# The asymptotic solutions of two-term linear fractional differential equations via Laplace transform (2023)

## Introduction

Due to the non-local nature of fractional derivatives, fractional differential equations have been widely applied in the mathematical modeling of different real phenomena, such as fluid mechanics, viscoelasticity and damping, diffusion, biology and engineering[10], [21], [24], [32], [37], [38]. In this paper, our motivation is to explore the asymptotic properties of the solution to the following two-term linear fractional differential equation with the Caputo derivatives $\left\{\begin{array}{c}{}_{0}^{C}{D}_{x}^{\alpha }u\left(x\right)+{c}_{1}{}_{0}^{C}{D}_{x}^{\beta }u\left(x\right)+{c}_{2}u\left(x\right)=f\left(x\right),\phantom{\rule{0ex}{0ex}}x>0,\hfill \\ u\left(0\right)={u}_{0},\phantom{\rule{0ex}{0ex}}{u}^{\prime }\left(0\right)={u}_{1},\hfill \end{array}\right\$where $0<\beta \le 1<\alpha \le 2$ or $1<\beta <\alpha \le 2$, ${c}_{1},{c}_{2}$ are two constants, and the force term $f\left(x\right)$ is given. As is well known, the $\alpha$-th order Caputo fractional derivative ${}_{0}^{C}{D}_{x}^{\alpha }u\left(x\right)$ is defined by[10], [21] ${}_{0}^{C}{D}_{x}^{\alpha }u\left(x\right)=\frac{1}{\Gamma \left(m-\alpha \right)}{\int }_{0}^{x}\frac{{u}^{\left(m\right)}\left(t\right)}{{\left(x-t\right)}^{\alpha +1-m}}\mathrm{d}t,$where $m=⌈\alpha ⌉$ is the smallest integer that is no less than $\alpha$.

ForEq.(1), a famous representive is the Bagley–Torvik equation[38], which was originally proposed to describe the motion state of structures containing elastic and viscoelastic materials, such as the motion of a rigid plate immersed in a Newtonian fluid[37]. The theory of existence and uniqueness of solutions to multi-term fractional differential equations has been developed[10], [11], [21], [32]. The stability analysis is another important topic for the initial value problems of fractional differential equations. Li and Zhang[26] gave a comprehensive survey on the stability theory of fractional differential equations. Čermák and Kisela[7] derived necessary and sufficient stability conditions for the homogeneous case ofEq.(1) when $\alpha /\beta$ is a rational number. Brandibur and Kaslik[1] discussed three-term homogeneous fractional differential equations and also derived necessary and sufficient stability conditions for the problem of real orders no more than $2$. Further, the asymptotic behavior of the solutions at infinity was discussed for some kinds of fractional differential equations when the solutions are stable[1], [7], [11], [29]. For the unstable solutions to fractional differential equations, Băleanu etal.[4] and Kassim etal.[18] also explored their asymptotic properties.

Except the investigation of qualitative properties for fractional differential equations, there are various methods to obtain analytical or numerical solutions, such as Adomian decomposition[17], fractional Green’s function[31], piecewise collocation[6], [34] or spectral collocation[12], [13] and some other methods[10], [14], [24]. She etal.[34] designed a novel scheme to solve a multi-term time-fractional initial–boundary value problem. A change of variable in the temporal direction is conducted to improve the regularity of the solution at initial time, so the $L1$ approach can be effectively used to discretize the Caputo derivatives. Li etal.[27] explored the long time numerical behaviors of nonlinear fractional pantograph equations and showed that the $L1$ scheme inherits the long time behavior of the underlying problems. The analytical or approximate solution of the fractional differential equation was also obtained by using the solution of the corresponding integer order differential equation[8], [9].

