Existence and uniqueness results for mixed derivative involving fractional operators

1.   Introduction

The pantograph equation is a special type of functional differential equations with proportional delay. The present study introduces a compound technique incorporating the perturbation method with a iteration algorithm to solve numerically the delay differential equation of pantograph type. The pantograph equations became a prime example of delay differential equation in the recent years. Over the last few years, the continuous and discrete cases of the pantograph equation have been extensively explored see ^[1,2,3].

Different authors discuss linear and non-linear pantograph equations. The solution to the simplest homogeneous linear pantograph cannot be expressed in terms of elementary functions. We can't solve even the simplest non-homogeneous linear pantograph problem using standard approaches like variation of constants and Laplace transformation. The existence and uniqueness of solutions for the linear pantograph equation's initial value problems change significantly depending on the beginning locations chosen. In general, the solution to the initial value problem may or may not exist, or may not be unique. some authors discuss linear pantograph Volterra delay-integro-differential equation and the multi-terms boundary value problem of fractional pantograph differential equations ^[4,5].

It is also possible to obtain additive, multiplicative, and functional separable solutions, as well as several additional precise solutions. Nonlinear pantograph-type PDEs of a more broad form, containing one or two arbitrary functions Polyanin et al. ^[6] examine Nonlinear pantograph-type diffusion PDEs, exact solutions and the principle of analogy. Recently, many research on fractional-order pantograph differential equations have recently been published, involving various operators ^[7,8,9], $\varphi$-Caputo derivative ^[10], Atangana-Baleanu-Caputo derivative ^[11,23,24].

Furthermore, several scientific scholars have produced results regarding the existence and uniqueness of solutions for various classes of fractional pantograph equations by applying various fixed-point theorems as like Shah et al. ^[12] discussed the dynamics and stability of-fractional pantograph equations and Houas et al. ^[13] studied the existence and Ulam stability of fractional pantograph differential equations with two Caputo-Hadamard derivatives.

Different authors like ^[14,15] work on the pantograph-catenary electrical contact system of high speed railways. The pantograph-catenary electrical contact system, which serves as the only power entrance, keeps the high-speed train's power transfer reliable and efficient. A pantograph-catenary system must take into account the wind, sand, rain, thunder, ice, and snow while designing it due to the rapid expansion of high-speed trains around the world. Commercialized lines are also being developed in China to cover isolated areas with severe environments. There are some recent results on the existence of solutions for fractional integro-differential and fractional differential equations ^[16,17].

The following fractional integro-differential equation of pantograph type is considered in this work, along with appropriate initial conditions.

Where $0 < \gamma < 1$ and $1 < \beta < 2$ as well as $\Lambda, q < 1$, $\phi:{\bf J}\times\mathfrak{R}\times\mathfrak{R}\rightarrow \mathfrak{R}$ and $k_i:{\bf J}\times{\bf J}\times\mathfrak{R}\rightarrow \mathfrak{R}$ are continuous for $i = 1, 2,$ and $^CD^\beta, \; ^CD^\gamma$ are the Caputo fractional derivatives.

We shall first look into the existence and uniqueness of solutions for (1.1). To do so, we turn the original problem into an equivalent integral equation, then establish the existence and uniqueness of the solutions using fixed point theorems.

The following is the outline for this paper. We look at some essential preliminaries in section $2$. We discuss the existence and uniqueness of problem (1.1) in section $3$. We explore a helpful application to represent our primary finding in section $4$. In section 5, we present some numerical methods. In section 6, we find some numerical results to show the applicability of our results.

2.   Preliminaries

We present some well-known definitions and lemma in this part.

