Fractional Quasi-Bessel Equations

We generalize the classical Bessel equation (1.1)

to a broad class of equations including multi-term fractional Bessel, Cauchy-Euler equations, and fractional differential equations with constant coefficients. We consider the Caputo

and Riemann-Liouville

fractional derivatives.

Equation (1.1) is a source for many generalizations and extensions. Numerous results for hyper-Bessel equations and corresponding operators, introduced by Dimovski, are obtained in [1, 4, 10, 12]. Paper [3] introduces another definition for the fractional order Bessel operator, which allows to construct an operational calculus applicable to Bessel equation with fractional derivatives. W. Okrasiński and L. Plociniczak [14] consider a fractional modification of the Bessel equation, and solve it in terms of the power series.

A natural extension of the classical Bessel equation (1.1) in terms of Caputo fractional derivatives

was analyzed by Rodrigues, Viera and Yakubovich [16], where a solution in a form of series for equation (1.4) was identified for a some specific values of $\nu$ depending on $\alpha$.

The multi-term fractional Bessel equation

with existence and uniqueness criteria was presented by the authors in [5] where we constructed solutions in the form of series

with coefficients $c_{n}$ as

The characteristic equation for equation (1.5)

must be solvable and all derivatives $D^{\alpha_{i}}x^{\gamma}$ should exist. Theorem 1 below states that series solution (1.6),(1.7) always exists in real numbers as long as all $d_{i}>0$ and $\nu$ satisfies a certain threshold condition. Our results for equation (1.5) are obtained for a broad class of parameters $\alpha_{i}$ and $\nu$. They include some results of ([10], sec. 3.4) and [16]. For example, if we deal with derivatives of integer orders $\alpha_{i}$ and $\nu=0$ in equation (1.5), then we arrive at the eigenvalue problem, which is solved in [10] and where the fundamental system of solutions (in terms of the hyper-Bessel functions of Delerue) has been determined. Besides fractional Bessel equations (1.5), the quasi-Bessel equations generalize Cauchy-Euler equidimensional equations as well and, thus, we extend further a number of results from the works of Kilbas and Zhukovskaya [9, 18, 19]. After a minor modification, our approach becomes also applicable to the equations with constant coefficients. In works [13], using the methods of [7], the initial value problem is solved for constant-coefficient fractional equations. In [2], the Cauchy problems for some linear constant-coefficient equations with real, including irrational, derivatives, are analyzed and solved. The presented method and its particular version for constant-coefficient equations also allow to treat a class of irrational derivatives. Additional extensive information regarding fractional differential equations can be found in many works on fractional calculus, e.g., in monographs [8, 10, 15, 17].

In this paper we analyze the next generalization of multi-term Bessel equations – the quasi-Bessel fractional equations

where $\alpha_{i}\in\mathbb{R^{+}}$ with $\alpha_{1}=\max\limits_{1\leq i\leq m}\{\alpha_{i}\}$ and $p_{1}=0$. Shifting indices $p_{i}$ (deviations with respect to (1.5)) should be non-negative and satisfy special almost-rational conditions defined in Section 2. If all additional indices $p_{i}$ are zero, then we arrive at the previously investigated in [5] multi-term fractional Bessel equation (1.5), where the powers of multipliers $x$ must match the order of derivatives. However, in this research the powers of $x$ have to match the order of only one derivative – the derivative of the highest order. As we show below, many other equations, including equations with constant coefficients, represent special cases of quasi-Bessel equations and, as just a corollary, we can derive the key points of the theory for fractional constant-coefficient equations.

The structure of this paper is as follows.
In Section 2 we present the fractional series solution for equation (1.9) and derive the steps to obtain it numerically.

In Section 3 we cite the existence theorem from [5], and prove our main existence result (Theorem 2), including the necessary condition claiming that the power of $x$ must match the highest fractional derivative.

Motivated by [16], Section 4 presents the uniqueness theorem for the initial value problem of equation (1.9).

Section 5 covers the use of the presented methodology to identify a series solution for homogeneous fractional equations with constant coefficients and its further extension into equations with power function factors. As a particular case of the constructed theory we have another derivation of the existence results for fractional differential equations with constant coefficients with multiple derivatives of any order.

