Nonlinear Fractional Partial Differential Equations Research Articles

A new identification of Lagrange multipliers to study solutions of nonlinear Caputo–Fabrizio fractional problems

This research aims to provide a new identification of Lagrange multipliers using a novel hybrid method based on the Khalouta transform method and the variational iteration method and to study the solutions of nonlinear fractional problems. This method is called Khalouta variational iteration method (KHVIM). The fractional derivative is considered in Caputo–Fabrizio sense. The uniqueness and convergence results are investigated using Banach fixed point theorem. The results are obtained in the form of successive approximations corresponding to the proposed problem. To demonstrate the accuracy and effectiveness of the proposed method, three different nonlinear fractional partial differential equations are provided. Approximate solutions of the fractional equations were obtained. These solutions quickly converged to exact solutions with lower computational cost. Furthermore, the method used in this study is more generalized and allows our results to be more extensive and cover several new and existing fractional problems in the literature.

Optical solutions for perturbed conformable Fokas–Lenells equation via Kudryashov auxiliary equation method

This paper is dedicated to the study of optical soliton solutions for the perturbed Fokas–Lenells equation with conformable derivative using the Kudryashov auxiliary equation method. The studied optical solutions include a class of categories, comprising dark, mixed dark-bright, multi bell-shaped, bell-shaped, and wave optical solutions. Furthermore, we analyzed the magnitude of the perturbed conformable Fokas–Lenells equation by investigating the impact of the conformable parameter and the effect of the time parameter on the novel optical solutions. It can be claimed that the current optical soliton solutions are novel and have not existed in the literature. The results obtained illustrate that the proposed method is robust, efficient, and readily applicable for constructing new solutions to a wide range of nonlinear fractional partial differential equations. The results of this study are expected to shed light on the field of soliton theory in nonlinear optics and mathematical physics.

A novel analytical technique for analyzing the (3+1)-dimensional fractional calogero- bogoyavlenskii-schiff equation: investigating solitary/shock waves and many others physical phenomena

This work presents a thorough analysis of soliton wave phenomena in the (3+1)-dimensional Fractional Calogero-Bogoyavlenskii-Schiff equation (FCBSE) with Caputo’s derivatives through the use of a novel analytical technique known as the modified Extended Direct Algebraic Method (mEDAM). By converting nonlinear Fractional Partial Differential equations (FPDE) into integer-order Nonlinear Ordinary Differential equations (NODE), and then using closed-form series solutions to translate the NODE into an algebraic system of equations, this method allows us to derive families of soliton solutions, which include kink waves, lump waves, breather waves, and periodic waves, exposing new insights into the behavior and distinctive features of soliton waves in the FCBSE. By including contour and 3D graphics, the behaviors of a few selected soliton solutions are well depicted, showcasing their amplitude, shape, and propagation characteristics. The results enhance our understanding of the FCBSE and show that the mEDAM is a valuable tool for studying soliton wave phenomena. This work creates new opportunities for studying wave phenomena in more intricately constructed nonlinear FPDEs (NFPDEs).

ON THE PERIODIC SOLITON SOLUTIONS FOR FRACTIONAL SCHRÖDINGER EQUATIONS

In this research, we use a novel version of the Extended Direct Algebraic Method (EDAM) namely generalized EDAM (gEDAM) to investigate periodic soliton solutions for nonlinear systems of fractional Schrödinger equations (FSEs) with conformable fractional derivatives. The FSEs, which is the fractional abstraction of the Schrödinger equation, grasp notable relevance in quantum mechanics. The proposed gEDAM technique entails creating nonlinear ordinary differential equations via a fractional complex transformation, which are then solved to acquire soliton solutions. Several 3D and contour graphs of the soliton solutions reveal periodicity in the wave profiles that offer crucial perspectives into the behavior of the system. The work sheds light on the dynamics of FSEs by displaying numerous families of periodic soliton solutions and their intricate relationships. These results hold significance not only for comprehending the dynamics of FSEs but also for nonlinear fractional partial differential equation applications.

