A comprehensive review of the recent numerical methods for solving FPDEs
-
Fahad Alsidrani
and
Norazak Senu
Abstract
Fractional partial differential equations (FPDEs) have gained significant attention in various scientific and engineering fields due to their ability to describe complex phenomena with memory and long-range interactions. Solving FPDEs analytically can be challenging, leading to a growing need for efficient numerical methods. This review article presents the recent analytical and numerical methods for solving FPDEs, where the fractional derivatives are assumed in Riemann-Liouville’s sense, Caputo’s sense, Atangana-Baleanu’s sense, and others. The primary objective of this study is to provide an overview of numerical techniques commonly used for FPDEs, focusing on appropriate choices of fractional derivatives and initial conditions. This article also briefly illustrates some FPDEs with exact solutions. It highlights various approaches utilized for solving these equations analytically and numerically, considering different fractional derivative concepts. The presented methods aim to expand the scope of analytical and numerical solutions available for time-FPDEs and improve the accuracy and efficiency of the techniques employed.
1 Introduction
In classical calculus, the integer-order derivative and integral are well-established mathematical operations with a clear physical and geometric interpretation, making them highly useful for solving practical problems in the scientific fields. The meaning of derivative is the rate of change in the relationship between the dependent and independent variables, and there is a well-accepted geometrical explanation, for example, in the relationship between the position and speed of an object. Unfortunately, the situation changes when we consider fractional-order differentiation or integration. The existing literature has no entirely acceptable geometrical interpretation for the fractional-order derivatives. However, fractional derivatives (FDs) capture a system’s history and memory in fractional modeling, allowing for a more accurate representation of complex phenomena. In contrast to integer-order derivatives, fractional-order derivatives are considered nonlocal operators, which implies that when computing a FD at a particular point, it relies on data in the vicinity of the current moment and information from the entire historical dataset. While this characteristic allows for the description of physical phenomena with memory features, it can pose challenges in numerical computations [1–5].
Furthermore, Podlubny [1] presented a geometric interpretation based on left-sided and right-sided Riemann-Liouville fractional integrals. Also, he suggested a physical interpretation for the Riemann-Liouville FD and for the Caputo FD. There is still a lack of geometric and physical interpretation of fractional integration and differentiation compared to simple interpretations of their integer order. Another geometrical interpretation based on the conformable FD of
Let
called the fractional cord of the curve
2 Fundamentals of fractional calculus
Fractional calculus stands out in the modeling problems involving nonlocality and memory effect concepts that are not well explained by integer-order calculus. The historical development of fractional calculus dates back to several centuries and involves contributions from multiple mathematicians. The concept of FDs has more than 300 years of history, yet it is still an exciting research topic, and interested researchers are actively working on it. It can be traced back to the letter from L’Hôpital to Leibnitz in 1695, which has the meaning of the derivative of a function of order 1/2. Later investigations and further developments were by other mathematicians, such as Euler in 1730, Lagrange in 1772, Laplace in 1812, Lacroix in 1819, Fourier in 1822, Abel in 1823–1826, Liouville in 1832–1873, Riemann in 1847, Holmgren in 1865–1867, and Grunwald in 1867–1872. These are early mathematicians who explored the possibility of extending differentiation and integration to noninteger orders [4,7–9].
However, the formalization of fractional calculus took several more years to develop. Since then, many researchers tried to put a helpful definition of a FD. Most of them used an integral form for the FD. We briefly introduce the commonly used FD in literature. In 1819, the first mention of FD in a published paper by Lacroix [10]. Starting with
In particular, he calculated the following derivative of fractional order, for
The next stage was taken form Fourier in 1822. He gave a definition of FD through the so-called Fourier transform by
Abel in 1823, discussed the FD as well.
