Some Partial Differential Equations In Ms. Khin Saw Mu

Some Partial Differential Equations In

ℝ

𝒏𝒏

Ms. Khin Saw Mu*

Department of Mathematics University of KyaukSe Kyaukse City, Myanmar

DOI: 10.29322/IJSRP.9.07.2019.p91113

http://dx.doi.org/10.29322/IJSRP.9.07.2019.p91113

Abstract- In this paper, the initial-value problems are studied and solved for homogeneous and non-homogeneous transport equations. Moreover, the fundamental solution and mean value theorem are used to derive Laplace's equation. Finally, general solution of one-dimensional wave equation is established to apply d'Alembert's formula.

.

Index Terms- homogeneous transport, non-homogenous transport, Laplace equation, Wave equation, d'Alembert's formula

I. INTRODUCTION

partial differential equation (PDE) is an equation involving an unknown function of two or more variables and certain of its partial derivatives.

Using the notation explained in a typical PDE, fix an integer k ≥ 1 and let U denote an open subset of Rn_.

An expression of the form

F(Dk_u(x),Dk-1u(x),……Du(x),u(x),x) = 0 (x ϵ U)

u called a k order partial differential equation, where

𝐹𝐹:𝑅𝑅𝑛𝑛𝑘𝑘

×𝑅𝑅𝑛𝑛𝑘𝑘−1

× … . .×𝑅𝑅𝑛𝑛_{×𝑅𝑅×𝑈𝑈 → 𝑅𝑅}

In this paper, three fundamental linear partial differential equations for which various explicit formulas for solutions are available. These are

the transport equation ut + b. Du = 0

Laplace’s equation ∆ u = 0

the wave equation utt - ∆ u = 0

The geometric notations are that

(i) A point in ℝ𝑛𝑛+1 will often be denoted as (x, t) = (x1,...,xn, t), and we usually interpret t = xn+1 as time.

A point𝑥𝑥 ∈ ℝ𝑖𝑖𝑛𝑛 will sometimes be written 𝑥𝑥= (𝑥𝑥′,𝑥𝑥_𝑛𝑛) for 𝑥𝑥′= (𝑥𝑥₁, … ,𝑥𝑥_𝑛𝑛−1)∈ ℝ𝑛𝑛−1

(ii) U, V and W usually denote open subsets of 𝑖𝑖𝑛𝑛 We write

𝑉𝑉 ⊂⊂ 𝑈𝑈 if 𝑉𝑉 ⊂ 𝑉𝑉� ⊂ 𝑈𝑈 𝑉𝑉�is compact, and say V is compactly contained in U.

(iii) 𝜕𝜕𝑈𝑈 = boundary of U,

𝑈𝑈�=𝑈𝑈𝑈𝑈𝜕𝜕𝑈𝑈 closure of U. (iv) 𝑈𝑈_𝑇𝑇 =𝑈𝑈× (0,𝑇𝑇)

(v) 𝐵𝐵0=�𝑦𝑦 ∈ ℝ𝑛𝑛�|𝑥𝑥 − 𝑦𝑦| <𝑟𝑟� = open ball with center x and radius r > 0.

(vi) B(x, r) = closed ball with center x and radius r > 0.

(vii) α(n) = volume of unit ball B(0, 1) in ℝ𝑛𝑛

=

𝜋𝜋 𝑛𝑛 2

Γ�𝑛𝑛₂+1�

where

Γ

(

𝑥𝑥

) =

∫ 𝑡𝑡

𝑥𝑥−1

𝑒𝑒

−𝑡𝑡

𝑑𝑑𝑡𝑡

∞

0

nα(n) = surface area of unit sphere ∂B(0, 1) in ℝ𝑛𝑛 (viii) If a = (a1,...,an) and b = (b1,...,bn) belong to 𝑖𝑖𝑛𝑛

𝑎𝑎.𝑏𝑏=∑ 𝑎𝑎𝑖𝑖𝑏𝑏𝑖𝑖, |𝑎𝑎| = (∑ 𝑎𝑎𝑛𝑛𝑖𝑖=1 𝑖𝑖2)

1

2, |𝑏𝑏| = (∑ 𝑏𝑏𝑛𝑛_{𝑖𝑖=1 𝑖𝑖}2)12

𝑛𝑛 𝑖𝑖=1

The notations of functions are that (i) If u: U→ ℝ we write

(𝑥𝑥) =�𝑢𝑢1_{(𝑥𝑥), … ,𝑢𝑢}𝑚𝑚_{(𝑥𝑥)� (𝑥𝑥 ∈ 𝑈𝑈)} we say u is smooth provided u is infinitely differentiable.