Lemma 2.1. ^[18,19] Suppose that $\beta > \varrho > 0$ and $\phi\in\mathcal{L}^1([b, d])$. At that point $D^\varrho I^\beta\phi(t) = I^{\beta-\varrho}\phi(t), \; t\in[b, d].$

Lemma 2.2. ^[18,19] For $\beta > 0$ and $\varrho > -1$, we get

For $\varrho = 0$ and $y = 0$, we get

Assume that $[b, d]\in\mathfrak{R}$ be a finite interval as well as suppose that $\beta, \gamma, \xi\in{\bf C}$ and $\mathfrak{R}(z) = Re(z)$ for $z\in{\bf C}$. The RL-fractional integral and derivative of order $\beta\in{\bf C}$ are defined by

and

On the interval $[b, d]$, the Caputo fractional derivative of order $\beta$ is defined by

When $b = 0$, $I^\beta_{b^+}z$ and $^CD^{\beta}_{b^+}z$ are denoted by $I^\beta z$ and $^CD^\beta z$. The semi-group features of the fractional integral operator $I^\beta_{b^+}$ as well as the fractional differentiation operator $D^\gamma_{b^+}$ are given by ^[19].

Lemma 2.3. Suppose that $\mathfrak{R}(\beta), \mathfrak{R}(\gamma) > 0$ as well as $\phi(y)\in{\bf C}[b, d]$. For $y\in [b, d]$ the following statements are true:

(i) $(I^\beta_{b^+}I^\gamma_{b^+}\phi)(y) = (I^{\beta+\gamma}_{b^+}\phi)(y)$.

(ii) $(D^{\beta}_{b^+}I^\beta_{b^+}\phi)(y) = \phi(y)$.

(iii) If $\mathfrak{R}(\beta) > \mathfrak{R}(\gamma)$ at that point

(iv) Suppose that $m = [\mathfrak{R}(\beta) > 0]+1$ for $\mathfrak{R}(\beta)\not\in{\bf N}$ and $\phi_{m-\beta}(y) = (I^{m-\beta}_{b^+}\phi)(y)\in{\bf C}^m[b, d],$ then

Suppose that ${\bf C}_{\xi}[b, d]$ be the space of function $\phi$ defined on $(b, d]$ in such a way that $(y-b)^\xi\phi(y)\in{\bf C}[b, d]$ along the norm $\|\phi\|_{{\bf C}_{\xi}} = \|(y-b)^\xi\phi(y)\|_{\bf C}: = \sup_{y\in[b, d]}|(y-b)^\xi\phi(y)|$. Note that for $\xi = 0, \; {\bf C}_\xi[b, d] = {\bf C}[b, d].$ The continuity of the fractional integral operator $I^\beta_{b^+}$ from the space ${\bf C}_\xi[b, d]$ into ${\bf C}[b, d]$ is discussed in the following lemma (^[19] Lemma 2.8 (a)).

Lemma 2.4. Suppose that $\mathfrak{R}(\beta) > 0$ and $1\geq\mathfrak{R}(\xi)\geq0.$ If $\mathfrak{R}(\xi)\leq\mathfrak{R}(\beta)$ at that point the fractional integral operator $I^\beta_{b^+}$ is bounded from ${\bf C}_\xi[b, d]$ into ${\bf C}[b, d]$

According to the following (^[19] Lemma 2.21, part (a)) when $\mathfrak{R}(\beta)\not\in{\bf N}_0$ the Riemann-Liouville fractional integral operator $I^\beta_{b^+}$ is the left inverse of the Caputo fractional differentiation operator $^CD^{\beta}_{b^+}$.

Lemma 2.5. Suppose that $\beta\in{\bf C}$ with $0 < \mathfrak{R}(\beta)$ as well as $z(y)\in{\bf C}[b, d].$ If $\mathfrak{R}(\beta)\not\in{\bf N}$, at that point

The fixed point theorem in ^[20], first presented by Krasnoselskii, is necessary to show that the existence of solution for (1.1).

Theorem 2.6. Assume that $E$ be a nonempty and convex closed subset of a Banach space $Y$. Let $S$ as well as $R$ be two operators such that

(i) when $v, w\in E$ then $Sv+Rw\in E$,

(ii) $S$ is continuous and compact,

(iii) $R$ be a contraction mapping.

At that point $y\in E$ must exist in such a way that $y = Sy + Ry$.

