + β 22 lt 1 γ 21 + γ 22 lt 1 Λ11 + Λ21 ge 0 and a solution of Problem 31 exists

then it is unique

Proof of this theorem will be done similarly as in the Theorem 21 Nontrivial solutions of

this problem can be written as follows

U ls (x y t) = E ls

1048616 2

n + 2

1048617 1n+2

micro1

2(n+2)

1l

radic xJ 1

n+2

10486162radic

micro1l

n + 2 x

n+22

1048617

timese

983131minus lnradic β21+β22minusi

983080arctan

β2β1

+sπ983081983133

ym+1

e

983131minus lnradic γ 21+γ 22minusi

983080arctan

γ2γ1

+sπ983081983133

tk+1

where E ls are constants

We note that above given nontrivial solutions exist only when

β1β21+β22

= eminusΛ11+micro1lm+1 cos Λ12

m+1

β2β21+β22

= eminusΛ11+micro1lm+1 sin Λ12

m+1

γ 1γ 21+γ 22

= eminusΛ21+micro1lk+1 cos Λ22

k+1

γ 2γ 21+γ 22

= eminusΛ21+micro1lk+1 sin λ22

k+1

Here

Λ11 = minusmicro1l + (m + 1)ln

radic β 21 + β 22 Λ12 = (m + 1)arctan

β 2

β 1

Λ21 = minusmicro1l + (k + 1)ln

radic γ 21 + γ 22 Λ22 = (k + 1) arctan

γ 2

γ 1

References

[1] A Friedman Fundamental solutions for degenerate parabolic equations Acta Math-

Djursholm 133(1) (1975) 171-217

[2] YuP Gorkov Construction of a fundamental solution of parabolic equation with degener-

ation Calcul Methods and Programming 6 (2005) 66-70

[3] A Hasanov Fundamental solutions of generalized bi-axially symmetric Helmholtz equation

Complex Var Elliptic Eqn 52(8) (2007) 673-683

47




[4] NN Shopolov Mixed problem with non-local initial condition for a heat conduction equa-

tion Reports of Bulgarian Academy of Sciences 3(7) (1981) 935-936

[5] MM Smirnov Degenerate Elliptic and Hyperbolic Equations Nauka Moscow 1966

[6] JM Rassias Uniqueness of quasi-regular solutions for a bi-parabolic elliptic bi-hyperbolic

Tricomi problem Complex Var Elliptic Eqn 47(8) (2002) 707-718

[7] JM Rassias Mixed Type Partial Differential Equations in Rn PhD Thesis University

of California Berkeley USA 1977

[8] JM Rassias Lecture Notes on Mixed Type Partial Differential Equations World Scientific

1990

[9] JM Rassias Mixed type partial differential equations with initial and boundary values influid mechanics Int J Appl Math Stat 13(J08) (2008) 77-107

[10] JM Rassias A Hasanov Fundamental solutions of two degenerated elliptic equations and

solutions of boundary value problems in infinite area Int J Appl Math Stat 8(M07)

(2007) 87-95

[11] JM Rassias Uniqueness of quasi-regular solutions for a parabolic elliptic-hyperbolic Tri-

comi problem Bull Inst Math Acad Sinica 25(4) (1997) 277-287

[12] GC Wen The mixed boundary-value problem for second order elliptic equations with

degenerate curve on the sides of an angle Math Nachr 279(13-14) (2006) 1602-1613

[13] JM Rassias ET Karimov Boundary-value problems with non-local initial condition for

parabolic equations with parameter European Journal of Pure and Applied Mathematics

3(6) (2010) 948-957

48




2 Boundary problem with nonlocal condition for parabolic

equations with three lines of degeneration

Let Ω be a simple-connected bounded domain in R3

with boundaries S

i (i = 1

6) Here







We consider a degenerate parabolic equation

xnymut = tkymuxx + tkxnuyy minusλtkxnymu (21)

in the domain Ω Here m gt 0 n gt 0 k gt 0 λ = λ1 + iλ2 λ1 λ2 isin R

Problem 21 To find a function u (x y t) satisfying the following conditions

1 u (x y t) isin C 852008

Ω852009 cap C 2

21xyt (Ω)

2 u (x y t) satisfies the equation (21) in Ω

3 u (x y t) satisfies boundary conditions

u (x y t) |S 2cupS 3cupS 4cupS 5 = 0 (22)