(ii) If u, v are two functions, we write 𝑢𝑢 ≡ 𝑣𝑣 to mean that u is identically equal to v; that is, the functions u, v agree for all values of their arguments. We use the notation 𝑢𝑢 ∶=𝑣𝑣 to define u as equaling v.

(iii) If u: U→ ℝ we write

𝑢𝑢(𝑥𝑥) =�𝑢𝑢1_{(𝑥𝑥), … ,𝑢𝑢}𝑚𝑚_{(𝑥𝑥)� (𝑥𝑥 ∈ 𝑈𝑈)}

The function uk _{is the k}th _{component to u, k = 1,...,m.}

(iv) Averages: ∱_{𝐵𝐵(𝑥𝑥,𝑟𝑟)}𝑓𝑓𝑑𝑑𝑦𝑦= 1

𝛼𝛼(𝑛𝑛)𝑟𝑟𝑛𝑛∱𝐵𝐵(𝑥𝑥,𝑟𝑟)𝑓𝑓𝑑𝑑𝑦𝑦 = average of f over the ball B(x, r)

and

∱_{𝐵𝐵(𝑥𝑥,𝑟𝑟)}𝑓𝑓𝑑𝑑𝑑𝑑 ∶= _{𝑛𝑛𝛼𝛼(𝑛𝑛)𝑟𝑟}1 𝑛𝑛−1∫𝜕𝜕𝐵𝐵(𝑥𝑥,𝑟𝑟)𝑓𝑓𝑑𝑑𝑑𝑑= average of f over the sphere ∂B(x, r).

Then, the partial differential equations are used to solve the transport equation, Laplace’s equation and the wave equation for initial value problems.

II. TRANSPORTEQUATION

One of the simplest partial differential equations is the transport equation with constant coefficients. This is the PDE

ut + b. D u = 0 in ℝ𝑛𝑛× (0,∞) (1)

where b is a fixed vector in ℝ𝑛𝑛,𝑏𝑏= (𝑏𝑏₁, … . ,𝑏𝑏₂)𝑎𝑎𝑎𝑎𝑑𝑑𝑢𝑢:ℝ𝑛𝑛×

[0,∞)→ ℝ is the unknown, u = u(x, t). Here 𝑥𝑥= (𝑥𝑥₁, … . ,𝑥𝑥₂)∈

ℝ𝑛𝑛 _{denotes a typical point in space, and}

t

≥

0

_{denotes a typical}

time. We write 𝐷𝐷𝑢𝑢=𝐷𝐷_𝑥𝑥𝑢𝑢= (𝑢𝑢_𝑥𝑥1, … . ,𝑢𝑢_{𝑥𝑥𝑛𝑛}) for the gradient of u with respect to spatial variables x.

For any point (𝑥𝑥,𝑡𝑡)∈ ℝ𝑛𝑛× (0,∞)we define

𝑧𝑧(𝑠𝑠)≔ 𝑢𝑢(𝑥𝑥+𝑠𝑠𝑏𝑏,𝑡𝑡+𝑠𝑠) (𝑠𝑠 ∈ ℝ)

𝑧𝑧(𝑠𝑠) =𝐷𝐷𝑢𝑢(𝑥𝑥+𝑠𝑠𝑏𝑏,𝑡𝑡+𝑠𝑠).𝑏𝑏+𝑢𝑢𝑡𝑡(𝑥𝑥+𝑠𝑠𝑏𝑏,𝑡𝑡+𝑠𝑠) = 0 the second equality holding owing to (1). Thus z(.) is a constant function of s, and consequently for each point (x, t), u is constant

on the line through (x, t) with the direction (𝑏𝑏, 1)∈ ℝ𝑛𝑛+1 Hence if we know the value of u at any point on each such line,know its value everywhere in ℝ𝑛𝑛× (0,∞)

III. INITIAL VALUE PROBLEM AND NONHOMOGENEOUS PROBLEM OF SOMEPARTIALDIFFERENT

EQUATIONS

Now it is the time to articulate the research work with ideas gathered in above steps by adopting any of below suitable approaches:

