A new class of special functions arising from the solution of differential equations involving multiple proportional delays

TL;DR

This paper introduces a new class of special functions derived from differential equations with multiple proportional delays, expanding the landscape of special functions with unique properties and identities.

Contribution

It presents a novel class of special functions arising from differential equations with multiple proportional delays, which are independent of existing special functions.

Findings

New class of special functions identified

Basic properties and identities analyzed

Independence from existing special functions established

Abstract

Proportional delay is a particular case of time dependent delay. In this article, we consider differential equations involving multiple delays. The series solution of this equation leads to a class of special functions. This class of special functions is independent from all the existing special functions obtained as a solution of differential equations. We analyze the basic properties of this class and discuss various identities and relations.

Equations

$$\Gamma (n,x)={\int }_x^{\infty }{t}^{n-1}{e}^{-t}dt\text{and}$$

$$\gamma (n,x)={\int }_0^x{t}^{n-1}{e}^{-t}dt\text{respectively}.$$

$${}_1{F}_1(a;c;x)=\sum _{n=0}^{\infty }\frac{(a{)}_n}{(c{)}_n}\frac{x^n}{n!},c\ne 0,-1,-2,\cdots \text{and}$$

$$U(a;c;x)=\frac{\pi }{\sin (\pi c)}\left(\frac{{}_1{F}_1(a;c;x)}{\Gamma (c)\Gamma (1+a-c)}-x^{1-c}\frac{{}_1{F}_1(1+a-c;2-c;x)}{\Gamma (a)\Gamma (2-c)}\right),$$

$$-\pi <arg(x)\le \pi .$$

$$L_n^{(\alpha )}(x)$$

$$=$$

$$I^{\mu }f(t)=\frac{1}{\Gamma (\mu )}{\int }_0^t(t-\tau {)}^{\mu -1}f(\tau )d\tau ,t>0.$$

$$D^{\mu }f(t)$$

$$I^{\alpha }(x-b{)}^{\beta }$$

$$\left(I^{\alpha }{D}^{\alpha }f\right)(t)$$

$$E_{\alpha }(x)=\sum _{n=0}^{\infty }\frac{x^n}{\Gamma (\alpha n+1)}$$

$$u=f+L(u)+N(u),$$

$$u=\sum _{i=0}^{\infty }{u}_i=f+\sum _{i=0}^{\infty }L(u_i)+\sum _{i=0}^{\infty }{G}_i$$

$$u_0=f,u_{m+1}=L(u_m)+G_m,m=0,1,2,\cdots .$$

$$y^{\prime }(x)=f\left(x,y(q_{0}x),y(q_{1}x),\cdots ,y(q_{n}x)\right),q_0=1\text{and}0<q_i<1,i=1,2,\cdots ,n.$$

$$y^{\prime }(x)=f\left(x,y\left(x-{\tau }_0(x)\right),y\left(x-{\tau }_1(x)\right),y\left(x-{\tau }_2(x)\right)\cdots y\left(x-{\tau }_n(x)\right)\right)$$

$$y=\sum _{i=0}^{\infty }{y}_i.$$

$$\mid f\left(x,y_0,y_1,\cdots ,y_n\right)-f\left(x,u_0,u_1,\cdots ,u_n\right)\mid \le \sum _{i=0}^n{L}_i\mid y_i-u_i\mid .$$

$$y(x)=1+{\int }_0^{x}f\left(x,y(q_{0}t),y(q_{1}t),y(q_{2}t),\cdots ,y(q_{n}t)\right)dt.$$

$$y_0(x)$$

$$y_1(x)$$

$$\Rightarrow \mid y_1(x)\mid$$

$$\Rightarrow \mid y_1(q_{i}x)\mid$$

$$y_2(x)$$

$$\Rightarrow \mid y_2(x)\mid$$

$$\Rightarrow \mid y_2(q_{i}x)\mid$$

$$\mid y_m(x)\mid$$

$$\left|\sum _{m=0}^{\infty }{y}_m\right|$$

$$y^{\prime }(x)=g(y\left(x\right),y\left(x-{\tau }_1(x)\right),y\left(x-{\tau }_2(x)\right)\cdots y\left(x-{\tau }_n(x)\right)),$$