The comprehensive development and formalization of fractional calculus occurred in the 19th and 20th centuries. Riemann and Liouville established the foundation of modern fractional calculus in the mid-19th century. Riemann introduced the concept of fractional integration, and Liouville further developed the theory by introducing the FD. Liouville made the progress work in FDs and successfully applied it into potential theory. If a function
and if the function
In 1870, the Riemann-Liouville derivative of order
The Riemann-Liouville FD is singular when
The
Hadamard in 1892 proposed a nonlocal FD with singular logarithmic function kernel with memory of order
The FD with a logarithmic function kernel of order
If
where
The main recent development in the field of fractional calculus was in 2016. Atangana and Baleanu announced a new definition of the FD with a nonlocal and no-singular kernel to overcome the weakness of the existence of the singular kernels involved in the previous fractional operators [15]. They presented some useful properties of the Atangana-Baleanu derivative and applied them to solve a fractional heat transfer model. The AB operator in the Caputo sense and Riemann-Liouville sense of order
and
where
Remark 2.1
Some significant advantages of the FDs
The relation between the AB-Caputo and AB-Riemann-Liouville operators is given by
(2.13)Both Caputo and Riemann-Liouville FDs have singular kernels.
Both ABC and ABR FDs have nonsingular and nonlocal kernels.
Both Caputo and ABC FDs of a constant function are zero.
Both Riemann-Liouville and ABR FDs of a constant function do not equal zero.
Proposition 2.1
For
Mathematical models established using FDs overlap better with experimental data than models based on integer-order derivatives. FD provides a mathematical tool to describe and analyze systems that exhibit memory or long-term dependence, which provides an understanding of real-world processes. These derivatives have been developed to deal with the solutions of significant equations in various scientific fields. The definitions of the fractional order derivative are not unique and can yield different behaviors for the FD, mainly when the order is noninteger. One practical reason for the different FD formulations in literature is that due to the necessity of using a model describing the behavior of viscoelastic materials, thermal media, or electromagnetic systems, which cannot be defined very well in the old approach. Moreover, each formulation in Tables 1 and 2 captures specific aspects of the viscoelastic response, such as memory effects, relaxation processes, or singularities, allowing researchers to choose the most appropriate derivative. Consequently, each application area may require a specific FD formulation. The choice of initial and boundary conditions can influence the definition of FDs depending on the problem. For example, in modeling anomalous diffusion phenomena behavior in physics, the Riemann-Liouville FD is commonly used in which the derivative captures the memory effect in the diffusion process. On the other hand, if the initial conditions are essential in control systems, anomalous diffusion phenomena, or wave equations, then the Caputo derivative is more suitable and often preferred. It is important to note that when the order is an integer, the behavior of FDs is the same as that of classical derivatives. However, their behavior can differ when the order is noninteger. For instance, in the case of noninteger orders, the Caputo derivative of a constant is zero, unlike the Riemann-Liouville derivative, which does not yield zero. The development of new derivatives, including FDs based on nonlocal kernel functions, arises from the need to enhance the modeling capabilities of various systems. The nonlocality of the new kernel allows a better description of the memory within structure and media with different scales [15]. The use of fractional partial differential equations (FPDEs) presents a more accurate description of diffusion processes in diverse scientific domains and is increasingly crucial in the representation of real-world situations.