A. Initial Value Problem

We consider the initial-value problem

𝑢𝑢𝑡𝑡+𝑏𝑏.𝐷𝐷𝑢𝑢= 0 𝑖𝑖𝑎𝑎𝑅𝑅𝑛𝑛 × (0,∞)

𝑢𝑢=𝑔𝑔𝑜𝑜𝑎𝑎𝑅𝑅𝑛𝑛 _{× (0,∞)} �. (2) Here 𝑏𝑏 ∈ 𝑖𝑖𝑛𝑛and𝑔𝑔:𝑖𝑖𝑛𝑛→ 𝑖𝑖 are known, and the problem is to compute u. Given (x, t) as above, the line through (x, t) with direction (b, 1) is represented parametrically by (𝑥𝑥+𝑠𝑠𝑏𝑏,𝑡𝑡+

𝑠𝑠) (𝑠𝑠 ∈ 𝑅𝑅).This line hits the plane 𝛤𝛤 ≔ ℝ𝑛𝑛× {𝑡𝑡= 0} when s = −t, at the point(𝑥𝑥 − 𝑡𝑡𝑏𝑏, 0) .Since u is constant on the line and

𝑢𝑢(𝑥𝑥 − 𝑡𝑡𝑏𝑏, 0) =𝑔𝑔(𝑥𝑥 − 𝑡𝑡𝑏𝑏), we deduce

𝑢𝑢(𝑥𝑥,𝑡𝑡) =𝑔𝑔(𝑥𝑥 − 𝑡𝑡𝑏𝑏) (𝑥𝑥 ∈ ℝ𝑛𝑛_{,𝑡𝑡 ≥0) (3)} So, if (2) has a sufficiently regular solution u, it must certainly be given by (3). And conversely, it is easy to check directly that if g is𝐶𝐶1 then u defined by (3) is indeed a solution of (2).

B. Nonhomogeneous Problem

Next we consider the associated nonhomogeneous problem

𝑢𝑢𝑡𝑡+𝑏𝑏.𝐷𝐷𝑢𝑢= 0 𝑖𝑖𝑎𝑎𝑅𝑅𝑛𝑛 × (0,∞)

𝑢𝑢=𝑔𝑔𝑜𝑜𝑎𝑎𝑅𝑅𝑛𝑛 _{× (0,∞)} �. (4) As before fix (𝑥𝑥,𝑡𝑡)∈ ℝ𝑛𝑛+1 and, inspired by the calculation above, set 𝑧𝑧(𝑠𝑠)≔ 𝑢𝑢(𝑥𝑥+𝑠𝑠𝑏𝑏,𝑡𝑡+𝑠𝑠) for 𝑠𝑠 ∈ ℝ Then

𝑧𝑧(𝑠𝑠) =𝐷𝐷𝑢𝑢(𝑥𝑥+𝑠𝑠𝑏𝑏,𝑡𝑡+𝑠𝑠).𝑏𝑏+𝑢𝑢𝑡𝑡(𝑥𝑥+𝑠𝑠𝑏𝑏,𝑡𝑡+𝑠𝑠) =

𝑓𝑓(𝑥𝑥+𝑠𝑠𝑏𝑏,𝑡𝑡+𝑠𝑠)

Consequently

𝑢𝑢(𝑥𝑥,𝑡𝑡)− 𝑔𝑔(𝑥𝑥 − 𝑡𝑡𝑏𝑏) =𝑧𝑧(0)− 𝑧𝑧(−𝑡𝑡)