4 u (x y t) satisfies a non-local condition

u (xy 0) = αu (xy 1) (23)

Here α = α1 + iα2 α1 α2 are real numbers and moreover α21 + α2

2 = 0

Theorem 21 If α21 + α2

2 lt 1 λ1 ge 0 and a solution of Problem 21 exists then it is unique

Proof Let us suppose that Problem 21 has two solutions u1 u2 Denoting u = u1 minus u2 we

claim that u equiv 0 in Ω

43






S 1ε =



minusε







Re852008

tkymuuxx

852009 = Re

852008tkymuux




Re (xnymuut) =

10486161

2xnym |u|2

1048617t


intint part Ω

Re1048667



dτ

=intintint Ω


983081dσ


Re991787991787 S 1

1

2xnym |u|2 dτ 1 +

991787991787 S 2


tkxnRe [uuy] dτ 3

minus991787991787 S 4


991787991787 S 5


991787991787 S 6

1

2xnym |u|2 dτ 6 (24)

=

991787991787991787 Ω


983081dσ


1

2

8520591 minus 852008α2

1 + α22

852009852061 1991787 0

1991787 0


+

991787991787991787 Ω


983081dσ = 0 (25)

44




Setting α21 + α2









X (0) = 0 X (1) = 0(27)

Y primeprime

(y) + micro2ym

Y (y) = 0

Y (0) = 0 Y (1) = 0(28)


T (0) = αT (1) (29)



micro1l =

1048616n + 2

2

micro1l

10486172

micro2 p =

1048616m + 2

2

micro2 p

10486172

(210)

X l (x) = Al

1048616 2

n + 2

1048617 1n+2

micro1

2(n+2)

1l x12 J 1

n+2

10486162radic

micro1l

n + 2 x

n+22

1048617 (211)

Y p (y) = B p

1048616 2

m + 2

1048617 1m+2

micro1

2(m+2)

2 p y12 J 1

m+2

10486162radic

micro2 p

m + 2xm+22

1048617 (212)


(x) = 0 and J 1m+2

(y) = 0 respec-



α1


cos λ2

k+1 α2

α21+α22= eminus


k+1



2 ln

852008α21 + α2

2

852009 λ2 = (k + 1)

983131arctan

α2

α1

+ sπ

983133 s isin Z + (213)

45





T s (t) = C se

983131minus ln


983080arctan

α2α1

+sπ983081983133

tk+1

(214)



ulps (x y t) = Dlps

1048616 2

n + 2

1048617 1n+2

1048616 2

m + 2

1048617 1m+2

micro1

2(n+2)

1l micro1

2(m+2)

2 p

radic xy

timesJ 1n+2

10486162radic

micro1l

n + 2 x

n+22

1048617middot J 1

m+2

10486162radic

micro2 p

m + 2ym+22

1048617e

983131minus ln


983080arctan

α2α1

+sπ983081983133

tk+1
















W =983163


Ω852009 cap C 2

11xyt (Ω)

983165


U (x y t) |S 2cupS 3 = 0 (32)

46




nonlocal conditions

U (x 0 t) = βU (x 1 t) (33)

U (xy 0) = γ U (xy 1)

Here γ = γ 1 + iγ 2 β = β 1 + iβ 2


then it is unique



U ls (x y t) = E ls

1048616 2

n + 2

1048617 1n+2

micro1

2(n+2)

1l

radic xJ 1

n+2

10486162radic

micro1l

n + 2 x

n+22

1048617

timese


983080arctan

β2β1

+sπ983081983133

ym+1

e


983080arctan

γ2γ1

+sπ983081983133

tk+1



β1β21+β22


m+1

β2β21+β22


m+1

γ 1γ 21+γ 22


k+1

γ 2γ 21+γ 22


k+1

Here



β 2

β 1



γ 2

γ 1

References


Djursholm 133(1) (1975) 171-217





47












1990




(2007) 87-95







3(6) (2010) 948-957

48






S 1ε =



minusε







Re852008

tkymuuxx

852009 = Re

852008tkymuux




Re (xnymuut) =

10486161

2xnym |u|2

1048617t


intint part Ω

Re1048667



dτ

=intintint Ω


983081dσ


Re991787991787 S 1

1

2xnym |u|2 dτ 1 +

991787991787 S 2


tkxnRe [uuy] dτ 3

minus991787991787 S 4


991787991787 S 5


991787991787 S 6

1

2xnym |u|2 dτ 6 (24)