It is noted that the Laplace transform is a powerful tool for solving linear differential or integral equations. For linear fractional differential equation, the validity of the Laplace transform has been confirmed[21], [25], [32]. The exact solutions of some special kinds of equations can be directly derived by using the Laplace transform[19]. Khalouta etal.[20] proposed the inverse fractional Shehu transform method to obtain the exact solution of fractional differential equation. Lin etal.[28] applied the Laplace transform of the fractional derivative and the binomial series to derive the explicit solution of the homogeneous fractional differential equations. Fukunaga[15] expressed the solutions of some linear multi-term fractional differential equations by the two parameter Mittag-Leffler functions on condition that the orders of Caputo derivatives are integer multiples of a common real number. Kukla etal.[22] presented a numerical-analytical method to obtain the approximate solutions to fractional-order linear commensurate multi-term differential equations with Caputo derivatives via Laplace transform. Since the inversion of the Laplace transform is usually difficult to obtain analytically, numerical inversion algorithms are used to find the approximate solution. Three numerical inverse Laplace transform algorithms were introduced and their effectiveness was compared and illustrated[3], [36]. In addition, we note that some special functions like the Mittag-Leffler type function often appeared to derive the explicit form of solutions based on Laplace transform[21], [23]. For instance, Salahshour etal.[33] developed a new technique to obtain the asymptotic solutions about the origin and infinity by using finite terms of Mittag-Leffler function, but the asymptotic expansion at infinity contains only one term. Recently, Wang etal.[39], [40] derived the asymptotic expansions of the inverse Laplace transform about zero and infinity by expanding the Laplace transform at infinity and zero, respectively. These expansions were called Puiseux series in these references, which are a generalization of the Taylor series including fractional exponents and logarithmic factors. They were formally known as psi-series[16].

ForEq.(1), we note that all the above-mentioned works almost focused on the series solution near zero, but the asymptotic expansion at infinity was seldom explicitly formulated except the discussion on the asymptotic properties[1], [7], [29], [33]. In this paper, we aim to formulate the asymptotic solutions forEq.(1) about the origin and infinity using Laplace transform[39], [40], [41]. Generally speaking, the finite term asymptotic expansions of the solution about the origin and infinity can approximate the solution very well when the variable is small and large, respectively. We can also use Padé approximation[2] to improve the accuracy of the asymptotic solutions. As an application of the asymptotic solution about the origin, we separate the singularity of the solution and design a high order Legendre collocation method to solve the equivalent Volterra integral counterpart ofEq.(1).

Compared with the results in literature, the proposed methods in this paper have some advantages, which make them attractive for solving fractional differential equations:

• Our asymptotic solutions about the origin and infinity are explicit and more general, which include logarithmic factors inheriting from the possible logarithmic singularity of the force term.

• The asymptotic expansion about the origin gives the complete singular information of the solution, which is helpful to design high order numerical methods.

• The explicit asymptotic expansion of the solution at infinity has many terms, which can accurately capture the long time behavior of the solution and reveal the stable or unstable property of the solution.

• The method can be applied to solve more general multi-term linear fractional differential equations with constant coefficients.

The paper is organized in the following way. In Section2, some preliminaries about the psi-series expansions of Laplace transform and the stability results of the homogeneous case ofEq.(1) are introduced. In Sections3 The series expansion for the solution about the origin, 4 The asymptotic expansion for the solution at infinity, we derive the asymptotic expansions for the solution ofEq.(1) about the origin and infinity, respectively, and analyze the conditions such that the solution is stable or unstable. In Section5, we briefly discuss a singularity-separation Legendre collocation method to solveEq.(1) and provide two numerical examples including the Bagley–Torvik equation to validate the asymptotic solutions by cross check. Section6 is a brief summary of this paper.

## Preliminary

In this section, we introduce some preliminaries about the Laplace transform and its inversion, as well as the stability results forEq.(1) with homogeneous right-hand side.

Given a function $f\left(x\right)$, its Laplace transform is defined by $F\left(s\right)=\mathcal{L}\left[f\left(x\right)\right]={\int }_{0}^{\infty }f\left(x\right){\mathrm{e}}^{-sx}\mathrm{d}x.$Further, the inversion formula of the Laplace transform (3) is $f\left(x\right)={\mathcal{L}}^{-1}\left[F\left(s\right)\right]=\frac{1}{2\pi \mathrm{i}}{\int }_{\sigma -\mathrm{i}\infty }^{\sigma +\mathrm{i}\infty }F\left(s\right){\mathrm{e}}^{sx}\mathrm{d}s,$where $\sigma$ is chosen such that $F\left(s\right)$ is analytic for all Re$\left(s\right)>\sigma$, and $\mathrm{i}=\sqrt{-1}$.