=∫ 𝑧𝑧_−𝑡𝑡0 (𝑠𝑠)𝑑𝑑𝑠𝑠

=∫ 𝑓𝑓_−𝑡𝑡0 (𝑥𝑥+𝑠𝑠𝑏𝑏,𝑡𝑡+𝑠𝑠)𝑑𝑑𝑠𝑠

By the change of variables t + s = s1, we get

𝑢𝑢(𝑥𝑥,𝑡𝑡)− 𝑔𝑔(𝑥𝑥 − 𝑡𝑡𝑏𝑏) =� 𝑓𝑓(𝑥𝑥+ (𝑠𝑠 − 𝑡𝑡)𝑏𝑏,𝑠𝑠1)𝑑𝑑𝑠𝑠1 𝑡𝑡

0

𝑢𝑢(𝑥𝑥,𝑡𝑡)− 𝑔𝑔(𝑥𝑥 − 𝑡𝑡𝑏𝑏) =∫ 𝑓𝑓₀𝑡𝑡 (𝑥𝑥+, (𝑠𝑠 − 𝑡𝑡)𝑏𝑏,𝑠𝑠)𝑑𝑑𝑠𝑠

and so

𝑢𝑢(𝑥𝑥,𝑡𝑡) =𝑔𝑔(𝑥𝑥 − 𝑡𝑡𝑏𝑏) +� 𝑓𝑓(𝑥𝑥+, (𝑠𝑠 − 𝑡𝑡)𝑏𝑏,𝑠𝑠)𝑑𝑑𝑠𝑠 𝑡𝑡

0

,

(

𝑥𝑥 ∈ ℝ

𝑛𝑛

,

𝑡𝑡 ≥

0)

(5)

Among the most important of all partial differential equations are undoubtedly Laplace’s equations

∆ u = 0 (6)

and Poisson's equation

−∆u = f. (7)

In both (6) and (7), 𝑥𝑥 ∈ 𝑈𝑈and the unknown is𝑢𝑢:𝑈𝑈� → ℝ u = u(x), where 𝑈𝑈 ⊂ 𝑅𝑅𝑛𝑛 is a given open set. In (7), the function

𝑓𝑓:𝑈𝑈 → ℝ is also given.

IV. DERIVATION OF SOLUTION FOR LAPLACE’S EQUATIONANDWAVEEQUATION Let u be a solution of Laplace's equation (6) in U = 𝑈𝑈=ℝ𝑛𝑛 of the form

u(x) = v(r),

where r = |x| = (x₁2+ L + x_n2)12and v is to be selected (if possible) so that ∆u = 0 holds. First note for i = 1,...,n that

We thus have

1

2 2 2 i i

1 n i

i

r

1

x

x

(x

x )

2x

.

x

2

x

r

−

∂

₌

₊

₊

_⋅

₌

₌

∂

L

i xi

i i

x

u

dv

r

u

v (r)

,

x

dr

x

r

∂

∂

_′

=

=

⋅

=

∂

∂

2

i i

x xi i i 2

x

v (r)

1

x

u

v (r)(

)

v (r)x (

)

r

r

r

r

′

′′

′

=

+

+

−

⋅

2 2

i i

2 3

x

1

x

v (r)

v (r)(

),

r

r

r

′′

′

Hence ∆u =

Since ∆u = 0, (8)

Integrating both sides,

If 𝑎𝑎 ≥3

where 𝑏𝑏= 𝑎𝑎 2−𝑛𝑛

If n =2, 𝑣𝑣′(𝑟𝑟) =𝑎𝑎_𝑟𝑟 v= a log r + c

𝑣𝑣(𝑟𝑟) =�𝑎𝑎log𝑏𝑏 𝑟𝑟+𝑐𝑐 (𝑎𝑎= 2 ) 𝑟𝑟𝑛𝑛−2+ 𝑐𝑐 (𝑎𝑎 ≥3) where b and c are constants.

The function is

∅(𝑥𝑥) =�−

1

2𝜋𝜋log|𝑥𝑥| (𝑎𝑎= 2 ) 1

𝑛𝑛(𝑛𝑛−2)𝛼𝛼(𝑛𝑛)|𝑥𝑥|𝑛𝑛−2 (𝑎𝑎 ≥3)

(9)

that defined for

𝑥𝑥 ∈ ℝ

𝑛𝑛, 𝑥𝑥 ≠0, is the fundamental solution of Laplace’s equations.

The wave equation is a simplified model for a vibrating string (n = 1), membrane (n = 2), or elastic solid (n = 3). In these physical interpretation u(x, t) represents the displacement in some

direction of the point x at time

t

≥

0.

the wave equation is utt −∆u = 0, (10)

and the nonhomogeneous wave equation is

utt −∆u = f (11)

subject to appropriate initial and boundary conditions. Hence t > 0 and 𝑥𝑥 ∈ 𝑈𝑈 where 𝑈𝑈 ⊂ 𝑅𝑅𝑛𝑛 is open. The unknown is 𝑢𝑢:𝑈𝑈�×

[0,∞)→ ℝ ,u = u(x, t), and the Laplacian ∆ is taken with respect to the spatial variables 𝑥𝑥= (𝑥𝑥₁, … ,𝑥𝑥_𝑛𝑛). In (11) the function

𝑓𝑓:𝑈𝑈× [0,∞)→ ℝis given.