3.   Existence and uniqueness of solution

Consider ${\bf C}({\bf J})$ be a Banach space along the norm $\|v\|_{\bf C} = \sup_{t\in{\bf J}}|v(t)|.$ We define

In the next lemma, we present an integral equation that corresponding to Eq (1.1).

Lemma 3.1. Suppose that $\phi:{\bf J}\times\mathfrak{R}\times\mathfrak{R}\rightarrow \mathfrak{R}$ as well as $k_i:{\bf J}\times{\bf J}\times\mathfrak{R}\rightarrow \mathfrak{R}$ are continuous functions. If and only if $v$ is a solution of the fractional integral equation, then the function $v\in {\bf C}({\bf J})$ fulfils problem (1.1).

Proof. Suppose that $v\in {\bf C}({\bf J})$ to solve the problem (1.1). Using the concept of the Caputo fractional derivative in Lemma (2.3) (d), we get

apply $I^\gamma_{0^+}$ on both sides

We define

The existence of (1.1) is investigated under the following conditions:

${\bf {H_1}}: {\phi:{\bf J}\times\mathfrak{R}\times\mathfrak{R}\times\mathfrak{R}}\rightarrow \mathfrak{R}$ is continuous and a continuous function exists, $b:E\rightarrow [0, \infty)$ in such a way that $t\in[0, \mathfrak{T}]$ and $v_i, w_i\in\mathfrak{R}, \; i = 1, 2.$

${\bf {H_2}}: {{k_i}:{\bf J}\times {\bf J}\times \mathfrak{R}\rightarrow \mathfrak{R}}, \; i = 1, 2$ be continuous and there exist $b_1:\Delta_\mathfrak{q}\rightarrow [0, \infty)$ and ${b_2}:\Delta\rightarrow [0, \infty)$ in such a way that $d_1(t): = \int_{0}^{\mathfrak{q}t}b_1(t, s)ds\in{{\bf C}} {({\bf J})}, \; d_2(t): = \int_{0}^{t}b_2(t, s)ds\in{{\bf C}} {({\bf J})}$

${\bf {H_3}}:$

Assume that the closed ball with radius $\mathfrak{r}_0$ and centre at $0$ is $\mathfrak{B}_{\mathfrak{r}_0}\subset {\bf C}({\bf J})$ as well as put

and

and define

Lemma 3.2. Suppose that $({\bf H}_1)$–$({\bf H}_3)$ be satisfied, at that point the operator $S$ maps $\mathfrak{B}_{\mathfrak{r}_0}$ into itself, and $S:\mathfrak{B}_{\mathfrak{r}_0}\rightarrow \mathfrak{B}_{\mathfrak{r}_0}$ is continuous and compact.

Proof. Step 1: We prove that $S(\mathfrak{B}_{\mathfrak{r}_0})\subset\mathfrak{B}_{\mathfrak{r}_0}$ where $\mathfrak{B}_{\mathfrak{r}_0} = \{v\in W: \|v\| \leq \mathfrak{r}_0\}$. For $v\in\mathfrak{B}_{\mathfrak{r}_0}$, for assumption $({\bf H}_1)$

For $t\in{\bf J}$ using assumptions $({\bf H}_2)$ and $({\bf H}_3)$ we have

Step 2: $S:\mathfrak{B}_{\mathfrak{r}_0}\rightarrow \mathfrak{B}_{\mathfrak{r}_0}$ is continuous. $\epsilon > 0$ be a fixed point, choose an arbitrary $v, w\in\mathfrak{B}_{\mathfrak{r}_0}$ in such a way that $\|v-w\|\leq\epsilon$. For $t\in {\bf J}$ we get

where

for $i = 1, 2$ using (3.3) we have

We see that $\omega_{\mathfrak{r}_0}(k_i, \epsilon)\rightarrow0, \; as\; \epsilon\rightarrow0$ from the uniform continuity of $k_i, \; i = 1, 2$ on bounded subsets of ${\bf J}\times\mathfrak{R}\times\mathfrak{R}$. As a result of the inequality (3.4) $S:\mathfrak{B}_{\mathfrak{r}_0}\rightarrow \mathfrak{B}_{\mathfrak{r}_0}$ is continuous.