Lemma 1

## [40]

Assume that $f\left(x\right)$ possesses the psi-series expansion about $x={0}^{+}$: $f\left(x\right)$

## The series expansion for the solution about the origin

In this section, we consider the fractional differential Eq.(1) with the orders satisfying $0<\beta \le 1<\alpha \le 2$. The series expansion for $u\left(x\right)$ about the origin is derived via Laplace transform. We also briefly discuss the case $1<\beta <\alpha \le 2$.

In order to solve the equation by Laplace transform, we assume that the explicit Laplace transform of $f\left(x\right)$ can be obtained by analytical methods or mathematical software. By performing the Laplace transform on the left and right-hand sides ofEq.(1) and using Lemma3, we

## The asymptotic expansion for the solution at infinity

Due to the limitation of convergence domain, the series solution about the origin forEq.(1) is accurate only when $x$ is near the origin, which means that it cannot recognize the behavior of $u\left(x\right)$ as $x$ tends to large. In this section, we further formulate the asymptotic expansion for $u\left(x\right)$ at $x=+\infty$. From Lemma2, in addition to the series expansion for $U\left(s\right)$ about $s={0}^{+}$, we also need to know all the non-zero singularities of $U\left(s\right)$ and the Laurent series expansions of $U\left(s\right)$ about them. Assume that $F\left(s\right)$

## Singularity-separation collocation method and experiments

As an application of the psi-series expansion for the solution about the origin, we show in this section that it can be used to design high order numerical methods, which is outlined as follows.

It has been shown thatEq.(1) can be converted to the following Volterra integral equation of the second kind[10] $u\left(x\right)={u}_{0}+{u}_{1}x-\frac{{c}_{1}}{\Gamma \left(\alpha -\beta \right)}{\int }_{0}^{x}\frac{u\left(t\right)-{u}_{0}-\left[\beta \right]{u}_{1}t}{{\left(x-t\right)}^{1-\alpha +\beta }}\mathrm{d}t-\phantom{\rule{0ex}{0ex}}\frac{{c}_{2}}{\Gamma \left(\alpha \right)}{\int }_{0}^{x}\frac{u\left(t\right)}{{\left(x-t\right)}^{1-\alpha }}\mathrm{d}t+\frac{1}{\Gamma \left(\alpha \right)}{\int }_{0}^{x}\frac{f\left(t\right)}{{\left(x-t\right)}^{1-\alpha }}\mathrm{d}t,$ where $\left[\beta \right]$ is the largest integer less than or equal to $\beta$. Denoting by ${\varphi }_{1}=\alpha -\beta ,\phantom{\rule{0ex}{0ex}}{\varphi }_{2}=\alpha$, ${c}_{k}^{\ast }=\frac{{c}_{k}}{\Gamma \left({\varphi }_{k}\right)}$

## Conclusion

In this paper, the asymptotic solutions about the origin and infinity for a two-term linear fractional differential equation are explicitly formulated in psi-series form via Laplace transform. Especially, we systematically discuss the asymptotic expansion of solution at infinity, which accurately describes the stable or unstable property of the solution. The expansion about the origin exhibits the singular behavior of the solution, which is accurate to the solution when the variable is near the

## Acknowledgment

This work was supported by the National Natural Science Foundation of China (grant No. 11971241), Natural Science Foundation of Jiangsu Province (grant No. BK20191375), the Program for Innovative Research Team in Universities of Tianjin(TD13-5078). The authors are very grateful to the editors and anonymous referees for their valuable comments and suggestions, which improve the quality of the paper.

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## FAQs

### What is the Laplace transform of a differential equation? ›

The Laplace transform will convert the equation from a differential equation in time to an algebraic (no derivatives) equation, where the new independent variable is the frequency. We can think of the Laplace transform as a black box. It eats functions and spits out functions in a new variable.

What is the Laplace transformation of the Caputo derivative? ›

The Laplace transform of constant proportional Caputo (CPC) derivative is found as [9], for Laplace transform of the derivatives see Table 1: L { 0 C P C D x α h ( x ) } = k 1 ( α ) s + k 0 ( α ) s α L { h ( x ) } − k 0 ( α ) s α − 1 h ( 0 ) .

What is the Elzaki transform method? ›

Elzaki transform method is a powerful device for constructing analytic approximate solution of scientific problems. It was initially introduced by Elzaki [5] in 2011 as a modification of the classical Sumudu transform.

