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Using Differentiation Term by Term Theorem to Study the Partial Differential Problems

Chii-Huei Yu

Department of Management and Information, Nan Jeon University of Science and Technology, Tainan City, Taiwan

Abstract: This article 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 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 propose two examples to do calculation practically. The research methods adopted in this study involved finding solutions through manual calculations and verifying our answers by using Maple.

Keywords: partial derivatives, infinite series forms, differentiation term by term theorem, Maple

Received November 17, 2013; Revised December 20, 2013; Accepted December 30, 2013

Citation: Chii-Huei Yu, Using Differentiation Term by Term Theorem to Study the Partial Differential Problems. Turkish Journal of Analysis and Number Theory, 1 (1). 63-68. January 2013. doi:10.12691/tjant-1-1-13

1. 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.

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 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 the 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

[picture] | (1)

[picture] | (2)

where α > 0, n is a positive integer, and λp are real numbers for all p = 1, …, n. We can obtain the infinite series forms of any order partial derivatives of these two types of multivariable functions using differentiation term by term theorem; these are the major results of this study (i.e., Theorem 1 and Theorem 2), and hence greatly reduce the difficulty of calculating their higher order partial derivative values. As for the study of related partial differential problems can refer to [1-13]. 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. Therefore, Maple provides insights and guidance regarding problem-solving methods.

2. Main Results

Firstly, we introduce some notations and formulas used in this study.

2.1. Notations

2.1.1. Suppose n is a positive integer, are λp real numbers for all p = 1, …, n. Define

and

2.1.2. Suppose r is any real number, m is any positive integer. Define

2.1.3. Suppose n is a positive integer, jp are non-negative integers for all p = 1, …, n. For the n-variables function f (x₁, x₂, …, xn), its jp-times partial derivative with respect to xp for all p = 1, …, n, forms a j₁ + j₂ + … + jn-th order partial derivative, and denoted by

2.2. Formulas

2.2.1. .

2.2.2. .

Next, we introduce an important theorem used in this paper.

2.3. Differentiation Term by Term Theorem ([14])

For all non-negative integers k, if the functions gk (a, b) → R satisfy the following three conditions：(i) there exists a point such that is convergent, (ii) all functions gk (x) are differentiable on open interval (a, b), (iii) is uniformly convergent on (a, b). Then is uniformly convergent and differentiable on (a, b). Moreover, its derivative .

Before deriving the first major result in this study, we need a lemma.

2.4. Lemma 1

Suppose y is a real number with y > -1. Then the logarithm function

Proof If -1 < y < 1, because the derivative of ln (1 + y),

[picture] | (5)

By Formula 2.2.1, we obtain

for all -1 < y ≤ 1.

If y ≥ 1, then

for all y ≥ 1.

Next, we determine the infinite series forms of any order partial derivatives of the n-variables function (1).

2.5. Theorem 1

Suppose n is a positive integer, α > 0, and λp are real numbers, jp are non-negative integers for all p = 1, …, n. If the n-variables function

satisfies exists, , and .

Case(1) If , then the j1 + j2 + … + jn-th order partial derivative of f (x₁, …, xn),

[picture] | (6)

Case(2) If , then

[picture] | (7)

Proof Case(1) If , because

[picture] | (8)

(By Eq. (3))

By differentiation term by term theorem, differentiating jp-times with respect to xp for all p = 1, …, n on both sides of Eq. (8), we obtain

Case(2) If , by Eq. (4),

[picture] | (9)

Also, by differentiation term by term theorem, differentiating jp-times with respect to xp for all p = 1, …, n on both sides of Eq. (9), we obtain

Before deriving the second major result in this paper, we also need a lemma.

2.6. Lemma 2

Suppose y is a real number. Then the inverse tangent function

Proof The derivative of tan⁻¹ y,

If -1<y<1, then

[picture] | (13)

By Formula 2.2.2 , we obtain

for all -1 ≤ y ≤ 1.

If , because

[picture] | (14)

By Formula 2.2.2, we have

for all y ≥ 1.

And

for all y ≤ -1.