Recent advances in employing the Laplace homotopy analysis method to nonlinear fractional models for evolution equations and heat-typed problems

This study stems from the growing need for effective mathematical tools to tackle nonlinear fractional evolution equations, which find wide applications in physics and engineering. Traditional methods often struggle to provide exact closed-form solutions for such equations, leading to the development of novel approaches. In this context, the combination of the homotopy analysis method with the Laplace transform presents a promising avenue for obtaining exact solutions to these complex equations. By exploring this hybrid approach, the study aims to address the existing challenges in solving nonlinear fractional models and provide a reliable method for obtaining closed-form solutions. In this way, this study focuses on rendering the coupling between the homotopy analysis approach and the Laplace transform in the search for exact closed-form solutions for the class of fractional evolution and heat-typed equations. In fact, this class of equations features some interesting nonlinear Caputo fractional partial differential equations that model dissimilar physical processes in physics and engineering. A generalized recurrent scheme is derived, which is applicable to all the members of the governing class, and further analyzed the application of this very scheme on some test fractional initial-value problems of much concern. Eventually, the method has been found to be advantageous, with various advantages, including convergence to available exact solutions. Thus, of the advantage of the current study is innovative combination of mathematical methods, the applicability of these methods to a special class of fractional equations, the development of a generalized scheme, the successful solution of prototype problems, and the practical advantages of the method.

Impressive Exact Solitons to the Space-Time Fractional Mathematical Physics Model via an Effective Method

A new class of truncated M-fractional exact soliton solutions for a mathematical physics model known as a truncated M-fractional (1+1)-dimensional nonlinear modified mixed-KdV model are achieved. We obtain these solutions by using a modified extended direct algebraic method. The obtained results consist of trigonometric, hyperbolic trigonometric and mixed functions. We also discuss the effect of fractional order derivative. To validate our results, we utilized the Mathematica software. Additionally, we depict some of the obtained kink, periodic, singular, and kink-singular wave solitons, using two and three dimensional graphs. The obtained results are useful in the fields of fluid dynamics, nonlinear optics, ocean engineering and others. Furthermore, these employed techniques are not only straightforward, but also highly effective when used to solve non-linear fractional partial differential equations (FPDEs).

Spectral shifted Chebyshev collocation technique with residual power series algorithm for time fractional problems.

In this paper, two problems involving nonlinear time fractional hyperbolic partial differential equations (PDEs) and time fractional pseudo hyperbolic PDEs with nonlocal conditions are presented. Collocation technique for shifted Chebyshev of the second kind with residual power series algorithm (CTSCSK-RPSA) is the main method for solving these problems. Moreover, error analysis theory is provided in detail. Numerical solutions provided using CTSCSK-RPSA are compared with existing techniques in literature. CTSCSK-RPSA is accurate, simple and convenient method for obtaining solutions of linear and nonlinear physical and engineering problems.

Utmost travelling wave phenomena to the fractional type nonlinear evolution equation in mathematical physics

In applied physics and engineering, non-linear fractional types of partial differential equations become increasingly prominent as an estimation technique to explain a wide variety of non-linear phenomena. Throughout this study, we choose to use an adaptable extended tanh-function scheme over a conformable derivative in order to construct a comprehensive closed-form traveling wave solution of the time fractional Cahn–Hilliard and modified Kawahara equations. The mentioned equations, have been used as frameworks for various phenomena such as quantum theory, geosciences, water wave mechanics, and solitary waves in shallow water in the porous medium. The different values of the fractional order of the model can be employed successfully to organize the wave velocity. We recognized numerous forms of solutions by using the computational software, namely solitons type, bell types, kink types, periodic types, multiple periodic, single soliton, singular kink, and other types of solutions that are illustrated using 3D, contour, and spherical. The attained results have effectively handled the previously stated phenomena in various fields of engineering and mathematical physics. The results originating in this study were confirmed by the computational software, which was used to rewrite them as non-linear fractional partial differential equations. We ensure that the technique is revised to make it more effective, pragmatic, and credible, and we pursue more comprehensive exact results for traveling waves, and then for solitary waves.