The classical FDs
| Name | Definition | Domain |
|---|---|---|
| Riemann-Liouville left-sided derivative |
|
|
| Riemann-Liouville right-sided derivative |
|
|
| Caputo left-sided derivative |
|
|
| Caputo right-sided derivative |
|
|
| Grünwald-Letnikov left-sided derivative |
|
|
| Grünwald-Letnikov right-sided derivative |
|
|
The commonly used FDs with nonsingular kernel and the order
| Name | Definition | Domain |
|---|---|---|
| Caputo-Fabrizio |
|
|
| Yang et al. |
|
|
| Atangana-Baleanu-Caputo |
|
|
| Atangana-Baleanu-Riemann-Liouville |
|
|
| Caputo-Fabrizio, Gaussian kernel |
|
|
| Sun-Hao-Zhang-Baleanu |
|
|
Over the past few years, researchers have demonstrated that mathematical models can successfully describe many phenomena in engineering, physics, biology, and chemistry using fractional calculus’s mathematical tools and solving different types of noninteger order partial differential equations. Indeed, determining an exact solution for a nonlinear fractional partial differential equation (FPDE) is challenging and tough to compute for some problems. Consequently, obtaining exact solutions for nonlinear FPDEs becomes crucial for evaluating and validating the accuracy and efficiency of numerical methods. Time-fractional differential equations involving FDs can be solved numerically using various methods. In the mathematical literature, there are many mathematical approaches of fractional order have been implemented to obtain approximate solutions for TFPDEs. These methods include the variational iteration method (VIM), adomian decomposition method (ADM), homotopy analysis method (HAM), homotopy perturbation method (HPM), Hermite wavelet method (HWM), optimal homotopy asymptotic method (OHAM), Shehu decomposition method (SDM), direct power series method (DPSM), and others. In modified numerical methods, an integral transform is used in the FD operators to enhance the performance and accuracy of the numerical methods. It is worth mentioning that there are different ways to define FDs, and thus using any of these definitions can be viewed as a fractional calculus.
3 Special functions of fractional calculus
The specialized mathematical functions in the field of mathematical physics have occurred to address the demands of applied sciences, offering solutions to integer-order differential equations derived from mathematical physics models. In this section, we provide some basic information on the most important functions in the theory of differentiation of arbitrary order and the theory of fractional differential equations including the Gamma and Beta functions and the Mittag-Leffler functions, and the readers may refer to [4,9,17,18].
3.1 Gamma function
The Gamma function
Let
is known as the Gamma function or the Euler integral of the second kind and converges absolutely. It is defined for all complex numbers except for the nonpositive integers. This function is a generalization of the factorial in the form defined by
It is obvious that by using integration by parts, we obtain the fundamental property of the Gamma function as follows:
Proposition 3.1
Some important properties of the Gamma function include
3.2 Beta function
The Beta function is a special function defined as follows:
where
Corollary 3.1
The Beta function can be written in the form of Gamma function as follows:
3.3 Mittag-Leffler function
The Mittag-Leffler function with two-parameters is a special function defined by
For
Corollary 3.2
The most known relationships for the Mittag-Leffler can be mentioned as follows [19]:
3.4 Integral transforms
Integral transforms have proven to be powerful tools in addressing FPDEs. In this exploration, we focus on employing a range of transforms, such as the Laplace transform, Sumudu transform, Elzaki transform, or Shehu transform to convert FPDEs into ordinary differential equations. This transformation simplifies the solution process, making it possible to apply well-established techniques. The modified methods depend on combining an analytical or numerical approach with an appropriate transformation operator, making it easier to achieve analytical or numerical solutions for FPDEs that involve fractional-order derivatives. In the following, we provide the most employed integral transforms in fractional calculus.
Defined a set of function:
If
(3.7)and the inverse Laplace transform of
(3.8)where the integration is done along the vertical line
The Sumudu integral transform of a function
(3.9)and the inverse of Sumudu integral transform is defined as:
(3.10)where
The Elzaki integral transform of a function
(3.12)
Proposition 3.2
The integral transforms have many important properties. Let
Linearity property: The Laplace transform with respect to
(3.13)where
Convolution property: The convolution property of the Laplace transform is given as follows:
(3.14)where the convolution of two functions
(3.15)
Proof
By setting
Proposition 3.3
[24] The Elzaki transform meets the following properties.
Linearity property. For
(3.16)Convolution property.
(3.17)
Proof
By setting
where
Proposition 3.4
Let
The Shehu transform is a linear operator.
(3.18)
Proof
where
The Shehu transform meets the following convolution equality.
(3.19)
Proof
By setting
where
The Sumudu transform of convolution is given by
(3.20)
Proof
By setting
where
Lemma 3.1
[25,26] For
Laplace transform of
(3.21)
Proof
Sumudu transform of
(3.22)
Proof
Shehu transform of
(3.23)
Proof
Elzaki transform of
(3.24)
Proof
3.5 A criterion for defining FD
How can we say whether a given operator is a FD? Determining whether an operator qualifies to be a FD involves several key criteria and mathematical principles. To establish a clear understanding of FDs, we select the wide sense criterion (WSC) and the strict sense criterion (SSC), which was proposed in [28].