=

991787991787991787 Ω


983081dσ


1

2

8520591 minus 852008α2

1 + α22

852009852061 1991787 0

1991787 0


+

991787991787991787 Ω


983081dσ = 0 (25)

44




Setting α21 + α2









X (0) = 0 X (1) = 0(27)

Y primeprime

(y) + micro2ym

Y (y) = 0

Y (0) = 0 Y (1) = 0(28)


T (0) = αT (1) (29)



micro1l =

1048616n + 2

2

micro1l

10486172

micro2 p =

1048616m + 2

2

micro2 p

10486172

(210)

X l (x) = Al

1048616 2

n + 2

1048617 1n+2

micro1

2(n+2)

1l x12 J 1

n+2

10486162radic

micro1l

n + 2 x

n+22

1048617 (211)

Y p (y) = B p

1048616 2

m + 2

1048617 1m+2

micro1

2(m+2)

2 p y12 J 1

m+2

10486162radic

micro2 p

m + 2xm+22

1048617 (212)


(x) = 0 and J 1m+2

(y) = 0 respec-



α1


cos λ2

k+1 α2

α21+α22= eminus


k+1



2 ln

852008α21 + α2

2

852009 λ2 = (k + 1)

983131arctan

α2

α1

+ sπ

983133 s isin Z + (213)

45





T s (t) = C se

983131minus ln


983080arctan

α2α1

+sπ983081983133

tk+1

(214)



ulps (x y t) = Dlps

1048616 2

n + 2

1048617 1n+2

1048616 2

m + 2

1048617 1m+2

micro1

2(n+2)

1l micro1

2(m+2)

2 p

radic xy

timesJ 1n+2

10486162radic

micro1l

n + 2 x

n+22

1048617middot J 1

m+2

10486162radic

micro2 p

m + 2ym+22

1048617e

983131minus ln


983080arctan

α2α1

+sπ983081983133

tk+1
















W =983163


Ω852009 cap C 2

11xyt (Ω)

983165


U (x y t) |S 2cupS 3 = 0 (32)

46




nonlocal conditions

U (x 0 t) = βU (x 1 t) (33)

U (xy 0) = γ U (xy 1)

Here γ = γ 1 + iγ 2 β = β 1 + iβ 2


then it is unique



U ls (x y t) = E ls

1048616 2

n + 2

1048617 1n+2

micro1

2(n+2)

1l

radic xJ 1

n+2

10486162radic

micro1l

n + 2 x

n+22

1048617

timese


983080arctan

β2β1

+sπ983081983133

ym+1

e


983080arctan

γ2γ1

+sπ983081983133

tk+1



β1β21+β22


m+1

β2β21+β22


m+1

γ 1γ 21+γ 22


k+1

γ 2γ 21+γ 22


k+1

Here



β 2

β 1



γ 2

γ 1

References


Djursholm 133(1) (1975) 171-217





47












1990




(2007) 87-95







3(6) (2010) 948-957

48




Setting α21 + α2









X (0) = 0 X (1) = 0(27)

Y primeprime

(y) + micro2ym

Y (y) = 0

Y (0) = 0 Y (1) = 0(28)


T (0) = αT (1) (29)



micro1l =

1048616n + 2

2

micro1l

10486172

micro2 p =

1048616m + 2

2

micro2 p

10486172

(210)

X l (x) = Al

1048616 2

n + 2

1048617 1n+2

micro1

2(n+2)

1l x12 J 1

n+2

10486162radic

micro1l

n + 2 x

n+22

1048617 (211)

Y p (y) = B p

1048616 2

m + 2

1048617 1m+2

micro1

2(m+2)

2 p y12 J 1

m+2

10486162radic

micro2 p

m + 2xm+22

1048617 (212)


(x) = 0 and J 1m+2

(y) = 0 respec-



α1


cos λ2

k+1 α2

α21+α22= eminus


k+1



2 ln

852008α21 + α2

2

852009 λ2 = (k + 1)

983131arctan

α2

α1

+ sπ

983133 s isin Z + (213)