On the optical wave structures to the fractional nonlinear integrable coupled Kuralay equation

This paper is mainly concerning the study of truncated M-fractional Kuralay equations that have applications in numerous fields, including nonlinear optics, ferromagnetic materials, signal processing, engineering fields and optical fibers. Due to its ability to clarify a wide range of sophisticated physical phenomena and reveal more dynamic structures of localized wave solutions, the Kuralay equation has captured a lot of attention in the research field. The newly designed integration methods, known as the modified Sardar subequation method and enhanced modified extended tanh expansion method are used as solving tools to validate the solutions. The goal of this study is to extract several kinds of optical solitons, such as mixed, dark, singular, bright-dark, bright, complex and combined solitons. Due to the many potential applications for superfast signal routing techniques and shorter light pulses in communications, the optical propagation of soliton in optical fibers is now a topic of significant interest. In nonlinear dispersive media, optical solitons are stretched electromagnetic waves that maintain their intensity due to a balance between the effects of dispersion and nonlinearity. In addition, exponential, periodic, hyperbolic solutions are generated. The applied approaches are efficient in explaining fractional nonlinear partial differential equations by providing pre-existing solutions and also producing new solutions by combining results from multiple processes. Additionally, we plot the contour, 2D, and 3D graphs with the associated parameter values to visualize the solutions. The results of this study show the effectiveness of the approaches adopted and help enhance comprehension of the nonlinear dynamical behavior of specific systems. We expect that a substantial amount of engineering model specialists will greatly benefit from our work. The findings demonstrate the efficacy, efficiency, and applicability of the computational method employed, particularly in dealing with intricate systems.

Separation method of semifixed variables together with integral bifurcation method for solving generalized time‐fractional thin‐film equations

It is well known that investigation on exact solutions of nonlinear fractional partial differential equations (PDEs) is a very difficult work compared with integer‐order nonlinear PDEs. In this paper, based on the separation method of semifixed variables and integral bifurcation method, a combinational method is proposed. By using this new method, a class of generalized time‐fractional thin‐film equations are studied. Under two kinds of definitions of fractional derivatives, exact solutions of two generalized time‐fractional thin‐film equations are investigated respectively. Different kinds of exact solutions are obtained and their dynamic properties are discussed. Compared to the results in the existing references, the types of solutions obtained in this paper are abundant and very different from those in the existing references. Investigation shows that the solutions of the model defined by Riemann–Liouville differential operator converge faster than those defined by Caputo differential operator. It is also found that the profiles of some solutions are very similar to solitons, but they are not true soliton solutions. In order to visually show the dynamic properties of these solutions, the profiles of some representative exact solutions are illustrated by 3D graphs.

An Efficient Technique to Study Time Fractional Extended Fisher-Kolmogorov Equation

Reaction-diffusion partial differential equations are among the most widely used equations in applied mathematical modelling. In this study, we examine the solutions of one such equation, namely, the time-fractional extended Fisher-Kolmogorov equation. This equation is widely used in the study of population growth and wave propagation dynamics. The technique we consider is a combination of the natural transform and the Adomian decomposition method. Fractional derivatives are considered with singular and non-singular kernels. The existence and uniqueness of the solutions are presented. We analyse two different cases of the proposed problem to determine the validity and efficacy of the proposed scheme. Additionally, numerical simulation is shown, and the nature of the achieved solution is captured in terms of plots for various fractional orders. The outcome demonstrates that the method is straightforward, efficient, and dependable. The proposed method does not require any predetermined assumptions, linearization, perturbation, or discretization, and it prevents rounding errors. Therefore, the technique is ready to be implemented for a variety of nonlinear time fractional partial differential equations.

On the approximations to fractional nonlinear damped Burger’s-type equations that arise in fluids and plasmas using Aboodh residual power series and Aboodh transform iteration methods

Damped Burger’s equation describes the characteristics of one-dimensional nonlinear shock waves in the presence of damping effects and is significant in fluid dynamics, plasma physics, and other fields. Due to the potential applications of this equation, thus the objective of this investigation is to solve and analyze the time fractional form of this equation using methods with precise efficiency, high accuracy, ease of application and calculation, and flexibility in dealing with more complicated equations, which are called the Aboodh residual power series method and the Aboodh transform iteration method (ATIM) within the Caputo operator framework. Also, this study intends to further our understanding of the dynamic characteristics of solutions to the Damped Burger’s equation and to assess the effectiveness of the proposed methods in addressing nonlinear fractional partial differential equations. The two proposed methods are highly effective mathematical techniques for studying more complicated nonlinear differential equations. They can produce precise approximate solutions for intricate evolution equations beyond the specific examined equation. In addition to the proposed methods, the fractional derivatives are processed using the Caputo operator. The Caputo operator enhances the representation of fractional derivatives by providing a more accurate portrayal of the underlying physical processes. Based on the proposed two approaches, a set of approximations to damped Burger’s equation are derived. These approximations are discussed graphically and numerically by presenting a set of two- and three-dimensional graphs. In addition, these approximations are analyzed numerically in several tables, including the absolute error for each approximate solution compared to the exact solution for the integer case. Furthermore, the effect of the fractional parameter on the behavior of the derived approximations is examined and discussed.