Section 6 provides a few numerical examples which support the constructed theory. Numerical method which utilizes substitution [6] is used to cross check the identified series solution in Example 1 and extend it further. Examples 2-4 confirm that our methodology works on previously analytically solved problems with one fractional derivative presented in monograph by Kilbas, Srivastava, and Trujillo [8].

Throughout this paper notation $D^{\alpha}u(x)$ is used for both, Riemann-Liouville and Caputo, types of fractional derivatives, whereas $D_{C}^{\alpha}u(x)$ refers to the Caputo derivative, and $D_{R}^{\alpha}u(x)$ is the Riemann-Liouville derivative.

We go beyond equation (1.5) and consider equation (1.9) in the following form:

It is worth mentioning its particular cases:

This section is applicable to equations with both Caputo and Riemann-Liouville derivatives. The only difference is the condition on acceptable $\gamma$ in (1.8) to generate a true solution. For Riemann-Liouville it is $\gamma>-1$ and for Caputo – $\gamma>\lceil\alpha_{1}\rceil-1$ to assure existence of derivatives.

Notation. Let $n_{\max}$ be the integer ceiling for the highest non-integer derivative, i.e., $n_{\max}=\max{\{n_{i}\}}$ and $n_{\min}$ be the integer ceiling for the lowest non-integer derivative, i.e., $n_{\min}=\min{\{n_{i}\}}$ where
$n_{i}-1<\alpha_{i}<n_{i},n_{i}\in\mathbb{N},i=1,...,m$.

In the multi-term fractional Bessel equation (1.5) it was possible to use $s=\beta$ as a step (increase) in the series solution because it allowed recursively express all $c_{n}$ (1.7). In the more general case of equation (2.1), it is not possible since some coefficients $c_{n}$ do not get ’balanced out’ and need to be forced to be zero, which leads to a trivial solution. Therefore, we need to find the largest possible step $s\leq\beta$ which allows us to balance out (express recursively) all coefficients. Thus, we proceed as follows.

Let us assume that the solution for equation (2.1) is

where $\gamma$ satisfies the equation

where $p_{i}=0,i=2,...,m_{1}$ and

It is clear that all roots of equation (2.3) belong to a bounded interval.
In addition to $p_{1}=0$, several other terms could also have $p_{i}=0$. Let us call them the pure Bessel terms. For these terms the power of the factor $x^{\alpha}$ matches the order of the derivative $D^{\alpha}u(x)$. Let $m_{1}$ be the number of pure Bessel terms in (2.1). Then, all other terms have strictly positive shifted powers $p_{i}>0$ for $i=m_{1}+1,...,m$.

By plugging expression (2.2) into equation (2.1), we obtain

or

If the step $s$ is such that $\displaystyle\frac{p_{i}}{s}=n_{p_{i}}\in\mathbb{N}$ and $\displaystyle\frac{\beta}{s}=n_{\beta}\in\mathbb{N}$ then we obtain

Step $s$ should be such that any powers of $x$ in quasi-Bessel equation (2.1) are included in the set $\gamma+sn$. We want to find the maximum possible step $s$ to minimize the number of steps and avoid having unnecessary zero coefficients $c_{n}$. Let us describe how step $s$ can be identified.

• •

  From the definition of quasi-Bessel equation (2.1) we have $\beta\in\mathbb{Q}_{r}^{+}$, $p_{i}\in\mathbb{Q}_{r}^{+}$. Therefore we first convert them into rational numbers:
  $\displaystyle\beta^{0}=\frac{\beta}{r}=\frac{a_{1}}{b_{1}}$; $\ \ \displaystyle p_{i}^{0}=\frac{0}{1}=\frac{a_{i}}{b_{i}},2\leq i\leq m_{1};\ \ p_{i}^{0}=\frac{p_{i}}{r}=\frac{a_{i}}{b_{i}}$, $m_{1}<i\leq m$. If $r\in\mathbb{Q^{+}}$, then $r=1$.

• •

  Find the lowest common denominator $N_{LCD}=\text{LCD}\{b_{i}\}$,
  $m_{1}<i\leq m$.

