Kernel of A Di¤usion Equation With Non-Constant Space-Time Di¤usion Coe¢ cient

Kernel of A Di¤usion Equation With Non-Constant Space-Time Di¤usion Coe¢ cient

McSylvester Ejighikeme Omaba

, Eze Raymond Nwaeze

Received 14 March 2019

Abstract

We compute a new solution of a di¤usion equation with space and time dependent variable di¤usion coe¢ cients in terms of a special function known as a con‡uent hypergeometric (Kummer) function. This new solution generalizes the already existing well-known fundamental solution of the constant di¤usion coe¢ cient heat equation (the Gaussian heat kernel).

1 Introduction

The study of di¤usion equation with di¤usion coe¢ cients is as old as Mathematics itself, see [2, 10, 16]

and their references. Fa and Lenzi in [5,6] and Fa [7] studied di¤usion equation with space-time-dependent di¤usion coe¢ cients. Speci…cally, in [5], they considered a power law di¤usion coe¢ cient and anomalous di¤usion equation in one-dimensional space of the type

@u(x; t)

@t =D @

@x jxj @u(x; t)

@x ; (1)

wheretis a scaled time and a function of t(that is placed inside the derivative). The solution to the above equation (1) is given by

u(x; t) =Ce

jxj2+

D(2+ )2t

t^{1=(2+ )} ; withC a normalization constant.

This research was motivated by a paper [11] which studied a fractional order di¤usion equation with a generalized di¤usion constant with its numerical and modeling applications:

@ P(x; t)

@t =D _; @ P(x; t)

@jxj ;

where P(x; t) is a di¤usion propagator, _@t^@ represents the Caputo time-fractional derivative for 0 <

1; _@^@_jxj represents the order of the Riesz space-fractional derivative for1< 2andD ; is the generalized di¤usion constant(distant =time ).

In what follows, therefore, we consider a di¤usion equation with both temporally and spatially-dependent (non-constant) di¤usion coe¢ cient

ut(x; t) (x; t) u(x; t) = 0; x2Rⁿ; t >0; (2) where (x; t) is a space-time di¤usion coe¢ cient. Di¤usion equations with temporally (time-dependent) and spatially (space-dependent) variables di¤usion coe¢ cients have many modeling applications. For the

Mathematics Sub ject Classi…cations: 35K05, 35K08, 33C15, 33C36.

yDepartment of Mathematics, College of Science, University of Hafr Al Batin, P. O Box 1803 Hafr Al Batin 31991, KSA

zDepartment of Mathematics and Computer Science, Alabama State University, Montgomery, AL 36101, USA

133

application of time-dependent di¤usion coe¢ cients see [3,13,15,18,19] and see [1,17,20] for the application of spatially variable di¤usion coe¢ cients. See also [8,9,14] for more applications. A special case of equation (2) is when (x; t) = a constant coe¢ cient, with the fundamental solution known as the usual heat kernel given by

u(x; t) =p(t; x) = 1

(4 t)ⁿ⁼²e ^jxj

2

4 t; x2Rⁿ; t >0; (3)

wherejxjrepresents the Euclidean norm ofxonRⁿ.

There is no known kernel (analytic solution) in literature, to the best of our knowledge, for the di¤usion coe¢ cient of non-constant term. We therefore seek to give a fundamental solution for the equation for a given space-time dependent di¤usion coe¢ cient

(x; t) =jxj^at^b; x2Rⁿ; t >0; a6= 2; b6= 1:

2 Remarks I

We assume thatxshould not be 0whenais negative. We thus obtain the following result.

Theorem 1 For (x; t) = jxj^at^b; x 2 Rⁿ; t > 0; a 6= 2; b 6= 1, then the general solution to (2) is a con‡uent hypergeometric (Kummer) function given by

u(x; t) = C₁ t^{n(1+b)2 a 1}

F₁ n

2 a;n a

2 a; 1 +b (2 a)²

jxj^{2 a} t^1+b + C2

a 2 a

n 2 a 2

a ^{n2 a2} 2 a 1 +b

n 2

2 a t^{(1+b)(n2 a 2)} jxj^{n 2}

1F₁ 2

2 a;1 n 2

2 a; 1 +b (2 a)²

jxj^{2 a}

t^1+b ; (4)

where C1 and C2 are some normalization constants (which can be determined uniquely by assuming some initial conditions on u(x; t)).