Step 3: An equi-continuous subset of ${\bf C}({\bf J}$) is $S(\mathfrak{B}_{\mathfrak{r}_0})$. Supposition $({\bf H}_2)$ states that for any $v\in\mathfrak{B}_{\mathfrak{r}_0}$ as well as $s\in{\bf J}$ we have

and similarly

Now, let $t_1, t_2\in{\bf J}$ and $t_1\leq t_2$.

By Eqs (3.5) and (3.6), we get

As $t_1\rightarrow t_2$, the right hand side of inequality (3.7) tends to zero. We can see from Steps 1–3 and the Arzela-Ascoli theorem that $S:\mathfrak{B}_{\mathfrak{r}_0}\rightarrow \mathfrak{B}_{\mathfrak{r}_0}$ is continuous and compact.

Theorem 3.3. In the space ${\bf C}({\bf J})$, problem (1.1) has at least one solution with assumptions $({\bf H}_1)$–$({\bf H}_3)$.

Proof. Define the $R$ operator on ${\bf C}({\bf J})$ as follows:

The operator $R$ is clearly defined and $Rv\in {\bf C}({\bf J})$ for some $v\in {\bf C}({\bf J})$ due to the continuity of $\phi$ and Lemma (2.3).

For any $v, w\in{\bf B}_{\mathfrak{r}_0}$, and $t\in {\bf J}$ based on assumptions $({\bf H}_1)$–$({\bf H}_3)$ and inequality (3.2).

As a result, $Sv+Rw\in\mathfrak{B}_{\mathfrak{r}_0}$ for every $v, w\in\mathfrak{B}_{\mathfrak{r}_0}$ We can also using $({\bf H}_1)$ for some $v, w\in {\bf C}({\bf J})$ we get

B is a contraction mapping, based on assumption $({\bf H}_3)$ and inequality (3.8). The assumptions of Theorem (2.6) are thus satisfied by Lemma (3.2).

Theorem 3.4. If $({\bf H}_1)$ and $({\bf H}_3)$ are true, then the following assumption is true.

${\bf H}_4$ : $k_i:{\bf J}\times{\bf J}\times\mathfrak{R}\rightarrow \mathfrak{R}, i = 1, 2$ is continuous as well as $b_i:{\bf J}\times{\bf J}\rightarrow [0, \infty), i = 1, 2$ exist in such a way that

and

Then, for ${\bf J}$, problem (1.1) has a unique solution.

Proof. It is sufficient to prove that the integral equation (3.1) has a unique solution using Lemma (3.1). Define the $\mathcal{F}$ operator on ${\bf C}({\bf J})$ as follows:

$\mathcal{F}v\in {\bf C}({\bf J})$ for any $v\in {\bf C}({\bf J})$ is simply found using the continuity of $\phi, k_1, k_2,$ and Lemma (2.2). $\mathcal{F}$ be the fixed point are the solution of (3.1). In the next section, we show that $\mathcal{F}$ is a contraction mapping, and $\mathcal{F}$ has a specific fixed point according to the Banach contraction principle. Suppose that $v, w\in {\bf C}({\bf J})$. According to $({\bf H}_1)$ and $({\bf H}_4)$, for any $t\in{\bf J}$ we get

Hence

By assumption $({\bf H}_3)$ it prove that $\mathcal{F}$ is a contraction mapping.

4.   Example

Example 4.1. Consider the integro-differential equation given below

Put

At that point

All above relations shows that $({\bf H}_1), \; ({\bf H}_4)$ and $({\bf H}_3)$ are fulfilled.