Investigating the new perspectives of Caudrey–Dodd–Gibbon equation arising in quantum field theory

The main purpose of the paper is obtaining the analytical results for beta fractional Caudrey–Dodd–Gibbon equation which is used to resolve complex problems in fluid dynamics, chemical kinetics, plasma physics, quantum field theory, crystal dislocations, and nonlinear optics by using auxiliary method. Beta derivative is a useful fractional operator due to satisfying basic properties of integer order derivative and also, allows us using chain rule and wave transform to turn nonlinear fractional partial differential equations into integer order ordinary differential equations. By the way many analytical methods can be applied to these equations. In order to understand the physical features of the solutions, 3D and 2D graphical illustrations are given. Finally, authors expect that the obtained solutions may give a deep insight for the explanation of physical phenomena in the fluid dynamics, chemical kinetics, plasma physics, quantum field theory, crystal dislocations, and nonlinear optics.

Thenovel numerical solutions for conformable fractional Kuramoto-Sivashinsky equations by using Cq-HATM and CHPETM

The present study used the novel conformable fractional methods to examine the nonlinear Kuramoto-Sivashinsky equations of the conformable fractional derivative. The Kuramoto-Sivashinsky equation is a mathematical model employed to elucidate a diverse range of chemical and physical phenomena, encompassing chemical reaction-diffusion processes, plasma instabilities, viscous flow issues, flame front propagation, and magnetized plasmas. The present inquiry provides an examination of convergence and error for the future scheme. The conformable q-homotopy analysis transform method (Cq-HATM) provides h-curves that illustrate the convergence range of the series solution achieved. In order to validate the effectiveness and suitability of the Cq-HATM, we examine separate instances. In this study, we introduce an application that demonstrates the potential benefits and effectiveness of the proposed approach. Furthermore, an error analysis is performed in order to verify the accuracy of the scheme. Numerical simulations are conducted to validate the precision of the forthcoming approach. The findings obtained from the numerical and graphical analyses are presented in this study. The method provided in this study showcases a notable degree of computational precision and ease in examining and resolving intricate phenomena related to conformable fractional nonlinear partial differential equations within the domains of science and technology.

Comparative study of fractional Newell–Whitehead–Segel equation using optimal auxiliary function method and a novel iterative approach

This research explores the solution of the time-fractional Newell–Whitehead–Segel equation using two separate methods: the optimal auxiliary function method and a new iterative method. The Newell–Whitehead–Segel equation holds significance in modeling nonlinear systems, particularly in delineating stripe patterns within two-dimensional systems. Employing the Caputo fractional derivative operator, we address two case study problems pertaining to this equation through our proposed methods. Comparative analysis between the numerical results obtained from our techniques and an exact solution reveals a strong alignment. Graphs and tables illustrate this alignment, showcasing the effectiveness of our methods. Notably, as the fractional orders vary, the results achieved at different fractional orders are compared, highlighting their convergence toward the exact solution as the fractional order approaches an integer. Demonstrating both interest and simplicity, our proposed methods exhibit high accuracy in resolving diverse nonlinear fractional order partial differential equations.

An optimization method for solving a general class of the inverse system of nonlinear fractional order PDEs

In this paper, we introduce a general class of the inverse system of nonlinear fractional order partial differential equations (GCISNF-PDEs) with initial-boundary and two overdetermination conditions. An optimization method is considered based on the generalized shifted Legendre polynomials (GSLPs) for solving GCISNV-FPDEs. The concept of the fractional order derivatives (F-Ds) is utilized in the Caputo type. Operational matrices (OMs) of classical derivatives and F-Ds of GSLPs are extracted. Making use of GSLPs, OMs, and Lagrange multipliers method, we reduce the given GCISNF-PDEs into an algebraic system of equations. The proposed approach achieves satisfactory results simply for a small number of the novel GSLPs. In this work, two mathematical examples are illustrated to analyse the introduced method convergence and test its validity as well as applicability.

