The Partial Differential Problem

In calculus and engineering mathematics courses, the evaluation of the partial derivatives of multivariable functions is important. This paper takes the mathematical software Maple as the auxiliary tool to study the partial differential problem of two types of multivariable functions. We can obtain the infinite series forms of any order partial derivatives of these two types of multivariable functions by using differentiation term by term theorem, and hence greatly reduce the difficulty of calculating their higher order partial derivative values. On the other hand, we provide two examples to do calculation practically. The research methods adopted in this study involved finding solutions through manual calculations and verifying these solutions by using Maple. This type of research method not only allows the discovery of calculation errors, but also helps modify the original directions of thinking from manual and Maple calculations. For this reason, Maple provides insights and guidance regarding problem-solving methods.


Introduction
The computer algebra system (CAS) has been widely employed in mathematical and scientific studies. The rapid computations and the visually appealing graphical interface of the program render creative research possible. Maple possesses significance among mathematical calculation systems and can be considered a leading tool in the CAS field. The superiority of Maple lies in its simple instructions and ease of use, which enable beginners to learn the operating techniques in a short period. In addition, through the numerical and symbolic computations performed by Maple, the logic of thinking can be converted into a series of instructions. The computation results of Maple can be used to modify our previous thinking directions, thereby forming direct and constructive feedback that can aid in improving understanding of problems and cultivating research interests. Inquiring through an online support system provided by Maple or browsing the Maple website (www.maplesoft.com) can facilitate further understanding of Maple and might provide unexpected insights. As for the instructions and operations of Maple, we can refer to [1][2][3][4][5][6][7].
In calculus and engineering mathematics curricula, the evaluation and numerical calculation of the partial derivatives of multivariable functions are important. For example, Laplace equation, wave equation, as well as some other important physical equations are involved the partial derivatives. On the other hand, evaluating the m -th order partial derivative value of a multivariable function at some point, in general, needs to go through two procedures: firstly determining the m -th order partial derivative of this function, and then taking the point into this m -th order partial derivative. These two procedures will make us face with increasingly complex calculations when calculating higher order partial derivative values (i.e. m is large), and hence to obtain the answers by manual calculations is not easy. In this paper, we study the partial differential problem of the following two types of n -variables functions Where n is a positive integer, k k b a , are real numbers for all n k ,.., 1 = .We can obtain the infinite series forms of any order partial derivatives of these two types of n -variables functions by using differentiation term by term theorem ; these are the major results of this study (i.e., Theorems 1 and 2), and hence greatly reduce the difficulty of calculating their higher order partial derivative values. For the study of related partial differential problems can refer to [8][9][10][11][12][13][14][15][16][17][18][19]. In addition, we propose two examples to do calculation practically. The research methods adopted in this study involved finding solutions through manual calculations and verifying these solutions by using Maple. This type of research method not only allows the discovery of calculation errors, but also helps modify the original directions of thinking from manual and Maple calculations. Therefore, Maple provides insights and guidance regarding 54 The Partial Differential Problem problem-solving methods.

Main Results
Firstly, we introduce some notations and formulas used in this paper.

Formulas
Next, we introduce an important theorem used in this study. The following is the first result in this study, we determine the infinite series forms of any order partial derivatives of the multivariable function (1).

Theorem 1 Suppose n is a Positive Integer
Are Real Numbers, and k j are Non-Negative Integers For All n k ,.., 1 = . If the n -Variables Function for all n k ,.., 1 = , By differentiation term by term theorem, differentiating k j -times with respect to k x ( n k ,.., 1 = ) on both sides of (12), we obtain the Using differentiation term by term theorem, differentiating k j -times with respect to k x ( n k ,.., 1 = ) on both sides of (13), we have ) , , , Using differentiation term by term theorem, differentiating k j -times with respect to k x ( n k ,.., 1 = ) on both sides of (14), we obtain ) , , , Next, we determine the infinite series forms of any order partial derivatives of the multivariable function (2).

Theorem 2Let the Assumptions be the Same as Theorem 1.Supposethe n -Variables Function
for all n k ,.., The Partial Differential Problem

Examples
In the following, for the partial differential problem of the two types of multivariable functions in this study, we provide two examples and use Theorems 1 and 2 to determine the infinite series forms of any order partial derivatives and some higher order partial derivative values of these functions. On the other hand, we employ Maple to calculate the approximations of these higher order partial derivative values and their solutions for verifying our answers.  8  3  5  2  3  /  11  1  1  2  /  7  3  6  2  4  /  3  1  3  2