3 Remarks II

1. The above solution (4) is not always positive or non-negative for all the parametersa; b; nand there- fore for now represents a mathematical result (since the negative solutions do not represent any known applicable di¤usion processes like density, temperature, probability density functions, etc). Thus the solution has known physical applications for the positive values of u(x; t)and possible future applica- tions whenu(x; t)assumes negative values.

2. Our solution (4) agrees with solution (3) when a=b = 0by the property of the con‡uent hypergeo- metric function in (5), that is,

C1

t^{n2 1}F1

n 2;n

2; 1 4

jxj²

t =p(t; x)

is the Gaussian kernel withC1a normalization constant and thus equation (4) generalizes solution (3).

We brie‡y consider the con‡uent hypergeometric function and their properties.

4 Preliminary

For second-order di¤erential equations of the standard form d²w

dz² +p(z)dw

dz +q(z)w(z) = 0

withp(z) andq(z)some given complex-valued functions, they have some special functions as its solutions.

In particular is the con‡uent hypergeometric equation zd²w

dz² + (b z)dw

dz aw(z) = 0 whose solution is given by

1F1(a;b;z) :=

X1 n=0

(a)n

n!(b)_nzⁿ; (b6= 0; 1; 2; :::)

known as the con‡uent hypergeometric function (Kummer function) with the following integral representa- tions

1F₁(a;b;z) = (b) (a) (b a)

Z 1 0

e^zuu^{a 1}(1 u)^{b a 1}du;

where<(b)><(a)>0; argu= arg(1 u) = 0; and

1F₁(a;b;z) = 1 2 i

(b) (a)

Z i1 i1

(a+s) ( s)

(b+s) ( z)^sds;

wherea6= 0; 1; 2; :::; jarg( z)j< ₂:The con‡uent hypergeometric function has the following properties relating to some special and elementary functions, thus,

1F1(a;a;z) =e^z; (5)

1F₁ +1

2; 2 + 1; 2z = z

2 e^zI (z);

1F₁(1;n+ 1;z) =nz ⁿe^z (n; z);

1F₁ 1 2;3

2; z² = p

2z erf(z)

withI (z)the modi…ed Bessel function, (n; z)the incomplete gamma function anderf(z)the error function, respectively de…ned by

I (z) =i J (iz) =i X1 k=0

1 k! ( +k+ 1)

iz 2

2k+

;

(n; z) = Z z

0

t^{n 1}e ^tdt;

erf(z) = 2 p2

Z z 0

e ^t2dt:

See [12] and its respective references for more on con‡uent hypergeometric function.

5 Proof of Theorem 1

We assume a solution of the structure type (see [4]) u(x; t) = u( x; t) = 1

t u x

t ;1 = 1 t v x

t ; x2Rⁿ; t >0;

with = ¹_t andv(y) =u(y;1), where the constants ; and the functionv:Rⁿ!Rare to be found. Then from (2), we have

ut(x; t) jxj^at^b u(x; t) =ut(x; t) jxj t

a

t^{b+ a} u(x; t) = 0

and thus fory=_t^x (andjyj=^j_t^xj),

t ^{( +1)}v(y) t ^{( +1)}y:Dv(y) t ^{( +2 )}t^{b+ a}jyj^a v(y) = 0;

which implies that

t ^{( +1)}v(y) + t ^{( +1)}y:Dv(y) +t ^{[( b)+ (2 a)]}jyj^a v(y) = 0:

To simplify the equation, we let( b) + (2 a) = + 1) =₂^1+b_a. Thus with =₂^1+b_a we have v(y) + 1 +b

2 a y:Dv(y) +jyj^a v(y) = 0:

We further letv be a radially symmetrical solution, that is, v(y) =w(jyj) for somew:R!R. Therefore forr=jyj;

w+ 1 +b

2 a rw⁰+r^aw⁰⁰+n 1

r^{1 a} w⁰= w+ n 1

r^{1 a} + 1 +b

2 a r w⁰+r^aw⁰⁰= 0:

Multiply through byr^{n 1}and let =n ^1+b_{2 a} to obtain 1 +b

2 a (rⁿw)⁰+ (r^{a+n 1}w⁰)⁰ ar^{a+n 2}w⁰ = 0:

