Fractional Order Diffusion Equation Research Articles

Numerical solution of two-dimensional fractional differential equations using Laplace transform with residual power series method

Abstract One of the efficient and reliable methods for resolving fractional order linear as well as non-linear differential equations is the Laplace transform with residual power series method. This approach is used in the current research to obtain the numerical solutions of the two-dimensional fractional differential equations, namely, the temporal fractional order diffusion equation and the fractional biological population equation. The unknown coefficients of the series solutions to these equations are determined using the proposed approach. The difference between exact and analytical-numerical solutions is presented for these equations in the form of errors. The advantage of the suggested method over alternative approaches is that it requires less computation to solve these two-dimensional differential equations of time-fractional order.

A meshless method based on the Laplace transform for multi-term time-space fractional diffusion equation

<abstract><p>Multi-term fractional diffusion equations can be regarded as a generalisation of fractional diffusion equations. In this paper, we develop an efficient meshless method for solving the multi-term time-space fractional diffusion equation. First, we use the Laplace transform method to deal with the multi-term time fractional operator, we transform the time into complex frequency domain by Laplace transform. The properties of the Laplace transform with respect to fractional-order operators are exploited to deal with multi-term time fractional-order operators, overcoming the dependence of fractional-order operators with respect to time and giving better results. Second, we proposed a meshless method to deal with space fractional operators on convex region based on quintic Hermite spline functions based on the theory of polynomial functions dense theorem. Meanwhile, the approximate solution of the equation is obtained through theory of the minimum residual approximate solution, and the error analysis are provided. Third, we obtain the numerical solution of the diffusion equation by inverse Laplace transform. Finally, we first experimented with a single space-time fractional-order diffusion equation to verify the validity of our method, and then experimented with a multi-term time equation with different parameters and regions and compared it with the previous method to illustrate the accuracy of our method.</p></abstract>

Based on Gradient Algorithm for the Inverse Source Problem of a Class of Time-space Fractional Diffusion Equations

Since entering the 21st century, the establishment of fractional-order diffusion equations in various fields has been of great value and has garnered widespread attention. This study focuses on inverse source term problem for time-space fractional diffusion equation (TSFDE) using given boundary data. First, the identification source problem is transformed into a functional minimization problem utilize the Tikhonov-type regularization method. Then, the sensitivity and the adjoint problem are derived, and the gradient of functional is obtained. The conjugate gradient algorithm is used to solve the minimization problem. Finally, three xamplel with different types of source terms are used to stated the effectiveness and stability, the impact of various parameters on the numerical results is analyzed.

Optimizing signal smoothing using HERS algorithm and time fractional diffusion equation

Signal processing is often affected by various sources of noise that can distort or modify the signals. Removing these noises from the original signal is a crucial step in signal processing, and researchers have proposed several approaches to address this issue. However, achieving an optimized solution remains a challenge. In this study, we introduce a novel approach called the Hybrid Ebola-based Reptile Search (HERS) model based on Time Fractional Diffusion Equation (TFDE). The TFDE is a conventional diffusion equation used for preserving the peak smoothness of spectra signals. In our proposed technique, we consider the processing spectrum of the signal as the reference signal, which serves as the design for the diffusion equation. By applying the diffusion function, we achieve signal peak preservation and smoothing, referred to as the filtering of diffusion. One potential challenge with the time fractional order diffusion equation is its susceptibility to variations in the time step size. To address this, we employ the HERS algorithm to select an optimal time step size that enables efficient signal smoothing. To validate the effectiveness of the proposed technique, we conduct simulations and compare the results with conventional techniques such as the wavelet model, Savitzky-Golay, and regularization techniques. The performance evaluation confirms the superiority of our proposed HERS-TFDE approach in noise removal and signal smoothing. This research aims to contribute to the development of an optimized solution for noise removal in signal processing, leveraging the Hybrid Ebola Reptile Search algorithm and TFDE. The findings have the potential to enhance various signal-processing applications where noise reduction is critical.

