Solving the Partial Differential Problems Using Maple

Abstract

This paper considers the partial differential problem of two types of multivariable functions and uses mathematical software Maple for verification. The infinite series forms of any order partial derivatives of these two types of multivariable functions can be obtained using binomial series and differentiation term by term theorem, which greatly reduce the difficulty of calculating their higher order partial derivative values. On the other hand, four examples are used to demonstrate the calculations.

1

Introduction

In calculus and engineering mathematics, the evaluation and numerical calculation of the partial derivatives of multivariable functions are important. The Laplace equation, the wave equation, and other important physical equations involve the partial derivatives. The evaluation of the m -th order partial derivative value of a multivariable function at some point, generally, requires two procedures: the determination of the m -th order partial derivative of the function, and the substitution of the point into the m -th order partial derivative. These two procedures become increasingly complex calculations for increasing order of partial derivative, thus manual calculations become difficult. The present study considers the partial differential problem of the following two types of n -variables functions

where n is a positive integer, a, b,r, β_i, λ_i are real numbers for all i = 1,..,n, a,b ≠ 0, and a^r, b^r exist. We can obtain the infinite series forms of any order partial derivatives of these two types of multivariable functions using binomial series and differentiation term by term theorem; these are the major results of this study (i.e., Theorems 1 and 2), which greatly reduce the difficulty of calculate their higher order partial derivative values. The study of partial differential problems can refer to [1-24]. The methods adopted in [1-5] are different from the methods used in this paper, and [6-24] studied the evaluation of the partial derivatives of different types of multivariable functions using differentiation term by term theorem and complex power series method. [25] considered two differential equations whose independent variables involve the partial derivatives. [26] discussed the distance functions whose expressions contain the partial derivatives, and [27] found the solutions of some type of partial differential equation. In this article, some examples are used to demonstrate the proposed calculations, and the manual calculations are verified using Maple.

2

Main Results

Some notations used in this paper are introduced below.

2.1

Notations

2.1.1

∏i=1nci=c1×c2×…×cn where n is a positive integer, c_i are real numbers for all i = 1,.., n.

2.1.2

Suppose that t is any real number, and m is any positive integer. Define

2.1.3

Suppose that n is a positive integer, ji are non-negative integers for all i = 1,.., n. For the n- variables function f(x₁, x₂, …, x_n), its ji times partial derivative with respect to x_i for all i = 1,.., n, forms a j1 + j2 + … + jn -th order partial derivative, denoted as ∂j1+j2+⋅⋅⋅jnf∂xnjn⋅⋅⋅∂x2j2∂x1j1x1,x2,⋅⋅⋅,xn

The followings are two important theorems used in this study.

2.2

Binomial series

(1+u)r=∑k=0∞(r)kk!uk, where u,r are real numbers, and |u|<1.

2.3

Differentiation term by term theorem ([28, p230]).

For all non-negative integers k, if the functions g_k:(a,b)→R satisfy the following three conditions: (i) there exists a point x₀ ∈ (a, b) such that ∑k=0∞gk(x0) is convergent, (ii) all functions g_k(x) are differentiable on the open interval (a, b), and (iii) ∑k=0∞ddxgk(x) is uniformly convergent on (a,b), then ∑k=0∞gk(x) is uniformly convergent and differentiable on (a,b). Moreover, its derivative ddx∑k=0∞gk(x)=∑k=0∞ddxgk(x).

The following is the first major result in this study, we determine the infinite series forms of any order partial derivatives of the n -variables function (1).

2.4

Theorem 1

Suppose that n is a positive integer, a, b,r,λ_i,β_i are real numbers for all i = 1,.., n, a, b≠0, and a^r,b^r exist. If the n -variables function

satisfies that xiβi,xiλi,xiλir exist, x_i ≠ 0 for all i = 1,.., n, and ∏i=1nxiλi≠±ab.

Case A. If ∏i=1nxiλij1+j2+…+jn-th order partial derivative of f(x₁, x₂, …, x_n)

Case B. If ∏i=1nxiλi>ab∂j1+j2+⋅⋅⋅jnf∂xnjn⋅⋅⋅∂x2j2∂x1j1x1,x2,⋅⋅⋅,xn

Proof Case A. ∏i=1nxiλi>ab

Because

(Because ∏i=1nxiλi

By differentiation term by term theorem, differentiating ji-times with respect to x_i (i = 1,..,n) on both sides of Equation (5), we have: the j1 + j2+ … + jn-th order partial derivative of

Case B. If ∏i=1nxiλi>ab

Because

Using differentiation term by term theorem, differentiating ji -times with respect to x_i (i = 1,..,n) on both sides of Equation (6), we obtain:

2.5

Remark 1

If ∏i=1nxiλi28, p193]) yields:

is uniformly convergent. Thus, we can use differentiation term by term theorem to prove Equation (3) holds. The same reason that we can employ differentiation term by term theorem to confirm Equation (4) holds.