We …rst consider the case of time-dependent di¤usion coe¢ cient,a= 0to obtain 1 +b

2 (rⁿw)⁰+ (r^{n 1}w⁰)⁰= 0 and

w(r) =C e ^{(1+b4 )r2}

and the fundamental solution given by u(x; t) = 1

t v(x t ) = C

t e ^{(1+b4 )jxj}

2

t2 = C

t^n(1+b)=2e ^4t1+b1+bjxj2 = 1

(4 t)^n(1+b)=2e ^4t1+b1+bjxj2;

with = ^n(1+b)₂ ; = ^1+b₂ and C a normalization constant. This solution corresponds to (3) with = 1 whenb= 0. Next for the general casea; b6= 0, we solve forwin the second order di¤erential equation

r^aw⁰⁰+ n 1

r^{1 a} + 1 +b

2 a r w⁰+n 1 +b

2 a w= 0

to have

w(r) = C11F1

n

2 a;n a

2 a; 1 +b (2 a)²r^{2 a} +C₂ a

2 a (ⁿa 2²)

a(ⁿ2 a²) 2 a 1 +b

(ⁿ2 a²)

r ^{(n 2)}

1F1

2

2 a;1 n 2

2 a; 1 +b (2 a)²r^{2 a}

and withu(x; t) = _t¹w(^j_t^xj); = ^n(1+b)_{2 a} ; = ^1+b_{2 a} to get

u(x; t) = C₁ t^{n(1+b)2 a 1}

F₁ n

2 a;n a

2 a; 1 +b (2 a)²

jxj^{2 a} t^1+b +C2

a 2 a

(ⁿa 2²)

a(ⁿ2 a²) 2 a 1 +b

(ⁿ2 a²)t^{(1+b)(n2 a 2)} jxj^{n 2}

1F₁ 2

2 a;1 n 2

2 a; 1 +b (2 a)²

jxj^{2 a} t^1+b ; whereC₁ andC₂ are normalization constants.

6 Example

Whena=b= 1andn= 2, we have the solution given by

u(x; t) =C1

t^{4 1}F1 2;1; 2jxj

t² +C2 1F1 2;1; 2jxj

t² = C1

t⁴ +C2 1F1 2;1; 2jxj t² :

Next we plot graphs (see Pages 6 & 7) of the solution forC₁=C₂= 1;2;3;4;5;6whent= 1 andt= 2for di¤erentxvalues in the following interval[ 1;1];[ 2;2]; [ 3;3];[ 4;4];[ 5;5];[ 10;10]and[ 100;100].

7 Remarks III

The above example is non-negative for alljxj ^t2², that is;

u(x; t) = C₁

t⁴ +C_{2 1}F₁ 2;1; 2jxj

t² 0; 82jxj

t² 1 , jxj t² 2:

This implies that for a …xedx2Rⁿ and larget (that is,t growing very large, tending to in…nity), then the solution becomes non-negative.

Figure 1: Graphs of the solutionu(x; t) = ^C_t4¹ +C2 1F1

h

2;1; 2^j_t^x2^j

i where C1=C2= 1;2; :::;6andt= 1:

Figure 2: Graphs of the solutionu(x; t) = ^C_t4¹ +C2 1F1

h

2;1; 2^j_t^x2^j

i where C1=C2= 1;2; :::;6andt= 1:

Figure 3: Graphs of the solutionu(x; t) = ^C_t4¹ +C2 1F1

h

2;1; 2^j_t^x2^j

i where C1=C2= 1;2; :::;6andt= 2:

Figure 4: Graphs of the solutionu(x; t) = ^C_t4¹ +C_{2 1}F₁h

2;1; 2^j_t^x2^j

i where C1=C2= 1;2; :::;6andt= 2:

8 Conclusion

This solution arose from the curiosity of what happens to the heat equation when the di¤usion coe¢ cient becomes both time and space dependent. The result gave a con‡uent hypergeometric function which appli- cation of its property agrees with established result and thus the classical heat kernel in (3) is a particular case of this new kernel.

Acknowledgment. We want to appreciate the anonymous referee whose invaluable contribution and insight improved the content and context of the paper. The …rst author would want to acknowledge the support of the University of Hafr Al Batin, KSA, and also for providing a research friendly environment.

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