Let V represent any smooth subregion of U. The acceleration within V is then

2

tt 2

V V

d

udx

u dx

dt

∫

=

∫

and the net contact force is

V

dS,

F

υ

∂

−

∫

⋅

where F denotes the force acting on V through ∂V and the mass density is taken to be unity.

Newton's law asserts that the mass times the acceleration equals the net force:

tt

v v

u dx

F

υ

dS.

∂

= −

⋅

∫

∫

This identity obtains for each subregion V and so utt = −div F.

For elastic bodies, F is a function of the displacement gradient Du, whence

utt + div F(Du) = 0

for small Du, the linearization

F

(Du)

≈

a Du

is often appropriate; and so

utt – a ∆u = 0.

This is the wave equation if a = 1.

This physical interpretation strongly suggests it will be mathematically appropriate to specify two initial conditions, on the displacement u and the velocity ut, at time t = 0.

V. DERIVATION OF D'ALEMBERT'S FORMULA We consider the initial-value problem for the one-dimensional wave equation in all of Ρ:

utt−uxx= 0 in Rn × (0,∞)

u = g, ut= h on Rn × (0,∞) (12) where g, h are given. We desire to derive a formula for u in terms of g and h.

Note that the PDE in (12) can be "factored" to read

u

0.

t

x

t

x

∂

∂

∂

∂



₊



₋



₌



_∂

_∂



_∂

_∂









₍₁₃₎

Let

v(x, t) :

u(x, t).

t

x

∂

∂





=

_

−

_

∂

∂





₍₁₄₎

Then (13) says

𝑣𝑣𝑡𝑡(𝑥𝑥,𝑡𝑡) +𝑣𝑣𝑥𝑥(𝑥𝑥,𝑡𝑡) = 0 (𝑥𝑥 ∈ ℝ,𝑡𝑡> 0)

This is a transport equation with constant coefficients. Applying formula (3) (with n = 1, b = 1), we find

v(x, t) = a (x – t) (15) for a(x)

:

=

v(x, 0).Combining now (13) – (15), we obtain

𝑢𝑢𝑡𝑡(𝑥𝑥,𝑡𝑡)− 𝑢𝑢𝑥𝑥(𝑥𝑥,𝑡𝑡) =𝑎𝑎(𝑥𝑥 − 𝑡𝑡) 𝑖𝑖𝑎𝑎 ℝ× (0,𝛼𝛼)

2 2

n n n

i i

x xi i 2 3

i 1 i 1 i 1

x

1

x

u

v (r)

v (r)(

)

r

r

r

= = =

′′

′

=

+

−

∑

∑

∑

2 2 2 2

1 n 1 n

2 3

v (r)

v (r)

v (r)

(x

x )

n

(x

x ).

r

r

r

′′

′

′

=

+ +

L

+

⋅ −

+ +

L

2 2

2 3

v (r)

v (r)

v (r)

r

n

r

r

r

r

′′

′

′

⋅ +

⋅ −

⋅

v (r)

v (r)

(n 1).

r

′

′′

=

+

−

n 1

v (r)

v (r)

0,

r

−

′′

+

′

=

1 n

v

v ,

r

−

′′

=

′

v

1 n

.

v

r

′′

−

=

′

log v

′ = −

(1 n) log r

+

log a,

1 n

v

′ =

ar

−

.

2 n

r

v

a

c

This is a nonhomogeneous transport equation; and so formula (5) (with n = 1, b = − 1, f(x, t) = a (x – t)) implies for 𝑏𝑏(𝑥𝑥); =

𝑢𝑢(𝑥𝑥, 0)that

t

0

u(x, t)

=

∫

a(x

+ − −

(t s) s)ds

+

b(x

+

t)

x t

x t

1

a(y)dy b(x

t).

2

+

−

=

∫

+

+

(16) We lastly invoke the initial conditions in (12) to compute a and b. The first initial condition in (12) gives

b(x) = g(x) 𝑥𝑥 ∈ ℝ

whereas the second initial condition and (26) imply a(x) = v(x, 0) = ut(x, 0) – ux(x, 0)

= ℎ(𝑥𝑥)− 𝑔𝑔′(𝑥𝑥) (𝑥𝑥 ∈ ℝ)

Our substituting into (16) now yields

x t

x t

1

u(x, t)

h(y) g (y)dy g(x

t).