An operator
Linearity: The operator
Identity: The zero-order derivative of a function is the function itself, that is,
(3.25)The index law holds for
(3.26)Backward compatibility: The FD gives the same result as the classical derivative when the order
(3.27)The generalized Leibnitz rule is valid.
(3.28)when
Remark 3.1
If the index law (3.26) modified to include the positive order, it will lead to the SSC. Therefore, the modified index law is
for any
Therefore, based on these properties, the operators Grünwald-Letnikov derivative
Remark 3.2
For problems involving time, the causal derivative definitions such as the Grünwald-Letnikov derivative and the Liouville derivative are commonly employed.
The Liouville derivative is defined by
(3.30)Similarly, the Liouville-Caputo derivative can be used and is defined by
(3.31)
For space problems, we can use these formulas again, but with the meaning of the two-sided left and right derivatives. These derivative definitions are equivalent to a function with a Laplace transform or Fourier transform. Consequently, we can consider both singular and nonsingular kernel functions of FD definitions as tools that can be useful and effective in tackling problems and applications across different scientific fields. Their applications range from physics to signal processing, machine learning, and engineering.
Ortigueira and Machado [29] suggested that Grünwald-Letnikov and regularized Liouville derivatives are the suitable derivatives for a signal-processing approach. Fractional differential equations are essential for modeling various physical phenomena, including anomalous diffusion and viscoelasticity.
4 Classical numerical approaches and their applications
4.1 Variational iteration method
The VIM [30,31] is a numerical technique that can be used to solve ordinary and partial differential equations. It has been an efficient scheme for handling analytically fractional differential equations. The main idea behind VIM is to find an appropriate correction functional, which is added to the given initial condition of the solution to generate a better approximation. The correction functional is determined by solving a linear variational equation, which is obtained by taking the variational derivative of a suitable functional with respect to the unknown solution. This method requires determining a general Lagrange multiplier
To illustrate the basic idea of the VIM, consider the following differential equation:
where
where
Consequently, the solution is given by
So, the corrections functional equation (4.2) will generate a sequence of approximations. The exact solution is obtained by taking the limit of these successive approximations. In [32], Prakash et al. employed the VIM to solve a fractional model of the Newell-Whitehead-Segel equation, which has the form
where
where
Three-dimensional surface for the behavior of the approximate solution
4.2 Adomian decomposition method
The ADM is a very effective approach for solving broad classes of nonlinear ordinary and partial differential equations, with essential applications in different fields of applied mathematics, engineering, physics, and biology. Adomian first introduced it to find an approximate solution to a differential equation. An advantage of the decomposition method is that it can provide an analytical approximation to a relatively wide class of nonlinear and stochastic equations without linearization, perturbation, closure approximations, or discretization methods, resulting in massive numerical computation [34,35]. With the presented decomposition method, the structure of the problem will not be changed, and no linearization will be made.
To illustrate the basic idea of the ADM, consider the differential equation:
where
Applying the inverse operator
4.3 Homotopy analysis method
In 1992, Liao introduced a novel and powerful analytical method for solving nonlinear problems, known as the HAM. This technique has proven to be a highly efficient and adaptive tool for finding analytical solutions to a wide range of nonlinear problems. One of the key advantages of the HAM is that it provides a straightforward way to adjust and control the convergence of the series solution. It offers several notable benefits. First, it is applicable even when the given nonlinear problem does not involve any small or large parameters. Second, it allows convenient adjustment and control of the convergence region and approximation rate of the series solution when necessary. Third, by selecting different sets of base functions, the HAM can be employed to efficiently approximate nonlinear problems [36,37].
