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The truncated M-fractional Kuralay (TMFK)-II equation is prevalent in the exploration of specific complex nonlinear wave phenomena. Such types of wave phenomena are more applicable in science and engineering. These equations could potentially provide insights into understanding the intricate dynamics of optical phenomena, encompassing solitons, nonlinear effects, and wave interactions. This study aims to uncover a diverse range of soliton solutions to the model, spanning trigonometric, hyperbolic, exponential, and rational expressions. These solutions are unveiled through the application of extended hyperbolic functions and improved F-expansion techniques, representing the primary objective of this research. The three-dimensional (3D) and two-dimensional (2D) combined charts are plotted for some of these solutions. The impact of the fractional parameters and time variations is also illustrated. Moreover, the models are converted into a planar dynamical system using a Galilean transformation, and the analysis of bifurcation is examined. This research underscores the versatility of the aforementioned techniques for exploring complex nonlinear phenomena across various engineering and scientific disciplines. Finally, the findings of this study hold significant implications for advancing our understanding and analysis of nonlinear wave dynamics in diverse physical systems. [ABSTRACT FROM AUTHOR]

The paper investigates fractional differential equations involving the Liouville derivative. Solution to these equations under a boundary condition inside the time interval are derived in explicit form, their uniqueness is established using integral transforms technique. [ABSTRACT FROM AUTHOR]

A fractional order problem arising in porous media is considered. Well-posedness as well as stability are discussed. Mittag-Leffler stability is proved in case of a strong fractional damping in the displacement component and a fractional frictional one in the volume fraction component. This extends an existing result from the integer-order (second-order) case to the non-integer case. In the absence of the fractional damping in the volume fraction component, it is shown a convergence to zero and a Lyapunov uniform stability. [ABSTRACT FROM AUTHOR]

In this paper, Lie symmetry analysis method is applied to ( 2 + 1 ) {(2+1)} -dimensional time fractional cubic Schrödinger equation. We obtain all the Lie symmetries and reduce the ( 2 + 1 ) {(2+1)} -dimensional fractional partial differential equations with Riemann–Liouville fractional derivative to ( 1 + 1 ) (1+1) -dimensional counterparts with Erdélyi–Kober fractional derivative. Then we obtain the power series solutions of the reduced equations and prove their convergence. In addition, the conservation laws for the governing model are constructed by the new conservation theorem and the generalization of Noether operators. [ABSTRACT FROM AUTHOR]

The researchers have developed numerous analytical and numerical techniques for solving fractional partial differential equations most of which provide approximate solutions. Exact solutions, however, are vitally important in a convenient conception of the qualitative properties of the concerned phenomena and processes. In this paper, the pseudo-parabolic-type equations with conformable fractional derivatives are reduced to conformable fractional nonlinear ordinary differential equations by implementing a simple wave transformation. An important benefit of the proposed transformation is that it yields analytical solutions of the conformable pseudo-parabolic type equations by applying the exponential rational function strategy. The sensitivity behaviour of the model has been mentioned thoroughly. [ABSTRACT FROM AUTHOR]

In this paper, we focus on studying a fractional Schrödinger equation of the form { (− Δ) s u + V (x) u = f (x , u) in Ω , u > 0 in Ω , u = 0 in R n ∖ Ω , where 0 < s < 1 , n > 2 s , Ω is a smooth bounded domain in R n , (− Δ) s denotes the fractional Laplacian of order s, f (x , t) is a function in C (Ω ‾ × R) , and f (x , t) / t is nondecreasing in t and converges uniformly to an L ∞ function q (x) as t approaches infinity. The potential energy V satisfies appropriate assumptions. In the first part of our study, we analyze the asymptotic linearity of the nonlinearity and investigate the occurrence of the bifurcation phenomenon. We employ variational techniques and a "mountain pass" approach in our proof, notable for not assuming the Ambrosetti–Rabinowitz condition or any replacement condition on the nonlinearity. Additionally, we extend our methods to handle cases where the function f (x , t) exhibits superlinearity in t at infinity, represented by q (x) ≡ + ∞ . [ABSTRACT FROM AUTHOR]

In this paper, based on the differential quadrature method (DQM), matrix operators are derived for fractional integration and Caputo differentiation. These operators generalize the efficient DQM to fractional calculus. The proposed fractional differential/integral quadrature method (FDIQM) is used to solve various types of fractional ordinary and partial differential equations. FDIQM unifies the solution of multi‐integer fractional‐order differential equations leading to significant simplification in the implementation. Numerous examples are presented to demonstrate the accuracy of the operators. Other examples are presented to solve various fractional differential equations including time‐fractional sub‐diffusion equation, linear/nonlinear, and multiorder fractional differential equations. In addition, numerous boundary conditions are considered including mixed fractional derivatives. Further, a nonlinear fractional eigenvalue problem is solved efficiently, and its bifurcation diagrams are obtained. Comparisons between the proposed method and the existing ones are included, showing the ease of implementation, efficiency, and applicability of FDIQM. [ABSTRACT FROM AUTHOR]