$$y^{\prime }=g(y(x),y(x-{\tau }_1^{\ast }),y(x-{\tau }_2^{\ast }),\cdots ,y(x-{\tau }_n^{\ast })),{\tau }_1^{\ast }={\tau }_1(x_0),{\tau }_2^{\ast }={\tau }_2(x_0),\cdots ,{\tau }_n^{\ast }={\tau }_n(x_0)$$

$$\Vert g(y(x),y(x-{\tau }_1(x)),\cdots ,y(x-{\tau }_n(x)))-g(y(x),y(x-{\tau }_1(x_1)),\cdots ,y(x-{\tau }_n(x_n)))\Vert <\sum _{i=0}^n{ϵ}_i|x-x_i|,$$

$$y^{\prime }(x)=\sum _{i=0}^n{a}_{i}y(q_{i}x),y(0)=1,$$

Full text

Keywords: Special functions, Successive approximation, proportional delay.

1 Introduction

Special functions are plying a vital role in the solutions of differential equations. Exponential, sine, cosine, hypergeometric and Mittag-Leffler are important classes of special functions arising as solutions of various classical and fractional differential equations. Though there is a huge literature devoted to the special functions arising from ordinary differential equations (ODEs), there is a lack of corresponding literature in delay differential equations (DDE). In [1, 2] we discussed the solutions of a class of DDE viz. proportional delay differential equations and provided the solutions in terms of new special functions. In this article, we generalize the DDEs considered in [2] to involve multiple delays. Since the differential equations without delay are inequivalent to the differential equations involving delay, the special functions proposed in this article are independent from the existing special functions.

2 Preliminaries

In this section, we discuss some basic definitions and results [3, 4].

Definition 2.1**.**

The upper and lower incomplete gamma functions are defined as

[TABLE]

[TABLE]

Definition 2.2**.**

Kummer’s confluent hyper-geometric functions ${}_{1}F_{1}(a;c;x)$ and $U(a;c;x)$ are defined as below

[TABLE]

[TABLE]

[TABLE]

Definition 2.3**.**

The generalized Laguerre polynomials are defined as

[TABLE]

Definition 2.4**.**

A real function $f(x)$, $x>0$, is said to be in space $C_{\alpha}$, $\alpha\in\mathbb{R}$, if there exists a real number $p(>\alpha)$, such that $f(x)=x^{p}f_{1}(x)$ where $f_{1}(x)\in C[0,\infty)$.

Definition 2.5**.**

A real function $f(x)$, $x>0$, is said to be in space $C^{m}_{\alpha}$, $m\in\mathbb{N}\cup\{0\}$, if $f^{(m)}\in C_{\alpha}$.

Definition 2.6**.**

Let $f\in C_{\alpha}$ and $\alpha\geq-1$, then the (left-sided) Riemann-Liouville integral of order $\mu,\mu>0$ is given by

[TABLE]

Definition 2.7**.**

The (left sided) Caputo fractional derivative of $f,f\in C_{-1}^{m},m\in\mathbb{N}\cup\{0\}$, is defined as:

[TABLE]

Note that for $0\leq m-1<\alpha\leq m$ and $\beta>-1$

[TABLE]

Definition 2.8**.**

Mittag-Leffler function of order $\alpha>0$ is defined by the series

[TABLE]

2.1 Daftardar-Gejji and Jafari Method

Daftardar-Gejji and Jafari Method (DJM) [5] is one of the popular methods applied to solve nonlinear equations of form

[TABLE]

where $L$ and $N$ are linear and nonlinear operators respectively and $f$ is known function.

In this case, the DJM provides the solution in the form of series

[TABLE]

where $G_{0}=N(u_{0})$ and $G_{i}=\left\{N\left(\sum_{j=0}^{i}u_{j}\right)-N\left(\sum_{j=0}^{i-1}u_{j}\right)\right\}$, $i\geq 1$.