Unconditional stability analysis of Grünwald Letnikov method for fractional-order delay differential equations

In this paper, we investigate the unconditional stability and the generally unconditional stability of the Grünwald Letnikov method for fractional-order delay differential equations (FDDEs), which is the generalization of P-stability and GP-stability for classical integer-order delay differential equations. Using the Z-transform, an equivalent form of the discrete Laplace transform, we first show the unconditional stability of the Grünwald Letnikov method for any delay and any constraint mesh. Secondly, we also derive the generally unconditional stability of the Grünwald Letnikov method with a linear interpolation for approximating the delay term under a general uniform mesh. It is shown that the Grünwald Letnikov method for FDDEs preserves the stability for the analytical solution and hence naturally inherits the α-dependence. Finally, two numerical examples for FDDEs and time fractional-order diffusion equations with delay are presented to demonstrate the validity and effectiveness of theoretical results.

Fractional view analysis of the diffusion equations via a natural Atangana-Baleanu operator

In this article, the natural transform decomposition method (NTDM) is applied to evaluate the fractional-order diffusion equations utilizing the Atangana-Baleanu operator. The method offers an efficient analytical solution by combining the natural transform methodology with a decomposition strategy. The suggested method incorporates non-local effects and memory qualities introduced by the Atangana-Baleanu operator while demonstrating adaptability and applicability to various fractional-order diffusion equations. In comparison to previous analytical techniques, the NTDM has demonstrated the lowest calculation volume and the highest rate of convergence. The obtained results are presented illustratively by using graphical simulations. The suggested method is therefore easily adaptable to other non-linear issues. The NTDM includes comparative analysis capabilities, practicality, and accurate results for researchers and practitioners. In order to solve fractional-order phenomena, particularly diffusion equations, NTDM is regarded as the best analytical technique.

Numerical Solution of Advection–Diffusion Equation of Fractional Order Using Chebyshev Collocation Method

This work presents a highly accurate method for the numerical solution of the advection–diffusion equation of fractional order. In our proposed method, we apply the Laplace transform to handle the time-fractional derivative and utilize the Chebyshev spectral collocation method for spatial discretization. The primary motivation for using the Laplace transform is its ability to avoid the classical time-stepping scheme and overcome the adverse effects of time steps on numerical accuracy and stability. Our method comprises three primary steps: (i) reducing the time-dependent equation to a time-independent equation via the Laplace transform, (ii) employing the Chebyshev spectral collocation method to approximate the solution of the transformed equation, and (iii) numerically inverting the Laplace transform. We discuss the convergence and stability of the method and assess its accuracy and efficiency by solving various problems in two dimensions.

Numerical Solution of Two-Dimensional Fractional Order Diffusion Equation by using Elzaki Transform with Residual Power Series Method

The Elzaki transform with RPS method is an efficient and reliable approach for solution of non-integer order linear as well as non-linear differential equations. The major endeavourof current work is to find the solution of two dimensional non-integer order diffusion equation by Elzaki transform with RPS method. Firstly, Elzaki is applied on this equation. Secondly, inverse Elzaki is taken on this equation for finding the expressionof solution. Thirdly, assumed approximate solution is substituted on this equation. Then unidentified coefficient functions are obtained by using residual function is equal to zero as well as combining its initial circumstances. At last, coefficient functions are substituted in power series form for finding finite approximate logical solutions. The judgment between exact solution and approximate analytic solutions with different number of terms of this equation are determined and compared for reliability. This method reduces the size of computational works of solution of non-integer order non-linear diffusion equation

Approximate solution of space fractional order diffusion equations by Gegenbauer collocation and compact finite difference scheme

In this paper, approximation of space fractional order diffusion equation are considered using compact finite difference technique to discretize the time derivative, which was then approximated via shifted Gegenbauer polynomials using zeros of (N - 1) degree shifted Gegenbauer polynomial as collocation points. The important feature in this approach is that it reduces the problems to algebraic linear system of equations together with the boundary conditions gives (N + 1) linear equations. Some theorems are given to establish the convergence and the stability of the proposed method. To validate the efficiency and the accuracy of the method, obtained results are compared with the existing results in the literature. The graphical representation are also displayed for various values of \beta Gegenbauer polynomials. It can be observe in the tables of the results and figures that the proposed method performs better than the existing one in the literature.