2

+

−

′

=

_∫

−

+

+

Hence

𝑢𝑢(𝑥𝑥,𝑡𝑡) =1₂[𝑔𝑔(𝑥𝑥+𝑡𝑡) +𝑔𝑔(𝑥𝑥 − 𝑡𝑡)] +1₂� ℎ𝑥𝑥+𝑡𝑡 (𝑦𝑦)𝑑𝑑𝑦𝑦

𝑥𝑥−𝑡𝑡 ,

( 𝑥𝑥 ∈ ℝ,𝑡𝑡 ≥0) (17)

This is d'Alembert's formula.

To solve the solution of wave equation for n = 1,

Assume 𝑔𝑔 ∈ 𝐶𝐶2(ℝ), ℎ ∈ 𝐶𝐶1(ℝ) and define u by d'Alembert's formula. Then

(i) 𝑢𝑢 ∈ 𝐶𝐶2(ℝ× [0,∞))

(ii) 𝑢𝑢_{𝑡𝑡𝑡𝑡}− 𝑢𝑢_{𝑥𝑥𝑥𝑥}= 0 𝑖𝑖𝑎𝑎ℝ× (0,∞) and

(iii)

0 0

t

0 0

( x,t ) ( x ,0) ( x,t ) ( x ,0)

t 0 t 0

lim

u(x, t)

g(x ),

lim

u (x, t)

h(x ),

→ →

> >

=

=

for each point 𝑥𝑥0∈ ℝ The proof is

We have

𝑢𝑢(𝑥𝑥,𝑡𝑡) =1₂[𝑔𝑔(𝑥𝑥+𝑡𝑡) +𝑔𝑔(𝑥𝑥 − 𝑡𝑡)] +1₂∫_{𝑥𝑥−𝑡𝑡}𝑥𝑥+𝑡𝑡ℎ(𝑦𝑦)𝑑𝑑𝑦𝑦

( 𝑥𝑥 ∈ ℝ,𝑡𝑡 ≥0) (18) since 𝑔𝑔 ∈ 𝐶𝐶2(ℝ), ℎ ∈ 𝐶𝐶1(ℝ),𝑢𝑢 ∈ 𝐶𝐶2(ℝ× [0,∞))).

Next, we differentiate (18) with respect to x, we get

[

] [

]

x

1

1

u

g (x

t)

g (x

t)

h(x

t)

h(x

t) .

2

′

′

2

=

+ +

−

+

+ −

−

Again,

[

] [

]

xx

1

1

u

g (x

t)

g (x

t)

h (x

t)

h (x

t) .

2

′′

′′

2

′

′

=

+ +

−

+

+ −

−

(19) In addition, we differentiate (18) with respect to t, we get

[

] [

]

t

1

1

u

g (x

t)

g (x

t)

h(x

t)

h(x

t) .

2

′

′

2

=

+ −

−

+

+ +

−

(20) Again,

[

] [

]

tt

1

1

u

g (x

t) g (x

t)

h (x

t) h (x

t) .

2

′′

′′

2

′

′

=

+ +

−

+

+ −

−

(21) So, we get

u

tt

−

u

xx

=

0.

Finally, we get

0 0 0

0 ( x, t ) ( x ,0)

1

lim

u(x, t)

u(x ,0)

2g(x )

g(x ).

2

→





=

=

_

_

=

Then, from (20),

0

t t

0 ( x,t ) ( x ,0)

t 0

lim

u (x, t)

u (x , 0)

→ >

=

0 0 0 0 0

1

1

g (x )

g (x )

h(x )

h(x )

h(x ).

2



′

′



2





=

_

−

_

+

_

+

_

=

VI. CONCLUSION

This paper contains a brief summary of the work done and some recommendations for future research directions. These some partial differential equations are based on the computation of inequalities and integration. Some of the application areas include these equations such as Laplace’s equation, Wave equation.

ACKNOWLEDGMENT

I would like to thank to my former teachers for their helpful suggestion and constructive advice. I also wish to express deeply grateful to my partners at Department of Mathematics, University of KyauKSe, who helped directly or indirectly towards the completion of this work.

REFERENCES

[1] [1] Evans, C., “Partial Differential Equations”, Second Edition, American Mathematical Society, USA, 2010.

[2] [2] Friedman, A., “Partial Differential Equations of Parabolic Type”, Prentice-Hall, Englewood Cliffs, New Jersey, 1964.

AUTHORS

First Author – Ms. Khin Saw Mu, Lecturer, M.Sc (Maths), Ph.D (thesis), Department of Mathematics, Universityof