On the dynamics of soliton solutions for the nonlinear fractional dynamical system: Application in ultrasound imaging

This article examines the third-order non-linear fractional Westervelt model by using a newly developed integration tool termed the new extended direct algebraic approach. The different wave structures to the considered model are secured in various forms, including bright, dark, and combo solitons. The application of wave structures is advantageous in the examination of sound wave propagation and high amplitude phenomena in the fields of medical imaging and therapy. These solutions are effective in facilitating ultrasound propagation in tissue, underwater acoustics, acoustic cavitation, acoustic levitation, and other related applications. The internal tissue of human beings can be visualized and studied through the use of ultrasound imaging technologies in the field of medicine. This technology possesses numerous uses in both industrial and medicinal sectors. There has been a growing interest in fractional nonlinear partial differential equations due to their ability to describe various complex occurrences and exhibit more dynamic architectures of localized wave solutions. On employing accurate parameter values, multiple graphical representations are generated to provide the visual depiction of the acquired results. The results of this study indicate that the chosen methodology effectively improves nonlinear dynamical processes. The findings indicate that the selected methodology demonstrates efficacy, feasibility, and versatility when applied to intricate systems across several domains, with a special emphasis on ultrasonic imaging. The findings indicate that the system exhibits a potentially significant abundance of soliton structures.

Usage of the Fuzzy Adomian Decomposition Method for Solving Some Fuzzy Fractional Partial Differential Equations

In this study, we examine the numerical solutions of nonlinear fuzzy fractional partial differential equations under the Caputo derivative utilizing the technique of fuzzy Adomian decomposition. This technique is used as an alternative method for obtaining approximate fuzzy solutions to various types of fractional differential equations and also investigated some new existence and uniqueness results of fuzzy solutions. Some examples are given to support the effectiveness of the proposed technique. We present the numerical results in graphical form for different values of fractional order and uncertainty γ∈0,1.

Efficient approximate analytical technique to solve nonlinear coupled Jaulent–Miodek system within a time-fractional order

<abstract><p>In this article, we considered the nonlinear time-fractional Jaulent–Miodek model (FJMM), which is applied to modeling many applications in basic sciences and engineering, especially physical phenomena such as plasma physics, fluid dynamics, electromagnetic waves in nonlinear media, and many other applications. The Caputo fractional derivative (CFD) was applied to express the fractional operator in the mathematical formalism of the FJMM. We implemented the modified generalized Mittag-Leffler method (MGMLFM) to show the analytical approximate solution of FJMM, which is represented by a set of coupled nonlinear fractional partial differential equations (FPDEs) with suitable initial conditions. The suggested method produced convergent series solutions with easily computable components. To demonstrate the accuracy and efficiency of the MGMLFM, a comparison was made between the solutions obtained by MGMLFM and the known exact solutions in some tables. Also, the absolute error was compared with the absolute error provided by some of the other famous methods found in the literature. Our findings confirmed that the presented method is easy, simple, reliable, competitive, and did not require complex calculations. Thus, it can be extensively applied to solve more linear and nonlinear FPDEs that have applications in various areas such as mathematics, engineering, and physics.</p></abstract>

Breather patterns and other soliton dynamics in (2+1)-dimensional conformable Broer-Kaup-Kupershmit system

<abstract><p>In this work, the Extended Direct Algebraic Method (EDAM) is utilized to analyze and solve the fractional (2+1)-dimensional Conformable Broer-Kaup-Kupershmit System (CBKKS) and investigate different types of traveling wave solutions and study the soliton like-solutions. Using the suggested method, the fractional nonlinear partial differential equation (FNPDE) is primarily reduced to an integer-order nonlinear ordinary differential equation (NODE) under the traveling wave transformation, yielding an algebraic system of nonlinear equations. The ensuing algebraic systems are then solved to construct some families of soliton-like solutions and many other physical solutions. Some derived solutions are numerically analyzed using suitable values for the related parameters. The discovered soliton solutions grasp vital importance in fluid mechanics as they offer significant insight into the nonlinear behavior of the targeted model, opening the way for a deeper comprehension of complex physical phenomena and offering valuable applications in the associated areas.</p></abstract>