Is Laplace equation a linear differential equation? ›

Because Laplace's equation is linear, the superposition of any two solutions is also a solution.

What are the advantages of Laplace transformation to solve differential equations? ›

The advantage of using the Laplace transform is that it converts an ODE into an algebraic equation of the same order that is simpler to solve, even though it is a function of a complex variable. The chapter discusses ways of solving ODEs using the phasor notation for sinusoidal signals.

What is the use of Laplace transform to solve partial differential equations? ›

Given a PDE in two independent variables x and t, we use the Laplace transform on one of the variables (taking the transform of everything in sight), and derivatives in that variable become multiplications by the transformed variable s. The PDE becomes an ODE, which we solve.

What is the formula used to solve Laplace equation? ›

Laplace's equation in spherical coordinates is: Consider the problem of finding solutions of the form f(r, θ, φ) = R(r) Y(θ, φ). By separation of variables, two differential equations result by imposing Laplace's equation: for some number m.

What does Laplace transform solve? ›

Laplace transform helps to solve the differential equations, where it reduces the differential equation into an algebraic problem.

What are differential transform methods? ›

The differential transform method is a procedure to obtain the coefficients of the Taylor expansion of the solution of differential and integral equations. So, one can obtain the Taylor expansion of the solution of arbitrary order and hence the solution of the given equation can be obtained with required accuracy.

What is the Mellin transform used for? ›

The Mellin Transform is widely used in computer science for the analysis of algorithms [clarification needed] because of its scale invariance property. The magnitude of the Mellin Transform of a scaled function is identical to the magnitude of the original function for purely imaginary inputs.

### What is linear Laplace transform? ›

The Laplace transform is an integral transform given by. It is a linear transformation which takes x to a new, in general, complex variable s. It is used to convert differential equations into purely algebraic equations.

Is The Laplace equation always true? ›

Not necessarily, the boundary conditions can be anything, not just constant.

What are the limitations of Laplace transform for differential equations? ›

• It is only used to solve complex differential equations like great methods.
• This method is only used to solve the differential equations using known constants.
• If the equation has unknown constants we cannot solve them using the Laplace Transform method.

What is the real life application of Laplace transform? ›

The Laplace transform is particularly useful in solving linear ordinary differential equations such as those arising in the analysis of electronic circuits, control systems etc. Data mining/machine learning: Machine learning focuses on prediction, based on known properties learned from the training data.

What are the two types of Laplace transform? ›

Laplace transform is divided into two types, namely one-sided Laplace transformation and two-sided Laplace transformation.

What are the advantages of the Laplace transform method of solving linear ordinary differential equations over the classical method? ›

Laplace transform makes the equations simpler to handle. When a higher order differential equation is given, Laplace transform is applied to it which converts the equation into an algebraic equation, thus making it easier to handle. Then we calculate the roots by simplification of this algebraic equation.

What is the importance of Laplace transform? ›

The Laplace transform can also be used to solve differential equations and is used extensively in mechanical engineering and electrical engineering. The Laplace transform reduces a linear differential equation to an algebraic equation, which can then be solved by the formal rules of algebra.

What is the importance of Laplace equation? ›

The Laplace equations are used to describe the steady-state conduction heat transfer without any heat sources or sinks. Laplace equations can be used to determine the potential at any point between two surfaces when the potential of both surfaces is known.

What are the number of conditions required to solve the Laplace's equation? ›

No initial conditions required.

Which kind of problems can be solved using Laplace transform? ›

Note that we can actually also use the Laplace Transform to solve Partial Differential Equations!

### What are the two methods of differential equation? ›

We can place all differential equation into two types: ordinary differential equation and partial differential equations.

Why do we need transforms? ›

The need for transform is most of the signals or images are time domain signal (ie) signals can be measured with a function of time. This representation is not always best.

What is the inverse Fourier transform? ›

The inverse Fourier transform is a mathematical formula that converts a signal in the frequency domain ω to one in the time (or spatial) domain t.

What is meant by bilateral Laplace transform? ›

The Bilateral Laplace Transform of a signal x(t) is defined as: The complex variable s = σ + jω, where ω is the frequency variable of the Fourier Transform (simply set σ = 0). The Laplace Transform converges for more functions than the Fourier Transform since it could converge off of the jω axis.