Finally, we find the infinite series forms of any order partial derivatives of the n-variables function (2).

2.7. Theorem 2

If the assumptions are the same as Theorem 1. Assume the n-variables function

satisfies exists, and .

Case(1) If , then the j₁ + j₂ + … + jn-th order partial derivative of g (x₁, …, xn),

[picture] | (15)

Case(2) If , then

[picture] | (16)

Proof

Case(1) If , by Eq. (10), we have

[picture] | (17)

Using differentiation term by term theorem, differentiating jp-times with respect to xp for all p = 1, …, n on both sides of Eq. (17), we have

Case(2) If , by Eqs. (11) and (12), and using differentiation term by term theorem, we obtain

3. 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, 2 to determine the infinite series forms of any order partial derivatives and some higher order partial derivative values of these functions. In addition, we employ Maple to calculate the approximations of these higher order partial derivative values and their solutions for verifying our answers.

3.1. Example 1

Suppose the two-variables function

[picture] | (18)

satisfies and .

Case (1) If , then by Eq. (6), we obtain any j₁ + j₂-th order partial derivative of f (x₁, x₂),

[picture] | (19)

Thus, we can determine the 12-th order partial derivative value of f (x₁, x₂) at ,

[picture] | (20)

Next, we use Maple to verify the correctness of Eq. (20).

>f:=(x1,x2)->ln(1+(3*x1-5*x2)^7);

>evalf(D[1$7,2$5](f)(1/3,2/15),18);

>evalf(3^7*(-5)^5*sum((-1)^(k-1)*product(7*k-j,j=0..

11)/k*(1/3)^(7*k-12),k=1..infinity),18);

Case (2) If , using Eq. (7), we obtain

[picture] | (21)

Hence, we obtain the 14-th order partial derivative value of f (x₁, x₂)at ,

[picture] | (22)

>evalf(D[1$6,2$8](f)(2,4/5),18);

>evalf(-7*13!*3^6*(-5)^8/2^14+3^6*(-5)^8*sum((-1)^(k-1)*product(-7*k-j,j=0..13)/k*2^(-7*k-14),k=1..infinity),

18);

3.2. Example 2

Assume the three-variables function

[picture] | (23)

satisfies .

Case (1) If , using Eq. (15), we obtain any j₁ + j₂ + j₃-th order partial derivative of g (x₁, x₂, x₃),

[picture] | (24)

Thus, we can evaluate the 15-th order partial derivative value of g (x₁, x₂, x₃) at ,

[picture] | (25)

We also use Maple to verify the correctness of Eq. (25).

>g:=(x1,x2,x3)->arctan((4*x1+9*x2-3*x3)^(4/5));

>evalf(D[1$4,2$6,3$5](g)(1/4,2/9,5/6),18);

>evalf(4^4*9^6*(-3)^5*sum((-1)^(k-1)*product((8*k-4)/5-j,j=0..14)/(2*k-1)*(1/2)^((8*k-79)/5),k=1..

infinity),18);

Case (2) If , by Eq. (16), we have

[picture] | (26)

Thus, we obtain the 17-th order partial derivative value of g (x₁, x₂, x₃) at (3, 1, 6),

[picture] | (27)

>evalf(D[1$7,2$4,3$6](g)(3,1,6),18);

>evalf(-4^7*9^4*(-3)^6*sum((-1)^(k-1)*product((-8*k+4

)/5-j,j=0..16)/(2*k-1)*3^((-8*k-81)/5),k=1..infinity),18);

4. Conclusion

In this paper, we provide a new technique to evaluate any order partial derivatives of two types of multivariable functions. We hope this technique can be applied to solve another partial differential problems. On the other hand, the differentiation term by term theorem plays a significant role in the theoretical inferences of this study. In fact, the applications of this theorem are extensive, and can be used to easily solve many difficult problems; we endeavor to conduct further studies on related applications. In addition, Maple also plays a vital assistive role in problem-solving. In the future, we will extend the research topic to other calculus and engineering mathematics problems and solve these problems by using Maple. These results will be used as teaching materials for Maple on education and research to enhance the connotations of calculus and engineering mathematics.

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