From Eq.(2.12), the DJM series terms are generated as bellow:

[TABLE]

3 Existence, uniqueness and convergence:Nonlinear Case

First, we consider the nonlinear equation

[TABLE]

The Eq. (3.1) is a particular case of time dependent delay differential equation (DDE)

[TABLE]

with $\tau_{i}(x)=(1-q_{i})x,\quad i=1,2,3,\cdots,n$. The DJM series solution of Eq. (3.1) is of the form

[TABLE]

We present convergence result of this series solution motivated from [6].

Theorem 3.1**.**

*Let f be a continuous function defined on a $(n+2)$ dimensional rectangle

$R=\{(x,y_{0},y_{1},\cdots,y_{n})|0\leq x\leq b,-\delta_{i}\leq y_{i}\leq\delta_{i},i=0,1,2,\cdots,n\}$ and $\mid f\mid\leq M$ on $R$. Suppose that f satisfies Lipschitz type condition

[TABLE]

Then the DJM series solution (3.2) of DDE (3.1) converges uniformly in the interval [0,b].

Proof.

Without loss of gerenality, assume that $y(0)=1$. The equivalent integral equation of (3.1) is

[TABLE]

Using DJM, we get

[TABLE]

$\textrm{Since}\;q_{0}=1,0<q_{i}<1,i=1,2,3,\cdots,n.,\;q_{i}b\leq b.$

[TABLE]

[TABLE]

Using induction, we get

[TABLE]

Taking summation over $m$, we get

[TABLE]

Thus, by [7], we can conclude that the series solution of (3.1) converges uniformly in the interval [0,b]. Hence existence of (3.1) is proved. The uniqueness is of solution is obvious by condition (3.3). ∎

4 Stability analysis

The following definitions and theorems are generalization of corresponding definition and theeorems given in [8].

Definition 4.1**.**

Consider the autonomus time-dependent delay differential equation (DDE),

[TABLE]

*where $g:\mathbb{R}^{n+1}\rightarrow\mathbb{R}$. The flow $\phi_{x}(x_{0})$ is the solution $y(x)$ of (4.1) with initial condition

$y(x)=x_{0},\,x\leq 0$. The point $y^{*}$ is called equilibrium solution of (4.1) if $g(y^{*},y^{*},\cdots,y^{*})=0$.

(a) If, for any $\epsilon>0$, there exist $\delta>0$ such that $|x_{0}-y^{*}|<\delta\Rightarrow|\phi_{x}(x_{0})-y^{*}|<\epsilon,$ then the system (4.1) is stable (in the Lyapunov sense) at the equilibrium $y^{*}$.

(b) If the system (4.1) is stable at $y^{*}$ and moreover,$\lim\limits_{x\rightarrow\infty}|\phi_{x}(x_{0})-y^{*}|=0$ then the system (4.1) is said to be asymptotically stable at $y^{*}$.

(c) If the system (4.1) is not stable then it is called unstable.*

Theorem 4.1**.**

Assume that the equilibrium solution $y^{*}$ of the equation

[TABLE]

is stable at equilibrium $y^{*}$ and

[TABLE]

for some $\epsilon_{i}>0$ and x,x_{i}\in[x_{0},x_{0}+c)$$(i=1,2,\cdots,n), c is a positive constant, then there exists $\bar{x}>0$ such that the equilibrium solution $y^{*}$ of Eq. (4.1) is stable at equilibrium $y^{*}$ on finite time interval $[x_{0},\bar{x})$.

Corollary 1**.**

If the real parts of all roots of $\lambda-\sum_{i=0}^{n}a_{i}e^{-\lambda\tau_{i}^{*}}$ are negative, where $a_{i}=\partial_{i}f$, $i=1,2,\cdots,n$ evaluated at equilibrium $y^{*}$, then Eq. (4.1) is stable at $y^{*}$ on finite time interval $[x_{0},\bar{x})$.