Computational Analysis of Fractional Diffusion Equations Occurring in Oil Pollution

The fractional model of diffusion equations is very important in the study of oil pollution in the water. The key objective of this article is to analyze a fractional modification of diffusion equations occurring in oil pollution associated with the Katugampola derivative in the Caputo sense. An effective and reliable computational method q-homotopy analysis generalized transform method is suggested to obtain the solutions of fractional order diffusion equations. The results of this research are demonstrated in graphical and tabular descriptions. This study shows that the applied computational technique is very effective, accurate, and beneficial for managing such kind of fractional order nonlinear models occurring in oil pollution.

Novel operational matrix method for the numerical solution of nonlinear reaction–advection–diffusion equation of fractional order

Novel operational matrix method for the numerical solution of nonlinear reaction–advection–diffusion equation of fractional order

Solving Fractional-Order Diffusion Equations in a Plasma and Fluids via a Novel Transform

Motivated by the importance of diffusion equations in many physical situations in general and in plasma physics in particular, therefore, in this study, we try to find some novel solutions to fractional-order diffusion equations to explain many of the ambiguities about the phenomena in plasma physics and many other fields. In this article, we implement two well-known analytical methods for the solution of diffusion equations. We suggest the modified form of homotopy perturbation method and Adomian decomposition methods using Jafari-Yang transform. Furthermore, illustrative examples are introduced to show the accuracy of the proposed methods. It is observed that the proposed method solution has the desire rate of convergence toward the exact solution. The suggested method’s main advantage is less number of calculations. The proposed methods give series form solution which converges quickly towards the exact solution. To show the reliability of the proposed method, we present some graphical representations of the exact and analytical results, which are in strong agreement with each other. The results we showed through graphs and tables for different fractional-order confirm that the results converge towards exact solution as the fractional-order tends towards integer-order. Moreover, it can solve physical problems having fractional order in different areas of applied sciences. Also, the proposed method helps many plasma physicists in modeling several nonlinear structures such as solitons, shocks, and rogue waves in different plasma systems.

A Comparative Study of Fractional-Order Diffusion Model within Atangana-Baleanu-Caputo Operator

We provided two different approaches for solving fractional-order diffusion equations in this article. The fractional Atangana-Baleanu derivative operator in addition to the Laplace transform is used to generate several new approximate-analytical solutions to the time-fractional diffusion equations. The implementation of a sophisticated and straightforward approach to solving diffusion equations having a fractional-order derivative is the motivation and uniqueness behind the current work. The solutions to some illustrative problems are calculated to ensure that the actual and approximate solutions to the targeted problems are in close contact. The results we obtained have a higher rate of convergence and provide a closed-form solution, according to analysis. The proposed method’s key advantage is the small amount of calculations required. The suggested techniques can be applied to nonlinear fractional-order problems in a variety of applied science areas due to their simple and straightforward implementation. It can be used to overcome specific fractional-order physical problems in a variety of fields of applied sciences.

