Partial -Differential Equation

ELISABETA PETI, ENKELEDA ZAJMI KOTONAJ, VERA HOXHA

Department of Mathematics,

University of Tirana,

Bulevardi Zogu I,

ALBANIA

(1) into an Euler-type equation. Since all of these problems are too difficult to solve in Classical Calculus, this

study aims to convert them into equations of this type for the ease of study in -Calculus. Then, we proposed a

transformation method for both the equation and the boundary conditions. Thus, the boundary value problem

consists of a second-order partial differential equation and boundary conditions dependent on the eigenvalues.

By using the procedure of -difference over a time scale , we obtained a second-order Euler -difference

equation with Dirichlet boundary conditions. Also, we have analyzed the exact number of eigenvalues for all

cases that arise from the study of our problem. Here, we have presented three theorems, two of which show the

correct number of eigenvalues of two issues with eigenvalues derived from our principal problem. The last one

shows a relation between two eigenvalues of a problem for 

The theory of quantum calculus, or calculus, has

been attracting the attention of many researchers,

and the interest in this subject is still growing for its

practical applications, especially in the physical

sciences, specifically within the domain

of “Quantum Physics.” It is seen as a connection

which corresponds to difference equations, than

in terms of continuous derivatives and continuous

differential equations. The basic rules and exciting

definitions of this calculus in comparison with

classical Newton-Leibnitz calculus were studied

among others in [1], [2], [3]. In particular, some

significant results of -calculus, where the

smoothness of a function is no longer a requirement,

are presented in [4].

At [5], the idea of generalizing the concept of -

calculus, we explore the concept of -derivative or

equations, -derivative or -difference equations),

which have applications across various

mathematical areas such as number theory, quantum

theory, combinatorics, orthogonal polynomials, and

essential hypergeometric functions. Recently, there

has been significant interest in applying quantum

calculus to differential transform methods to obtain

differential equations. The reduced transform

techniques were presented in the literature to

approach several different linear or nonlinear

differential systems, such as ordinary differential

equations, functional differential equations,

impulsive differential equations, and partial

differential equations, among others [6], [7].

Mainly when the solutions correspond to the

classical version of the provided initial value

On the other hand, the study of -derivatives on

discrete, continuous, and, more generally, on an

arbitrary nonempty closed set (i.e., a time scale) is a

well-known subject under current solid

development. For an introduction to the theory of

calculus on time scales, we refer to [10], [11], [12],

[13], [14] and [15]. As time scale calculus has

evolved, many authors have focused on integrating

methodologies from both time-scale

and delta calculus on a time scale .

Due to the importance of this quantum calculus

on time scales and taking into account that the

oscillatory or asymptotic properties of the solutions

of -difference equations are essential to

understanding several physics phenomena better,

[15], [16], [17] introduced in the literature the

concept of determining the eigenvalues and their

count for the resulting eigenvalue problem defined

on the quantum time scale.

describe a lot of PDEs based on the nature of

solution technique:

constant coefficients if 󰇛󰇜 is a constant.

b) If 󰇛󰇜 is not constant, in most cases the

equation can not be solved analiticaly because it

Since 󰇡

mathematical modeling, especially in complex

biological models.

The organization of this paper is as follows: in

cases and transforms it case by case into an Euler-

type difference equation. In the end, in Section

4, we determine the eigenvalues and their count for

the resulting eigenvalue problem and provide

examples to illustrate the effectiveness of the

proposed theorems.

2 Preliminaries

This section introduces some of the notations

ordinary derivative, the action of taking the

other words, for any constants  and , we have:

and a higher order of delta derivatives is as

follows:

problem. Additionally, by-calculus for a function

of  concerning the first and the second variable,

respectively, by:

and

(8)

(9)

Note that,

We start with an analysis according to the cases of

an equation with partial derivatives of the second

order and, case by case, transform it into the

following Euler-type:

Let  and consider the boundary value

with boundary conditions:

Its associated second-order -difference

equation, together with Dirichlet boundary

Dirichlet boundary conditions.

First, let us solve the problem of constructing

the function  such that equation (10) is of the Euler

and depending on the constants

divide the problem into three cases:

If  then function 󰇡

The equation (12) is now written in the form:

system 

which has a unique solution 󰇛󰇜󰇛󰇜. We

apply the following transformation:

where  

Let us recall a relation between the derivatives from

classic calculus,

moreover, we take the boundary value problem in

ordinary calculus as follows:

Let us construct its associated equation in -calculus

Let us now assume that , so there

exists , such that , .

Since,

the problem (10)-(11) is now written in the form:

obtain that:

Hence, from (26) and (27), the eigenvalue problem

From the sign of 󰇡

following three cases:

are linearly independent, so

󰇛󰇜 (30)

Our next step is to find the eigenvalues of (29).

We apply the first Dirichlet condition 󰇛󰇜 to

(30), and we obtain that:

so that 

we shall discuss the following conditions for 

This can occur for  or for even  

For  we have that

i.e., 󰇡

so that:

which leads us to , contradicting the

this case.

Case II. 󰇡

so 

condition, which gives us the following equality

For this to be true either  or ,

from which we do not obtain any eigenvalues.

Hence, there are no eigenvalues.

Case III. 󰇡

From substitutions 󰇛󰇜,

this leads to the subsequent result

where 󰇛󰇜󰇛󰇜 and

We let , and apply the other Dirichlet

condition to obtain:

Note that

contradiction. So, we consider only that 󰇛󰇜

values where

Since the cosine function is an even function,

Hence, we obtain the following main result.

Let . The problem (14) has

function, eigenvalues, and they can be calculated

from the formula (36). The corresponding

eigenfunctions are given by

For  the eigenvalue and its corresponding

eigenfunction are:

result as the above argument when the values of

specific eigenvalue problem. So, let 󰇛󰇜

i.e., 󰇡

which is not a valid value for  in Case I.

where we obtain that, , contradicting the fact