To illustrate the basic idea of the HAM, consider the differential equation:
where
where
where
Therefore, after simplification, we obtain the solution of
where
Recently, in the work by Qu et al. [39], the HAM was employed to solve three types of fractional-order partial differential equations: fractional Cauchy-Riemann equations, the fractional acoustic wave equation, and a two-dimensional space partial differential equation with time-fractional order. The derived analytical solution
![Figure 2
Three-dimensional surface for the behavior of the approximate solution when
μ
=
α
=
0.9
\mu =\alpha =0.9
, and
μ
=
α
=
1
\mu =\alpha =1
and the exact solution
ψ
(
ν
,
ς
)
\psi \left(\nu ,\varsigma )
with respect to
ν
\nu
and
ς
\varsigma
, respectively, for the model 1 given in [39]. (a)
ψ
Approx
{\psi }_{{\rm{Approx}}}
for
μ
=
α
=
0.9
\mu =\alpha =0.9
, (b)
ψ
Approx
{\psi }_{{\rm{Approx}}}
for
μ
=
α
=
1
\mu =\alpha =1
, and (c) Exact.](/document/doi/10.1515/math-2024-0036/asset/graphic/j_math-2024-0036_fig_002.jpg)
Three-dimensional surface for the behavior of the approximate solution when
4.4 Hermite wavelet method
The Hermite polynomials
The Hermite polynomials
The Hermite wavelets are defined on interval
where
To illustrate the basic idea of the HWM of FPDEs, consider the following differential equation
where
The two-dimensional Hermite wavelets are defined in the following formula:
where
and the nonlinear term
Thus, applying the inverse operator
The nonlinear time-fractional modified Fornberg-Whitham equation
where
5 Recent numerical approaches and their applications
5.1 Direct power series method
A novel approach called the direct power series method was proposed by Salah et al. [43] to solve fractional initial value problems. This method offers a straightforward and efficient way to obtain better approximate solutions for linear and nonlinear fractional differential equations, including both ordinary and partial differential equations. The core principle behind the power series approach is to express the solution as an infinite series and subsequently determine the coefficients of the series.
To illustrate the basic idea of the direct power series method, consider the following differential equation:
where
The
Therefore, we can obtain the following:
Replace each term of the target equation by its suitable coefficient
Simplify the obtained series representations to get a general term of the coefficients.
Substitute the values of
The temporal-fractional partial Burger equation:
(5.4)where
The time-fractional partial Phi-4 equation
(5.5)where
5.2 Galerkin method
In [46], Bin Jebreen and Cattani proposed a numerical scheme based on the Galerkin method for solving the time-FPDEs and using the operational matrix for the Caputo FD. The obtained results were compared with the radial basis functions approximation method by [47].
The nonhomogeneous time-fractional partial differential Burger’s equation:
(5.6)where
The nonhomogeneous time-fractional partial differential wave equation:
(5.7)where
5.3 Variational iteration transform method (VITM)
This modified method combines the VIM with a convenient transformation operator, including the Laplace transform, Shehu transform, Sumudu transform, or Elzaki transform, integrating suitable FDs. The goal of this integration is to achieve analytical or numerical solutions for time-FPDEs. As an illustration, Shah et al. [49], have successfully combined the Shehu transform and the VIM to obtain an approximate analytical solution for the time-fractional Fornberg-Whitham equation.