Numerical Method to Modify the Fractional-Order Diffusion Equation

Time or space or time-space fractional-order diffusion equations (FODEs) are widely used to describe anomalous diffusion processes in many physical and biological systems. In recent years, many authors have proposed different numerical methods to solve the modified fractional-order diffusion equations, and some achievements have been obtained. However, to our knowledge of the literature, up to date, all the proposed numerical methods to modify FODE have achieved at most a second-order time accuracy. In this study, we focus mainly on the numerical methods based on numerical integration in order to modify the fractional-order diffusion equation: 1 + 1 / 12 δ x 2 p j k − 1 + 1 / 12 δ x 2 p j k − 1 = μ α ∑ l = 0 k λ α l δ x 2 p j k − l + μ β ∑ l = 0 k λ β l δ x 2 p j k − l + τ / 2 1 + 1 / 12 δ x 2 f j k − 1 + f j k , k = 1,2 , … , K ; j = 1,2 , … , j − 1 , p j 0 = ω j , j = 0,1 , … , J , p 0 k = φ t k , p j k = ψ t k , k = 0,1 , … , K , f j l = f x j , t l , ω j = ω x j . Accordingly, numerical methods can be built to modify FODE with second-order time accuracy and fourth-order spatial accuracy in ∂ p x , t / ∂ t = ∂ 1 − α / ∂ t 1 − α + B ∂ 1 − β / ∂ t 1 − β ∂ 2 p x , t / ∂ x 2 + f x , t , 0 < t ≤ 1,0 < x < 1 , p x , 0 = 0,0 ≤ x ≤ 1 . p 0 , t = t 2 , p 1 , t = e t 2 , 0 ≤ t ≤ 1 . Our suggested method can improve the time precision with a certain value.

An efficient technique for solving fractional-order diffusion equations arising in oil pollution

In this article, non-linear time-fractional diffusion equations are considered to describe oil pollution in the water. The latest technique, fractional reduced differential transform method (FRDTM), is used to acquire approximate solutions of the time fractional-order diffusion equation and two cases of Allen–Cahn equations. The acquired results are collated with the exact solutions and other results from literature for integer-order α, which reveal that the proposed method is effective. Hence, FRDTM can be employed to obtain solutions for different types of nonlinear fractional-order IVPs arising in engineering and science.

Front Propagation of Exponentially Truncated Fractional-Order Epidemics

The existence of landscape constraints in the home range of living organisms that adopt Lévy-flight movement patterns, prevents them from making arbitrarily large displacements. Their random movements indeed occur in a finite space with an upper bound. In order to make realistic models, by introducing exponentially truncated Lévy flights, such an upper bound can thus be taken into account in the reaction-diffusion models. In this work, we have investigated the influence of the λ-truncated fractional-order diffusion operator on the spatial propagation of the epidemics caused by infectious diseases, where λ is the truncation parameter. Analytical and numerical simulations show that depending on the value of λ, different asymptotic behaviours of the travelling-wave solutions can be identified. For small values of λ (λ≳0), the tails of the infective waves can decay algebraically leading to an exponential growth of the epidemic speed. In that case, the truncation has no impact on the superdiffusive epidemics. By increasing the value of λ, the algebraic decaying tails can be tamed leading to either an upper bound on the epidemic speed representing the maximum speed value or the generation of the infective waves of a constant shape propagating at a minimum constant speed as observed in the classical models (second-order diffusion epidemic models). Our findings suggest that the truncated fractional-order diffusion equations have the potential to model the epidemics of animals performing Lévy flights, as the animal diseases can spread more smoothly than the exponential acceleration of the human disease epidemics.

A fractional diffusion model of CD8+T cells response to parasitic infection in the brain