45





T s (t) = C se

983131minus ln


983080arctan

α2α1

+sπ983081983133

tk+1

(214)



ulps (x y t) = Dlps

1048616 2

n + 2

1048617 1n+2

1048616 2

m + 2

1048617 1m+2

micro1

2(n+2)

1l micro1

2(m+2)

2 p

radic xy

timesJ 1n+2

10486162radic

micro1l

n + 2 x

n+22

1048617middot J 1

m+2

10486162radic

micro2 p

m + 2ym+22

1048617e

983131minus ln


983080arctan

α2α1

+sπ983081983133

tk+1
















W =983163


Ω852009 cap C 2

11xyt (Ω)

983165


U (x y t) |S 2cupS 3 = 0 (32)

46




nonlocal conditions

U (x 0 t) = βU (x 1 t) (33)

U (xy 0) = γ U (xy 1)

Here γ = γ 1 + iγ 2 β = β 1 + iβ 2


then it is unique



U ls (x y t) = E ls

1048616 2

n + 2

1048617 1n+2

micro1

2(n+2)

1l

radic xJ 1

n+2

10486162radic

micro1l

n + 2 x

n+22

1048617

timese


983080arctan

β2β1

+sπ983081983133

ym+1

e


983080arctan

γ2γ1

+sπ983081983133

tk+1



β1β21+β22


m+1

β2β21+β22


m+1

γ 1γ 21+γ 22


k+1

γ 2γ 21+γ 22


k+1

Here



β 2

β 1



γ 2

γ 1

References


Djursholm 133(1) (1975) 171-217





47












1990




(2007) 87-95







3(6) (2010) 948-957

48





T s (t) = C se

983131minus ln


983080arctan

α2α1

+sπ983081983133

tk+1

(214)



ulps (x y t) = Dlps

1048616 2

n + 2

1048617 1n+2

1048616 2

m + 2

1048617 1m+2

micro1

2(n+2)

1l micro1

2(m+2)

2 p

radic xy

timesJ 1n+2

10486162radic

micro1l

n + 2 x

n+22

1048617middot J 1

m+2

10486162radic

micro2 p

m + 2ym+22

1048617e

983131minus ln


983080arctan

α2α1

+sπ983081983133

tk+1
















W =983163


Ω852009 cap C 2

11xyt (Ω)

983165


U (x y t) |S 2cupS 3 = 0 (32)

46




nonlocal conditions

U (x 0 t) = βU (x 1 t) (33)

U (xy 0) = γ U (xy 1)

Here γ = γ 1 + iγ 2 β = β 1 + iβ 2


then it is unique



U ls (x y t) = E ls

1048616 2

n + 2

1048617 1n+2

micro1

2(n+2)

1l

radic xJ 1

n+2

10486162radic

micro1l

n + 2 x

n+22

1048617

timese


983080arctan

β2β1

+sπ983081983133

ym+1

e


983080arctan

γ2γ1

+sπ983081983133

tk+1



β1β21+β22


m+1

β2β21+β22


m+1

γ 1γ 21+γ 22


k+1

γ 2γ 21+γ 22


k+1

Here



β 2

β 1



γ 2

γ 1

References


Djursholm 133(1) (1975) 171-217





47












1990




(2007) 87-95







3(6) (2010) 948-957

48




nonlocal conditions

U (x 0 t) = βU (x 1 t) (33)

U (xy 0) = γ U (xy 1)

Here γ = γ 1 + iγ 2 β = β 1 + iβ 2


then it is unique



U ls (x y t) = E ls

1048616 2

n + 2

1048617 1n+2

micro1

2(n+2)

1l

radic xJ 1

n+2

10486162radic

micro1l

n + 2 x

n+22

1048617

timese


983080arctan

β2β1

+sπ983081983133

ym+1

e


983080arctan

γ2γ1

+sπ983081983133

tk+1



β1β21+β22


m+1

β2β21+β22


m+1

γ 1γ 21+γ 22


k+1

γ 2γ 21+γ 22


k+1

Here



β 2

β 1



γ 2

γ 1

References


Djursholm 133(1) (1975) 171-217





47












1990




(2007) 87-95







3(6) (2010) 948-957

48












1990




(2007) 87-95







3(6) (2010) 948-957

48

Boundary-value problems with non-local condition for degenerateparabolic equations

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Transcript of Boundary-value problems with non-local condition for degenerateparabolic equations