The nonlinear time-fractional Fornberg-Whitham equation:
(5.8)where
The nonlinear time-fractional Fornberg-Whitham equation:
(5.9)where
The linear homogeneous time-fractional Emden-Fowler heat equation:
(5.10)where
The linear nonhomogeneous time-fractional Emden-Fowler heat equation:
(5.11)where
![Figure 3
Three-dimensional surface for the behavior of the approximate solution
ψ
(
ν
,
ς
)
\psi \left(\nu ,\varsigma )
when
μ
=
0.4
,
0.6
\mu =0.4,0.6
and 0.8 with respect to
ν
\nu
and
ς
\varsigma
, respectively, for equation (5.11) given in [51]. (a)
ψ
Approx
{\psi }_{{\rm{Approx}}}
for
μ
=
0.4
\mu =0.4
, (b)
ψ
Approx
{\psi }_{{\rm{Approx}}}
for
μ
=
0.6
\mu =0.6
, and (c)
ψ
Approx
{\psi }_{{\rm{Approx}}}
for
μ
=
0.8
\mu =0.8
.](/document/doi/10.1515/math-2024-0036/asset/graphic/j_math-2024-0036_fig_003.jpg)
In this method, the Lagrange multiplier
Therefore, the iteration formula for
In [52], Iqbal et al. have successfully applied the modified VIM involving Atangana-Baleanu derivative of fractional-order to investigate the approximate solutions of fractional Fornberg-Whitham equations (5.8) and (5.9). This method is a mix of variational iteration technique and the Laplace transform approach. The LVIM approach has been successfully applied to a wide range of fractional differential equations, including those arising in physics, engineering, and other fields. In this method, the Lagrange multiplier
Therefore, the iteration formula for
In [24], Haroon et al. implemented the variational iteration transform involving fractional-order derivatives with the Atangana-Baleanu derivative. The Elzaki transformation is used in the Atangana-Baleanu-Caputo derivative to find the solution to the Fornberg-Whitham Equations (5.8) and (5.9). In this method, the Lagrange multiplier
where
Therefore, the iteration formula for
5.4 Adomian decomposition transform method
This modified method is based on a combination of the ADM with an appropriate transform operator such as Laplace transform, Shehu transform, Sumudu transform, or Elzaki transform with suitable FDs to achieve an analytical or numerical solutions of the time-FPDEs. In [24], Haroon et al. implemented the Adomian decomposition transform involving fractional-order derivatives with the Atangana-Baleanu-Caputo derivative. The Elzaki transformation is used in the Atangana-Baleanu-Caputo derivative to find the solution to the FW equations (5.8) and (5.9).
By using ADM procedure and Elzaki transform on the Caputo FD, we obtain the following recursive relations for
In [49], Shah et al. implemented modified techniques, namely, the SDM to achieve an approximate analytical solution for the fractional Fornberg-Whitham equations (5.8) and (5.9). The FD is considered in the Caputo sense. Figure 4 and Table 3, illustrate the graphical representation and numerical values of the derived analytical solution
![Figure 4
Three-dimensional surface for the behavior of the approximate solution
ψ
(
ν
,
ς
)
\psi \left(\nu ,\varsigma )
when
μ
=
0.4
,
0.6
\mu =0.4,0.6
and 0.8 with respect to
ν
\nu
and
ς
\varsigma
, respectively, for equation (5.9) given in [49]. (a)
ψ
Approx
{\psi }_{{\rm{Approx}}}
for
μ
=
0.4
\mu =0.4
, (b)
ψ
Approx
{\psi }_{{\rm{Approx}}}
for
μ
=
0.6
\mu =0.6
, and (c)
ψ
Approx
{\psi }_{{\rm{Approx}}}
for
μ
=
0.8
\mu =0.8
.](/document/doi/10.1515/math-2024-0036/asset/graphic/j_math-2024-0036_fig_004.jpg)
This table exhibits the absolute errors
|
|
|
|
|
|
|---|---|---|---|---|
| 0 | 0.028059 | 0.008048 | 0.001012 | 0.001216 |
| 0.3 | 0.009379 | 0.001055 | 0.001779 | 0.000018 |
| 0.6 | 0.009167 | 0.010082 | 0.004500 | 0.001261 |
| 0.9 | 0.027565 | 0.018968 | 0.007046 | 0.002746 |
| 1.2 | 0.045797 | 0.027637 | 0.009310 | 0.004568 |
| 1.5 | 0.063834 | 0.036011 | 0.011175 | 0.006869 |
| 1.8 | 0.081638 | 0.043994 | 0.012509 | 0.009800 |
| 2.1 | 0.099156 | 0.051480 | 0.013169 | 0.013533 |
| 2.4 | 0.116315 | 0.058338 | 0.012986 | 0.018258 |
| 2.7 | 0.133021 | 0.064418 | 0.011768 | 0.024193 |
| 3 | 0.149152 | 0.069536 | 0.009294 | 0.031588 |
By using ADM procedure and Shehu transform on the Caputo FD, we obtain the following recursive relations for
In [52], Iqbal et al. have successfully applied the modified decomposition method involving Atangana-Baleanu derivative of fractional-order to investigate the approximate solutions of fractional Fornberg-Whitham equations (5.8) and (5.9). This method is a mix of Adomian decomposition technique and the Laplace transform approach. In addition, in [53], Kumar et al. solved equation (5.8), involving the Atangana-Baleanu FD by using the Adomian decomposition technique and the Laplace transform approach.