Toxoplasma gondii(T. gondii) is a parasitic pathogen that causes serious brain diseases in fetuses and patients with immunodeficiency, particularly AIDS patients. In the field of immunology, a large number of studies have shown that effector CD8+T cells can respond toT. gondiiinfection in the brain tissue through controlling the proliferation of intracellular parasites and killing infected brain cells. These protective mechanisms do not occur without T cell movement and searching for infected cells, as a fundamental feature of the immune system. Following infection with a pathogen in a tissue, in their search for infected cells, CD8+T cells can perform different stochastic searches, including Lévy and Brownian random walks. Statistical analysis of CD8+T cell movement in the brain ofT. gondii-infected mouse has determined that the search strategy of CD8+T cells in response to infected brain cells could be described by a Lévy random walk. In this work, by considering a Lévy distribution for the displacements, we propose a space fractional-order diffusion equation for the T cell density in the infected brain tissue. Furthermore, we derive a mathematical model representing CD8+T cell response to infected brain cells. By solving the model equations numerically, we perform a comparison between Lévy and Brownian search strategies. we demonstrate that the Lévy search pattern enables CD8+T cells to spread over the whole brain tissue and hence they can rapidly destroy infected cells distributed throughout the brain tissue. However, with the Brownian motion assumption, CD8+T cells travel through the brain tissue more slowly, leading to a slower decline of the infected cells faraway from the source of T cells. Our results show that a Lévy search pattern aids CD8+T cells in accelerating the elimination of infected cells distributed broadly within the brain tissue. We suggest that a Lévy search strategy could be the result of natural evolution, as CD8+T cells learn to enhance the immune system efficiency against pathogens.

On a Reconstruction of a Solely Time-Dependent Source in a Time-Fractional Diffusion Equation with Non-smooth Solutions

An inverse source problem for a non-automonous time fractional diffusion equation of order \((0<\beta <1)\) is considered in a bounded Lipschitz domain in \(\mathbb {R}^d\). The missing solely time-dependent source is recovered from an additional integral measurement. The existence, uniqueness and regularity of a weak solution is studied. We design two numerical algorithms based on Rothe’s method over uniform and graded grids, derive a priori estimates and prove convergence of iterates towards the exact solution. An essential feature of the fractional subdiffusion problem is that the solution lacks the smoothness near the initial time, although it would be smooth away from \(t = 0\). Rothe’s method on a uniform grid addresses the existence of a such a solution (non-smooth with \(t^\gamma \) term where \(1>\gamma > \beta \)) under low regularity assumptions, whilst Rothe’s method over graded grids has the advantage to cope better with the behaviour at \(t=0\) (also here \(t^\beta \) is included in the class of admissible solutions) for the considered problems. The theoretical obtained results are supported by numerical experiments and stay valid in case of smooth solutions to the problem.

NUMERICAL SIMULATIONS FOR THE VARIABLE ORDER TWO-DIMENSIONAL REACTION SUB-DIFFUSION EQUATION: LINEAR AND NONLINEAR

The applications and the fields that use the anomalous sub-diffusion equations cannot be easily listed due to their wide area. Sure, one of the main physical reasons for using and researching fractional order diffusion equations is to explain anomalous diffusion that occurs in transport processes through complex and/or disordered structures, such as fractal media. One of the important applications is their use in chemical reactions, where a single material continues to shift from a high concentration area to a low concentration area until the concentration across the space is equal. The mathematical model that describes these physical-chemical phenomena is called the reaction sub-diffusion equation. In our study, we try to solve the 2D variable order version of these equations (2DVORSE) (linear and nonlinear) by using an accurate numerical technique which is the variable weighted average finite difference method (WAFDM). We will analyze the stability of the resulting scheme by using a modified suitable version of the John Von Neumann procedure. Specific stability conditions that occur or some parameters in the resulting schemes are derived and checked. At the end of the study, numerical examples are simulated to check the stability and the accuracy of the proposed technique.

A high-order L2 type difference scheme for the time-fractional diffusion equation

The present paper is devoted to constructing L2 type difference analog of the Caputo fractional derivative. The fundamental features of this difference operator are studied and it is used to construct difference schemes generating approximations of the second and fourth order in space and the (3−α)th-order in time for the time fractional diffusion equation with variable coefficients. Difference schemes were also constructed for the variable-order diffusion equation and the generalized fractional-order diffusion equation of the Sobolev type. Stability of the schemes under consideration as well as their convergence with the rate equal to the order of the approximation error are proven. The received results are supported by the numerical computations performed for some test problems.