What is the caputo derivative? ›

The Caputo derivative is of use to modeling phenomena which takes account of interactions within the past and also problems with nonlocal properties. In this sense, one can think of the equation as having “memory.”

What is the L1 scheme for Caputo derivative? ›

The L1 approximation of the Caputo derivative is constructed by dividing the interval [0,x] to subintervals of equal length h and approximating the first derivative on each subinterval using a second-order central difference approximation. = (k + 1)1−α - 2k1−α + (k - 1)1−α, (k = 2, ··· ,n - 1).

What is Caputo Fabrizio derivative? ›

The Caputo–Fabrizio fractional-order derivative (CF) is defined as follows [43] (2) 0 C F D t α u t = 1 1 − α ∫ 0 t u ′ τ exp − α t − τ 1 − α d τ , 0 < α ≤ 1 , in this definition, the derivative of a constant is equal to zero, but unlike the usual Liouville–Caputo definition (1), the kernel does not have a singularity ...

What is the Caputo derivative of a constant? ›

From these definitions, it is clear that the Caputo fractional derivative of a constant is zero, which is false when we consider the Riemann–Liouville fractional derivative.

Why do we use Caputo fractional derivative? ›

The Caputo and Atangana Baleanu fractional derivative operators are employed to get the solution of the system of fractional differential equations. The qualitative analysis has been made for the fractional-order model. Fixed-point theorem and an iterative schemes are used to get the existence and uniqueness.

What are the advantages of Caputo fractional derivative? ›

One of the great advantages of the Caputo fractional derivative is that it allows traditional initial and boundary conditions to be included in the formulation of the problem [4, 19, 37, 38]. In addition, its derivative for a constant is zero.

### What is difference between Riemann Liouville fractional derivative and Caputo fractional derivative? ›

3) Another difference between Riemann-Liouville and Caputo fractional derivative is in terms of interchange of integer order derivative and fractional order derivative. The interchange of integer order derivative and fractional order derivative is allowed under different conditions.

What is caputo fractional derivative of order α? ›

Another definition of fractional derivative introduced by Caputo, is defined as,(4) ( D α f ) ( t ) = 1 Γ ( m - α ) ∫ 0 t ( t - τ ) m - α - 1 f ( m ) ( τ ) d τ , for, m - 1 < α ≤ m where, is an integer. For more details, we refer the readers to Podlubny, 1999, Kilbas et al., 2006.

What is first principle derivative limits? ›

Formula for First principle of Derivatives:

y = f(x) with respect to its variable x. If this limit exists and is finite, then we say that: Wherever the limit exists is defined to be the derivative of f at x. This definition is also called the first principle of derivative.

What is theorem 1 of derivatives? ›

2.1 Theorem 1: The derivative of the sum of two functions is the sum of the derivatives of the functions. 2.2 Theorem 2: The derivative of the difference of two functions is the difference of the derivatives of the functions. 2.3 Theorem 3: The derivative of the product of two functions is given by the Product Rule.

What is Atangana Baleanu fractional derivative? ›

The Atangana–Baleanu derivative is a nonlocal fractional derivative with nonsingular kernel which is connected with variety of applications, see [5], [7], [9], [10], [15], [20]. Definition 2.1. Let p ∈ [1, ∞) and Ω be an open subset of the Sobolev space Hp(Ω) is defined by.

What is Caputo Nuvola? ›

Caputo Nuvola flour is a soft variety to use when making poolish and biga pizza doughs. Researched and developed in-house by Caputo, the '0' flour is made with 100% natural raw materials, with no added additives or preservatives.

What is Grunwald Letnikov fractional derivative? ›

In mathematics, the Grünwald–Letnikov derivative is a basic extension of the derivative in fractional calculus that allows one to take the derivative a non-integer number of times.

What are the advantages of fractional derivatives? ›

The main advantages of fractional derivatives are flexibility and non-locality. Since these derivatives are of fractional order, they can approximate real data with more flexibility than classical derivatives. Furthermore, they also take into consideration non-locality, something that classical derivatives cannot do.

What are the different types of fractional order derivatives? ›

The three most commonly used fractional derivatives are Riemann–Liouville, Caputo, and Grünwald–Letnikov.

What is rule 3 the derivative of a constant times a function? ›

The Constant multiple rule says the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function. The Constant rule says the derivative of any constant function is always 0.

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