The following is the second major result in this study, we obtain the infinite series forms of any order partial derivatives of the n -variables function (2).

2.6

Theorem 2

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

satisfies that a+bexp∑i=1nλixir exists, and

Case A. If exp∑i=1nλixij1+j2+…+jn-th order partial derivative of g (x₁,x₂,…,x_n),

Case B. If exp∑i=1nλixi>ab, then

Proof Case A. exp∑i=1nλixi

Because

By differentiation term by term theorem, differentiating ji-times with respect to x_i (i = 1,..,n) on both sides of Equation (9), we obtain: the j1 + j2 + … + jn -th order partial derivative of g(x₁,x₂,…,x_n),

Case B. exp∑i=1nλixi>ab

Because

Using differentiation term by term theorem, differentiating ji-times with respect to x_i (i = 1,..,n) on both sides of Equation (10), we obtain:

2.7

Remark 2

The same reason as that in Remark 1, we can use differentiation term by term theorem to prove Equations (7) and (8) hold.

3

Examples

For the partial differential problem of the multivariable functions in this study, four examples are proposed. Theorems 1 and 2 are used to obtain the infinite series forms of any order partial derivatives of these functions, and to evaluate some of their higher order partial derivative values. Additionally, Maple is used to calculate the approximations of these higher order partial derivative values to verify the manual calculations.

3.1

Example 1

Suppose that the domain of the two-variables function

is

3.1.1

If x13x24>92. Using Case A of Theorem 1 yields: any j1 +j2-th order partial derivative of f₁(x₁, x₂),

Therefore, the 13-th order partial derivative value of f₁(x₁,x₂) at 32,45

Next, we use Maple to verify the correctness of Equation (13).

>f1:=(x1,x2)->x1^(7/3)*x2^(5/2)*(9+2*x1^3*x2^4) ^(11/5);

>evalf(D[1$6,2$7](f1)(3/2,4/5),22);

7.123326797821678044703.10¹¹

>evalf(9^(11/5)*sum(product(11/5-j,j=0..(k-1))/k!* (2/9)^k*product(3*k+7/3-p,p=0..5)*product(4*k+5/2-q,q=0..6)*(3/2)^(3*k-11/3)*(4/5)^(4*k-9/2),k=0.. infinity),22);

7.123326797821678044705. 10¹¹

3.1.2

If x13x24>92. By Case B of Theorem 1, the j1 +j2 -th order partial derivative of f₁(x₁,x₂),

Thus, the 8-th order partial derivative value of f₁(x₁,x₂) at (2,3),

>evalf(D[1$4,2$4](f1)(2,3),18);

1.11843022977422795.10¹³

>evalf(2^(11/5)*sum(product(11/5-j,j=0..(k-1))/k!*(9/2)^k*product(-3*k+134/15-p,p=0..3)*product(-4* k+113/10-q,q=0..3)*2^(-3*k+74/15)*3^(-4*k+73/10), k=0..infinity),18);

1.11843022977422796.10¹³

3.2

Example 2

Assume that the domain of the three-variables function

is (x1,x2,x3)∈R3x1>0,x2>0,x3≠0,x12x25x34≠±113

3.2.1

If x12x25x34<113. Using Case A of Theorem 1, we obtain: any j1 + j2 + j3-th order partial derivative of f₂(x₁,x₂,x₃),

Thus, the 15-th order partial derivative value of f₂(x₁, x₂, x₃) at 16,23,34

>f2:=(x1,x2,x3)->x1^(11/6)*x2^(13/4)*x3^(8/5)/((11-3*x1^2*x2^5*x3^4)^2)^(1/3);

>evalf(D[1$4,2$6,3$5](f2)(1/6,2/3,3/4),14);

5.3261668131349.10⁷

>evalf(11^(-2/3)*sum(product(-2/3-j,j=0..(k-1))/k!*(-3/11)^k*product(2*k+11/6-p,p=0..3)*product(5*k+ 13/4-q,q=0..5)*product(4*k+8/5-s,s=0..4)*(1/6)^(2* k-13/6)*(2/3)^(5*k-11/4)*(3/4)^(4*k-17/5),k=0.. infinity),14);