By using ADM procedure and Laplace transform on the Atangana-Baleanu derivative, we obtain the following recursive relations for
Recently in [54], Ganie et al. implemented an efficient analytical technique EADM within the Caputo operator to investigate the solutions of some FPDEs. The method is based on a combination of ADM and Elzaki transformation. The transformation of Elzaki is a very useful and powerful method for solving the integral equation that cannot be solved by the Sumudu transformation method.
Consider the following equations:
The nonlinear fractional differential equation:
(5.21)where
The diffusion fractional differential equation:
(5.22)where
By using ADM procedure and Elzaki transform on the Caputo FD, we obtain the following recursive relations for
Furthermore, in [55], Shah et al. developed a novel technique known as LADM, to derive a series type approximate semi-analytical solution for the Benney equation in the context of Caputo-Febrizio FDs.
Consider the following nonlinear equations.
The second-order fractional Benney equation with the CF derivative
(5.24)The fractional third-order Benney equation with the CF derivative
(5.25)where
By using ADM procedure and Laplace transform on the Caputo-Febrizio FDs, we obtain the following recursive relations for
5.5 Homotopy analysis transform method
This modified method is based on a combination of the HAM with an appropriate transform operator in order to achieve an analytical or numerical solutions of the time-FPDEs. In [56], Sunitha et al. used the q-homotopy analysis method combined with the Elzaki transform to investigate the two-dimensional advection-dispersion problem. These equations are mainly used to describe the fate of pollutants in aquifers. Recently, in [57], Shah et al. have obtained approximate solutions for the nonlinear partial differential Benney equation by applying a combination of the HAM and Laplace transformation, with the consideration given to the fractional-order Atangana-Baleanu derivative in the Riemann-Liouville sense. The Benney equation applications arise in different areas of physics and engineering, including dynamics and fluid mechanics.
Consider the following nonlinear equations:
The fractional order Benney equation with the ABR derivative
(5.27)The fractional order Benney equation with the ABR derivative
(5.28)where
where
6 Computational challenges
The review provides analytical methods for solving FPDEs, including the ability to obtain exact solutions, gain an understanding of the underlying physical phenomena, and reduce computational costs compared to numerical methods. Solving FPDEs poses several challenges that are not present in classical PDEs. These challenges include nonlocality, singularities, memory effects, and long-range interactions. However, in the following, we address some of the challenges and limitations of numerical or analytical methods when applied to FPDEs.
Dealing with fractional nonlinear partial differential equations presents a significant limitation due to their complex nature and the absence of readily available solutions in closed forms. Analytical techniques are confined to specific systems, frequently necessitating the use of approximation methods or numerical simulations. In contrast, numerical methods offer broader applicability but demand substantial computational resources and strict validation to ensure accuracy and stability.
FPDEs often have nonlocal properties, where the FD at a point depends on the function values at all other points in the domain. Classical methods, which mostly depend on local information, struggle to capture this nonlocal behavior efficiently.
Due to the nonlocality of FDs, numerical methods for FPDEs may require keeping a large amount of historical data, making them memory-intensive and unusable for problems with long-time histories.
Many standard numerical software libraries and packages are designed for integer-order PDEs. Adapting them for FPDEs can be challenging, and dedicated software tools and libraries for FPDEs are less prevalent.