5.   Numerical method

Here, we want to use the Sinc collocation method to approximation the solution of (1.1). Therefore, the Sinc basis functions must be defined. The following definition is given for the translated Sinc base functions

where Sinc(t) on the complete real line $(-\infty, \infty)$ is given below

Let $q$ and $m$ be the two integers, with the help of the previous basis function, we may approximate the following function $\phi(t)$ on the real line:

where $e$ represent as step size. Furthermore, we can calculate the integrals on $\mathfrak{R}$ in the following manner using the Sinc quadrature rule:

Consider the single exponential transformation

Consider the inverse function

creating the infinite strip

the eye shaped domain

By using $\varphi_{b, d}$, consider $\Gamma$ be the image of the real line

We shall define the collocation points in $[b, d]$ as the image of the equidistance points $pe$ for some constant $e$ in the following manner in order to employ the Sinc collocation method

We will be able to approximate the function $\phi(t)$ at the finite interval $[b, d]$ using the transformations $\chi_{b, d}$ as well as $\varphi_{b, d}$.

where $e$ is a constant, $q$ and $m$ be the non-negative integers. Furthermore, the Sinc quadrature rule in the finite interval can be defined as

Since $\varphi^\prime_{b, d}(\zeta) = \frac{(\zeta-b)(d-\zeta)}{d-b}$ and $\varphi_{b, d}(pe) = (d-b)y_e+b,$ we get

Theorem 5.1. Suppose that $\phi\in\mathfrak{L}_{\rho-1, \sigma-1}(E)$ with

where $\mathfrak{L} = \{aj:|a| < l\}$ and

Taking $e = \sqrt{\frac{2\Pi l}{\sigma m}}$ as well as $q = [(\frac{\sigma}{\rho})m]+1$, we get

where $C_1$ is a constant which depends on $\phi, l, \rho,$ and $\sigma.$

The following theorems prove that for some constants $\rho$ and $\sigma$, the relation (5.1) and (5.2) attain exponential rates of convergence when $\phi(t)$ belongs to $\mathfrak{L}_{\rho, \sigma}(E)$. For the solution (1.1) using Sinc collocation method, near the boundary points $b$ and $d$ the solution tends to $0$. Define $K(t)$

where $\mu_b(t)$ and $\mu_d(t)$ can be written as

We have

So, by using Sinc basis functions, the function $K(t)$ can be approximated as shown below

Using (5.4) and (5.5) we must define the approximate solution given below

Substituting the solution $w_m(t)$ in (3.9), we get

Define some operators

since, relation (5.7) can be written as

Define $r_p\in[0, 1]$ as

with $\bar{e}\sqrt{\frac{2\Pi l}{(\beta+\gamma)q}}$ and $m = [(\beta+\gamma)q]+1$, using relation (5.3) with $r_p$, we get

for $(Rv)(t)$, we get

For $(Pv)(t)$, we have

when (5.9)–(5.11) are substituted into (5.8), we obtain

Define the points $t_k\in[0, {\bf T}]$

where $e = \sqrt{\frac{\Pi l}{(\beta+\gamma)q}}$. By definition of $\varphi_{b, d}(t)$, we obtain

Using (5.13), we have

Newton iteration method used to solve the above relationship.

6.   Numerical results

In this part, we find the numerical results for Example (4.1) to check the applicability of Sinc collocation method. In Table 1, SE means single exponential. If we indicate by $E_1$ as well as $E_2$ the greatest absolute errors calculated with $q = q_1$ and $q_2$. The practical orders of convergence can be obtained by using the following formula

In order to compare our method with other ones given in ^[21,22]. For different values of $q = 2, 4, 8, 16,$ we solved Example (4.1) and tabulated the results at specific places in Table 1. Furthermore, we displayed the largest absolute errors at equidistant points in Table 2.

7.   Conclusions

Our manuscript is mainly focused on mixed derivative for fractional differential equations of order $1 < \beta < 2$ and $0 < \gamma < 1$. Applying the main tools from the fractional calculus, fixed point theorem, integro-differential equation, we propose the definition of $\alpha$-mild solutions and obtain the existence and uniqueness dependence of the solution. Furthermore, we construct some important supposition to prove some important results. Finally, we provide an application to show the applicability of our main points.

Acknowledgements

This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia [Grant No. 1388], through its KFU Research Summer initiative.

Conflict of interest

The authors declare that they have no conflicts of interest.