5 Linear equation: Exact solution

Consider the differential equation involving multiple proportional delays,

[TABLE]

where $q_{0}=1$, $0<q_{i}<1$ and $a_{0},a_{i}\in\mathbb{R}$, $i=1,2,3,\cdots,n$. The equation (5.1) has applications in Science and Engineering [9, 10].

Integrating (5.1), we get

[TABLE]

Using successive approximation, we obtain

[TABLE]

[TABLE]

$\therefore$ The exact solution of (5.1) is

[TABLE]

If we define

[TABLE]

then

[TABLE]

This solution of (5.1) provides a novel special function

[TABLE]

Note: We use a brief notations $\mathcal{R}(\bar{a};\bar{q};x)$ for the special function $\mathcal{R}(a_{0},a_{1},\cdots,a_{n};q_{0},q_{1},\cdots,q_{n},x)$ and $(\bar{a};\bar{q})_{m}$ for $\prod_{j=0}^{m-1}\left(\sum_{i=0}^{n}a_{i}q^{j}_{i}\right)$.

6 Analysis

Theorem 6.1**.**

The power series

[TABLE]

has infinite radius of convergence.

Proof.

Suppose

[TABLE]

If $R$ is radius of convergence of (6.1) then by using ratio test [11]

[TABLE]

Thus the series has infinite radius of convergence. ∎

Corollary 2**.**

The power series (5.4) is absolutely convergent for all $x$ and hence it is uniformly convergent on any compact interval on $\mathbb{R}$.

Theorem 6.2**.**

For $r\in\mathbb{N}\cup\{0\}$, we have

[TABLE]

Theorem 6.3**.**

(Addition Theorem)

[TABLE]

Proof.

We have

[TABLE]

∎

Note: If $x+iy=z\in\mathbb{C}$, then

[TABLE]

Theorem 6.4**.**

If $z=re^{i\theta}\in\mathbb{C}$, then

[TABLE]

where,

[TABLE]

Theorem 6.5**.**

For $q_{0}=1$, $0<q_{i}<1$ and $a_{0},a_{i}\geq 0$, $i=0,1,2,3,\cdots,n$. The function $\mathcal{R}(\bar{a};\bar{q};x)$ satisfies the following inequality

[TABLE]

Proof.

Since $q_{0}=1$, $0<q_{i}<1$ and $a_{0},a_{i}\geq 0$, $i=0,1,2,3,\cdots,n$, we have

[TABLE]

Similarly, we have

[TABLE]

From (6.2) and (6.3), we get

[TABLE]

∎

7 Generalization to fractional order DDE

Consider the fractional delay differential equation involving multiple proportional delays,

[TABLE]

where $q_{0}=1$, $0<q_{i}<1$ and $a_{0},a_{i}\in\mathbb{R}$, $i=1,2,3,\cdots,n$.

The exact solution of (7.1) is

[TABLE]

where $(\bar{a};\bar{q})_{\alpha,m}=\prod_{j=0}^{m-1}\left(\sum_{i=0}^{n}a_{i}q^{\alpha j}_{i}\right)$

Theorem 7.1**.**

The power series (7) is convergent for all finite values of $x$.

Theorem 7.2**.**

For $q_{0}=1$, $0<q_{i}<1$ and $a_{0},a_{i}\geq 0$, $i=0,1,2,3,\cdots,n$. The function $\mathcal{R}_{\alpha}(\bar{a};\bar{q};x)$ satisfies the following inequality

[TABLE]

8 Conclusions

In this paper, we have obtained a new special function arising from differential equation involving multiple proportional delays. The solution is obtained by applying the successive approximation method. The existence, uniqueness, stability and convergence results for the time dependent delay differential equations are presented in this paper. The new special function exhibit different properties and relations. The generalization to fractional order case is also presented.

**Acknowledgements:

**S. Bhalekar acknowledges the Science and Engineering Research Board (SERB), New Delhi, India for the Research Grant (Ref. MTR/2017/000068) under Mathematical Research Impact Centric Support (MATRICS) Scheme.