Several oscillation results are proposed including necessary and sufficient conditions for the oscillation of fractional-order delay differential equations with constant coefficients, the sufficient or necessary and sufficient conditions for the oscillation of fractional-order delay differential equations by analysis method, and the sufficient or necessary and sufficient conditions for the oscillation of delay partial differential equation with three different boundary conditions. For this,
For the past three centuries, fractional calculus has been dealt with by mathematicians. In the recent decades, it is widely used in the fields of engineering, science, and economy. Recently, fractional differential systems have gained scholar’s attention [
In [
For the purpose, known methods are expanded for our results.
We first give some definitions and make some preliminaries. The fractional-order derivative
Riemann-Liouville’s fractional order derivative [
Caputo’s fractional order derivative is defined by
The derivative
Mittag-Lefflerfunction [
The first order delay differential equation including fractional-order derivative [
Equation (
The following statements are equivalent. Every solution of ( The characteristic equation (
In this paper, the main results are derived from the application of the basic theory in [
The general form of the fractional-order linear delay differential equation is
Since
The detailed derivation can be found in Appendices
The following statements are equivalent. Every solution of ( The characteristic equation (
In order to use Corollary
If If
(1) If
As
So
(2) In the same way, we get
If If
The equation If If
We consider delay differential equation with fractional-order derivatives
All the conditions of Theorem
We consider the following fractional-order linear delay differential equation:
Assume that
We may assume that (
We consider the eventually positive solution of the following inequality:
Assume the inequality (
If
We consider oscillation of solutions of the fractional-order delay differential equation with oscillating coefficients:
Let
We have the sufficient conditions for oscillation of all solutions of (
If
Assume that (
Similar to the proof process of Theorem
We consider the following fractional-order linear delay differential equations:
Their characteristic equations are expressed by (
Based on Lemma
The following statements are equivalent. Every solution of ( The characteristic equations of (
To research the oscillation properties of solutions for delay partial differential equations, we consider the delay parabolic differential equations [
The hypotheses are always true as follows.
Consider the boundary conditions as follows:
If there are no special instructions in this section, we only discuss the oscillation of solutions of the fractional-order delay partial differential equation with respect to time variable
Assume that As As
Let
By using Green’s formula and boundary conditions
Let
Based on Corollary
Consider the parabolic equation:
Because of
For the more general equation of (
There are the boundary conditions as follows:
Assume that
Some sufficient conditions for oscillation of all solutions of (
For the Dirichlet problem,
Assume that
Let
By using Green’s formula and boundary condition
Based on Corollary
Assume that
Let
By using Green’s formula and boundary condition
Let
The detail of the proof process is similar to the proof process of Theorem
Assume that
Let
By using Green’s formula and boundary condition
The detail of the proof process is similar to the proof process of Theorem
The delay parabolic differential equations are expressed by
Consider the boundary conditions, as shown in
Assume that
Let
The inequality (
By using Green’s formula and boundary conditions
Substituting (
By using Theorem
Figure
Values of
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0.0881072 | 0.1176864 | 0.1774485 | 0.2383790 | 0.2694375 | 0.3659126 |
|
0.0887969 | 0.1185872 | 0.1787170 | 0.2399068 | 0.2710383 | 0.3674039 |
|
0.089205 | 0.1191138 | 0.1794311 | 0.2407091 | 0.2718332 |
|
|
0.0892548 | 0.1191774 | 0.1795119 | 0.2407885 | 0.2719021 | 0.3678611 |
|
0.0892949 | 0.1192275 | 0.1795730 | 0.240842 | 0.2719425 | 0.3678068 |
|
0.0893251 | 0.1192646 | 0.1796147 | 0.2408705 |
|
0.3677171 |
|
0.0893456 | 0.1192889 | 0.1796375 |
|
0.2719395 | 0.3675928 |
|
0.0893567 |
|
|
0.2408533 | 0.2718973 | 0.3674346 |
|
|
0.1193001 | 0.1796277 | 0.2408087 | 0.2718286 | 0.3672431 |
|
0.0893512 | 0.1192875 | 0.1795959 | 0.2407409 | 0.2717341 | 0.3670191 |
|
0.0893351 | 0.1192631 | 0.1795465 | 0.2406501 | 0.2716142 | 0.3667631 |
|
0.0893102 | 0.1192271 | 0.1794801 | 0.2405370 | 0.2714694 | 0.3664759 |
|
0.0892768 | 0.1191797 | 0.1793968 | 0.2404020 | 0.2713003 | 0.3661581 |
|
0.0889875 | 0.1187809 | 0.17874 | 0.2394135 | 0.2701071 | 0.364132285 |
|
0.0885098 | 0.1181318 | 0.1777141 | 0.2379416 | 0.2683777 | 0.361433054 |
Curves of function
The curves of function
Curves of function
The Laplace transforms for two fractional derivatives with
The fractional Laplace transform [
In the most general form
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors thank Mr. Weisheng Xu for useful discussions on the proof process in the paper.