Spectral and singular value symbols are valuable tools to analyse the eigenvalue or singular value distributions of matrix-sequences in the Weyl sense. More recently, Generalized Locally Toeplitz (GLT) sequences of matrices have been introduced for the spectral/singular value study of the numerical approximations of differential operators in several contexts. As an example, such matrix-sequences stem from the large linear systems approximating Partial Differential Equations (PDEs), Fractional Differential Equations (FDEs), Integro Differential Equations (IDEs), using any discretization on reasonable grids via local methods, such as Finite Differences, Finite Elements, Finite Volumes, Isogeometric Analysis, Discontinuous Galerkin etc. Studying the asymptotic spectral behaviour of GLT sequences is useful in analysing classical techniques for the solution of the corresponding PDEs/FDEs/IDEs and in designing novel fast and efficient methods for the corresponding large linear systems or related large eigenvalue problems. The theory of GLT sequences, in combination with the results concerning the asymptotic spectral distribution of perturbed sequences of matrices, is one of the most powerful and successful tools for computing the spectral symbol f. In this regard, it would be beneficial to design an automatic procedure to compute the spectral symbols of such matrix-sequences and Ahmed Ratnani partially pursued it. Here, in the case of one-dimensional and two-dimensional differential problems, we continue in this direction by proposing an automatic procedure for computing the symbol of the underlying sequences of matrices, assuming that it is a GLT sequence satisfying mild conditions. [ABSTRACT FROM AUTHOR]

This study employs the Lie symmetry technique to explore the symmetry features of the time fractional Kupershmidt equation. Specifically, we use the Lie symmetry technique to derive the symmetry generators for this equation, which incorporates a conformal fractional derivative. We use the symmetry generators to transform the fractional partial differential equation into a fractional ordinary differential equation, thereby simplifying the analysis. The obtained reduced equation is of fourth order nonlinear ordinary differential equation. To find the wave solutions, F/G‐expansion process has been used to obatin different types of solutions of the time‐fractional Kuperschmidt equation. The obtained wave solutions are hyperbolic and trigonometric in nature. We then use Maple software to visually depict these wave solutions for specific parameter values, providing insights into the behaviour of the system under investigation. Peak and kink wave solutions are achieved for the given problem. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for a system of partial differential equations of fractional order D t B U(t, x) + A (D)U(t, x) = H(t, x), where U and H are vector-valued functions and the m × m-matrix of differential operators A (D) is triangular with elliptic operators on the diagonal. The main feature of this system is that the vector order B has different components βj ∈ (0, 1] which are not necessarily rational. We find sufficient (and necessary in some cases) conditions on the initial function and right-hand side of the equation that guarantee the existence of the classical solution. [ABSTRACT FROM AUTHOR]

The authors have proved a unique solvability of an inverse-initial problem for a time-fractional degenerate partial differential equation. Using a method of separation of variables, they obtained the Cauchy problem for fractional differential equation involving the bi-ordinal Hilfer derivative in time-variable. The authors present the solution to this Cauchy problem in an explicit form via the Kilbas–Saigo function. Further, using the upper and lower bounds of the function, they prove uniform convergence of infinite series corresponding to the solution of the formulated inverse-initial problem. [ABSTRACT FROM AUTHOR]

In the present article, the method which was obtained from a combination of the conformable fractional double Laplace transform method (CFDLTM) and the homotopy perturbation method (HPM) was successfully applied to solve linear and nonlinear conformable fractional partial differential equations (CFPDEs). We included three examples to help our presented technique. Moreover, the results show that the proposed method is efficient, dependable, and easy to use for certain problems in PDEs compared with existing methods. The solution graphs show close contact between the exact and CFDLTM solutions. The outcome obtained by the conformable fractional double Laplace transform method is symmetrical to the gain using the double Laplace transform. [ABSTRACT FROM AUTHOR]

The space fractional advection–diffusion equation is a crucial type of fractional partial differential equation, widely used for its ability to more accurately describe natural phenomena. Due to the complexity of analytical approaches, this paper focuses on its numerical investigation. A lattice Boltzmann model for the spatial fractional convection–diffusion equation is developed, and an error analysis is carried out. The spatial fractional convection–diffusion equation is solved for several examples. The validity of the model is confirmed by comparing its numerical solutions with those obtained from other methods The results demonstrate that the lattice Boltzmann method is an effective tool for solving the space fractional convection–diffusion equation. [ABSTRACT FROM AUTHOR]

We consider the fully nonlocal diffusion equations with nonnegative L1$L^1$‐data. Based on the approximation and energy methods, we prove the existence and uniqueness of nonnegative entropy solutions for such problems. In particular, our results are valid for the time‐space fractional Laplacian equations. [ABSTRACT FROM AUTHOR]

