The Iterative Solution of Taylor Formula for Partial Differential Equation

Yang Gao^*
*Correspondence: Yang Gao, Department of Science, Binzhou University, Binzhou, China, Email:

Keywords

Taylor's formula • Iteration method • Nonlinear • Partial differential equation

Introduction

This paper introduces that Taylor formula iteration method resolve partial differential equation. In this paper, six examples are used to introduce Taylor formula iteration method to solve partial differential equation. This paper also introduce that Generalized Taylor's formula can solve fractional partial differential equation. The iterative method of Taylor formula is an important and useful method to solve partial differential equation [1-10]. The solution of Taylor formula iteration method belongs to C^∞.

Variable coefficient problem

We consider equation as following:

We solve (1) by Taylor formula iteration method as following:

And we have:

By Taylor's formula, we get as following:

Two dimensional heat conduction equation solution

We study the equation as following:

Next, we solve (19) by Taylor formula iteration method,

On equation (21), finding 1-order partial derivative of t on both sides,

We have:

On equation (23), finding 1-order partial derivative of t on both sides,

We have:

The third problem with boundary values

We consider following equation:

By Taylor formula iteration method, we have:

Fractional partial differential equation

We consider following fractional partial differential equation:

Where q(x,y)is known integral polynomial.

The definition of Caputo fractional derivative about t:

q1(x,y) is known function.

We consider On equation (62), finding α -order partial derivative of t on both sides,

We have:

q2(x,y) is known function.

We consider on equation (62), finding 2α -order partial derivative of t on both sides,

We have:

Where

q3(x,y) is known function. We consider on equation (62), finding 3α -order partial derivative of t on both sides,

We have:

Where

q4(x, y) is known function. We have:

q_n(x, y) is known function.

By Generalized Taylor's formula:

So we have the solution of (62):

Nonlinear KdV equation

We consider the wave equation as following:

On equation (88), finding 1-order partial derivative of t on both sides,

We have:

On equation (94), finding 1-order partial derivative of t on both sides,

We have:

By Taylor's formula, we get as following:

Nonlinear sine-Gordon equation

We consider following constant coefficient equation

On equation (110), finding 1-order partial derivative of t on both sides,

We have:

On equation (114), finding 1-order partial derivative of t on both sides,

We have:

u_ttttt(x,0),u_tttttt(x,0)…is known function.

By Taylor's formula, we get as following:

So we can get the solution of (107).

Conclusion

Iterative solution of partial differential equations by Taylor formula is important and good methods that solve linear and nonlinear partial differential equations. And the method also can solve fractional partial differential equations.

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