• •

  Calculate the acceptable step and corresponding shifts for $\beta$ and $p_{i}$:

  $$s^{0}=\frac{1}{N_{LCD}};\ n_{\beta}^{0}=\frac{\beta^{0}}{s^{0}}\in\mathbb{N};\ n_{p_{i}}^{0}=\frac{p_{i}^{0}}{s^{0}}\in\mathbb{N},\ m_{1}<i\leq m.$$

  (2.7)

• •

  The identified parameters $\beta^{0},p_{i}^{0}$, $m_{1}<i\leq m$ in (2.7) can still have common factors. To maximize step $s$ we need to identify their greatest common factor ($N_{gcf}$), adjust step $s$ and each parameter. Then, finally, we obtain:

  $$s=s^{0}\cdot N_{gcf};\ n_{\beta}=\frac{n_{\beta}^{0}}{N_{gcf}};\ n_{p_{i}}=\frac{n_{p_{i}}^{0}}{N_{gcf}},\ m_{1}<i\leq m.$$

  (2.8)

As an example, for equation

we have $d_{1}=2$, $d_{2}=-3$, $d_{3}=1$, $\alpha_{1}=2.4$, $\alpha_{2}=1.8$, $\alpha_{3}=0.4$. Then $\beta=3$, $p_{2}=0.3$, $p_{3}=0.6$, and we obtain $b_{1}=1$, $b_{2}=10$, $b_{3}=5$, their $N_{LCD}=10$. Thus,

Since $N_{gsf}={\rm GCF}(30,3,6)=3$, then, finally,

Other examples are presented in Sections 5 and 6.

Equation (2) can be re-written to identify more clearly the recursive relationship of the coefficients $c_{n}$:

As long as condition (1.8) for $x^{\gamma}$ at $n=0$ is satisfied, we can make $c_{0}$ equal to any number. For simplicity we assume $c_{0}=1$. Then, to make equation (2) valid, we need to offset $c_{n}$ by the sum of coefficients $c_{n}^{p_{i}}$ and $c_{n}^{\beta}$

Coefficient $c_{n}$ should compensate previous like terms, the terms which are $n_{\beta}$ steps before $c_{n}$ together with the terms which are $n_{p_{i}}$ steps before $c_{n}$ from (2).

These coefficients can be expressed as

where $d_{p_{i}}$ and $\alpha_{p_{i}}$ correspond to $p_{i},m_{1}<i\leq m$ from the original equation (2.1) and

These coefficients (2.10) and (2.11) are equal to zero up to the point $n_{p_{i}}$ for the terms $c_{n}^{p_{i}}$ and up to $n_{\beta}$ for $c_{n}^{\beta}$. Hence, we combine (2.10), (2.11) and obtain the following formula for coefficients $c_{n}$:

where $U(n-k)=\begin{cases}1\text{ if }n\geq k\\ 0\text{ if }n<k\end{cases}$ and $Q(r,p),s,n_{p_{i}},n_{\beta}$ are defined in (2.4), (2.8). This algorithm provides the fractional series solution for equation (2.1).

In this section we prove that the constructed fractional series solution and matching power $x^{\alpha}$ and the highest derivative (i.e., $p_{1}=0$) is the necessary and sufficient condition to obtain the series solution (2.2) where coefficients are defined in (2). Based on the formulas in the previous section it looks like the described logic provides a solution to equation (2.1) if at least one $p_{i}=0,i=1,...,m$ (not necessarily $p_{1}=0$). However, this is not the case as shown in Theorem 2 below.

Let $n_{m_{0}}=\displaystyle\max_{1\leq i\leq m_{0}}{\{n_{i}\}}$ where $n_{i}-1<\alpha_{i}<n_{i},\ n_{i}\in\mathbb{N},i=1,...,m_{0}$. Here $m_{0}\leq m_{1}$ is the number of fractional derivatives and $m_{1}-m_{0}$ is the number of integer derivatives with $p_{i}=0$ in equation (2.1) or the number of integer derivatives in multi-term fractional Bessel equation (1.5).

Thus, if the characteristic equation (1.8) has several roots, then all roots except for the largest root need to be checked for validity.