7 Conclusion
This comprehensive review explored various techniques and approaches revolving around fractional calculus, integral transforms, and fractional differential operators. We highlighted their applications and advantages in deriving solutions for FPDEs. These equations extend conventional partial differential equations by incorporating FDs, enabling the modeling of memory and nonlocal effects in diverse fields such as physics, biology, and engineering. Solving FPDEs analytically poses significant challenges due to the nonlocal nature of FDs. However, recent years have witnessed remarkable progress in analytical methods for tackling these equations, providing valuable tools for understanding complex phenomena involving memory and nonlocal effects. By combining semi-analytical techniques with appropriate transformations, researchers can expedite the process of obtaining solutions for nonlinear partial differential equations that model real-world phenomena. As FPDEs continue to gain importance in modeling complex phenomena, developing efficient numerical approaches remains crucial. With the ongoing research and exploration, the future of solving FPDEs analytically looks promising, paving the way for potential applications across diverse scientific and engineering disciplines.
Acknowledgement
The researchers would like to thank the Deanship of Scientific Research, Qassim University, for funding the publication of this project.
-
Author contributions: All authors have read and agreed to the published version of the manuscript. Methodology: FA; investigation: FA; validation: AK; writing-original draft: FA; writing–review and editing: FA and AK; supervision: AK and NS.
-
Conflict of interest: The authors declare no conflict of interest.
References
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- The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
- Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces
- Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method
- Finite groups with gcd(χ(1), χc (1)) a prime
- Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
- The hull-kernel topology on prime ideals in ordered semigroups
- ℐ-sn-metrizable spaces and the images of semi-metric spaces
- Strong laws for weighted sums of widely orthant dependent random variables and applications
- An extension of Schweitzer's inequality to Riemann-Liouville fractional integral
- Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
- Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
- Note on stability estimation of stochastic difference equations
- Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
- Purity and hybridness of two tensors on a real hypersurface in complex projective space
- Classification of positive solutions for a weighted integral system on the half-space
- A quasi-reversibility method for solving nonhomogeneous sideways heat equation
- Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
- Noetherian rings of composite generalized power series
- On generalized shifts of the Mellin transform of the Riemann zeta-function
- Further results on enumeration of perfect matchings of Cartesian product graphs
- A new extended Mulholland's inequality involving one partial sum
- Power vector inequalities for operator pairs in Hilbert spaces and their applications
- On the common zeros of quasi-modular forms for Γ+0(N) of level N = 1, 2, 3
- One special kind of Kloosterman sum and its fourth-power mean
- The stability of high ring homomorphisms and derivations on fuzzy Banach algebras
- Integral mean estimates of Turán-type inequalities for the polar derivative of a polynomial with restricted zeros
- Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
- Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
- The curvature entropy inequalities of convex bodies
- Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
- High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
-
Riemannian invariants for warped product submanifolds in
- Generalized quadratic Gauss sums and their 2mth power mean
- Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
- Enochs conjecture for cotorsion pairs over recollements of exact categories
- Zeros distribution and interlacing property for certain polynomial sequences
- Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
- Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
- Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
- A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
- On the δ-chromatic numbers of the Cartesian products of graphs
- Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
- Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
- System of degenerate parabolic p-Laplacian
- Stochastic stability and instability of rumor model
- Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
- Stability of an additive-quadratic functional equation in modular spaces
- Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
- On k-prime graphs
- On the existence of tripartite graphs and n-partite graphs
- Classifying pentavalent symmetric graphs of order 12pq
- Almost periodic functions on time scales and their properties
- Some results on uniqueness and higher order difference equations
- Coding of hypersurfaces in Euclidean spaces by a constant vector
- Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
- Degrees of (L, M)-fuzzy bornologies
- A matrix approach to determine optimal predictors in a constrained linear mixed model
- On ideals of affine semigroups and affine semigroups with maximal embedding dimension
- Solutions of linear control systems on Lie groups
- A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
- On prime spaces of neutrosophic extended triplet groups
- On a generalized Krasnoselskii fixed point theorem
- On the relation between one-sided duoness and commutators
- Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
- Erratum
- Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
- Corrigendum
- Corrigendum to “Matrix stretching”
- Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