5.3261668131350.10⁷

3.2.2

If x12x25x34>113. Using Case B of Theorem 1 yields: any j1+j2+j3-th order partial derivative of f₂(x₁,x₂,x₃),

Hence, the 11-th order partial derivative value of f₂(x₁,x₂,x₃) at (4,2,5),

>evalf(D[1$3,2$2,3$6](f2)(4,2,5),14);

-0.000019108587954897

>evalf(3^(-2/3)*sum(product(-2/3-j,j=0..(k-1))/k!*(-11/3)^k*product(-2*k+1/2-p,p=0..2)*product(-5*k-1/12-q,q=0..1)*product(-4*k-16/15-s,s=0..5)*4^(-2*k-5/2)*2^(-5*k-25/12)*5^(-4*k-106/15),k=0.. infinity),14);

-0.000019108587954897

3.3

Example 3

If the domain of the two-variables function

3.3.1

If exp(5x1+8x2)<79. By Case A of Theorem 2, we obtain: any j1+j2-th order partial derivative of g₁(x₁,x₂),

Therefore, the 12-th order partial derivative value of g₁(x₁,x₂) at −14,−25

>g1:=(x1,x2)->exp(2*x1+3*x2)*(7+9*exp(5*x1+8* x2))^(13/3);

>evalf(D[1$5,2$7](g1)(-1/4,-2/5),22);

5.855345231176575279854.10¹⁴

>evalf(7^(13/3)*sum(product(13/3-j,j=0..(k-1))/k!* (9/7)^k*(5*k+2)^5*(8*k+3)^7*exp(-89/20*k-17/10), k=0..infinity),22);

5.855345231176575279879.10¹⁴

3.3.2

If exp (5x1+8x2)>79.. Using Case B of Theorem 2 yields: any j1 + j2-th order partial derivative of g₁(x₁,x₂),

Therefore, we can determine the 17-th order partial derivative value of g₁(x₁, x₂)at 72,16,

Maple was used to verify the correctness of Equation (25):

>evalf(D[1$8,2$9](g1)(7/2,1/6),22);

1.028551772875461091101.10⁶⁸

>evalf(9^(13/3)*sum(product(13/3-j,j=0..(k-1))/k!*(7 /9)^k*(-5*k+71/3)^8*(-8*k+113/3)^9*exp(-113/6*k+ 802/9),k=0..infinity),22);

1.028551772875461091120.10⁶⁸

3.4

Example 4

Suppose that the domain of the three-variables function

3.4.1

If exp(7x1−4x2+2x3)<145. By Case A of Theorem 2, we obtain: any j1 + j2 + j3-th order partial derivative of g₂(x₁, x₂, x₃),

Thus, the 14-th order partial derivative value of g₂(x₁,x₂,x₃) at −17,23,−12,

>g2:=(x1,x2,x3)->exp(6*x1+11*x2-8*x3)/((14-5*exp (7*x1-4*x2+2*x3))^8)^(1/9);

>evalf(D[1$6,2$5,3$3](g2)(-1/7,2/3,-1/2),22);

-1.324276036594542259720.10¹⁶

>evalf(14^(-8/9)*sum(product(-8/9-j,j=0..(k-1))/k!*(-5/14)^k*(7*k+6)^6*(-4*k+11)^5*(2*k-8)^3*exp(-14/3 *k+220/21),k=0..infinity),22);

-1.324276036594542259721.10¹⁶

3.4.2

If exp(7x1−4x2+2x3)>145. Using Case B of Theorem 2 yields: any j1 +j2 + j3 -th order partial derivative of g₂(x₁,x₂,x₃),

Hence, the 16-th order partial derivative value of g₂(x₁,x₂,x₃)at 17,14,1,

>evalf(D[1$5,2$7,3$4](g2)(1/7,1/4,1),40);

-7.22751822579461952474808.10²⁰

>evalf(5^(-8/9)*sum(product(-8/9-j,j=0..(k-1))/k!*(-14/5)^k*(-7*k-2/9)^5*(4*k+131/9)^7*(-2k-88/9)^4* exp(-2*k-1555/252), k=0..infinity),22);

-7.227518225794619524748.10²⁰

4

Conclusion

This article proposed two methods (i.e., the binomial series and the differentiation term by term theorem) to solve the partial differential problem of some multivariable functions. The two methods can be applied to evaluate any order partial derivatives of general multivariable functions. Further studies on related applications will be conducted in the future. Moreover, other calculus and engineering mathematics problems will be considered and solved using Maple.