We introduce an efficient variational hybrid quantum-classical algorithm designed for solving Caputo time-fractional partial differential equations. Our method employs an iterable cost function incorporating a linear combination of overlap history states. The proposed algorithm is not only efficient in terms of time complexity but also has lower memory costs compared to classical methods. Our results indicate that solution fidelity is insensitive to the fractional index and that gradient evaluation costs scale economically with the number of time steps. As a proof of concept, we apply our algorithm to solve a range of fractional partial differential equations commonly encountered in engineering applications, such as the subdiffusion equation, the nonlinear Burgers' equation, and a coupled diffusive epidemic model. We assess quantum hardware performance under realistic noise conditions, further validating the practical utility of our algorithm. [ABSTRACT FROM AUTHOR]

A unique solvability of the inverse initial problem for a time-degenerate fractional partial differential equation is proved. Using the method of variable separation, we obtain the Cauchy problem for the fractional differential equation involving the bi-ordinal Hilfer derivative in the time variable. The authors present the solution to this Cauchy problem in an explicit form via the Kilbas–Saigo function. Further, using the upper and lower bounds of this function, the authors prove the uniform convergence of the infinite series corresponding to the solution of the formulated inverse initial problem. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the fractal nature of the local fractional Landau–Ginzburg–Higgs Equation (LFLGHE) describing nonlinear waves with weak scattering in a fractal medium. The main goal of the paper is to introduce and apply the Local Fractional Elzaki Variational Iteration Method (LFEVIM) for solution of LFLGHE. Convergence analysis of LFEVIM solution for general nonlinear local fractional partial differential equation is also provided. Two examples of the local fractional LFLGHE are considered to demonstrate the applicability of the proposed technique with numerical simulations on Cantor set. [ABSTRACT FROM AUTHOR]

In this article, we generalized Mittag‐Leffler‐type functions F¯A(3),F¯B(3),F¯C(3)$$ {\overline{F}}_A&#x0005E;{(3)},{\overline{F}}_B&#x0005E;{(3)},{\overline{F}}_C&#x0005E;{(3)} $$, and F¯D(3)$$ {\overline{F}}_D&#x0005E;{(3)} $$, which correspond, respectively, to the familiar Lauricella hypergeometric functions FA(3),FB(3),FC(3)$$ {F}_A&#x0005E;{(3)},{F}_B&#x0005E;{(3)},{F}_C&#x0005E;{(3)} $$, and FD(3)$$ {F}_D&#x0005E;{(3)} $$ of three variables. Initially, from the Mittag‐Leffler type function in the simplest form to the functions we are studying, necessary information about the development history, study, and importance of this and hypergeometric type functions will be introduced. Among the various properties and characteristics of these three‐variable Mittag‐Leffler‐type function F¯D(3)$$ {\overline{F}}_D&#x0005E;{(3)} $$, which we investigate in the article, include their relationships with other extensions and generalizations of the classical Mittag‐Leffler functions, their three‐dimensional convergence regions, their Euler‐type integral representations, their Laplace transforms, and their connections with the Riemann‐Liouville operators of fractional calculus. The link of three‐variable Mittag‐Leffler function with fractional differential equation systems involving different fractional orders is necessary on certain applications in physics.Therefore, we provide the systems of partial differential equations which are associated with them. [ABSTRACT FROM AUTHOR]

This paper aims to study the existence and stability results concerning a fractional partial differential equation with variable exponent source functions. The local existence result for α ∈ (0 , 1) is established with the help of the α -resolvent kernel and the Schauder-fixed point theorem. The non-continuation theorem is proved by the fixed point technique and accordingly the global existence of solution is achieved. The uniqueness of the solution is obtained using the contraction principle and the stability results are discussed by means of Ulam-Hyers and generalized Ulam-Hyers-Rassias stability concepts via the Picard operator. Examples are provided to illustrate the results. [ABSTRACT FROM AUTHOR]

In the context of forward uncertainty quantification, we investigate the convergence of the Galerkin projections for random fractional differential equations. The governing system is formed by a finite set of independent input random parameters (a germ) and by a fractional derivative in the Caputo sense. Input uncertainty arises from biased measurements, and a fractional derivative, defined by a convolution, takes past history into account. While numerical experiments on the gPC-based Galerkin method are already available in the literature for random ordinary, partial and fractional differential equations, a theoretical analysis of mean-square convergence is still lacking for the fractional case. The aim of this contribution is to fill this gap, by establishing new inequalities and results and by raising new open problems. [ABSTRACT FROM AUTHOR]

This paper introduces the fractional Fourier series on the fractional torus and proceeds to investigate several fundamental aspects. Our study includes topics such as fractional convolution, fractional approximation, fractional Fourier inversion, and the Poisson summation formula. We also explore the relationship between the decay of its fractional Fourier coefficients and the smoothness of a function. Additionally, we establish the convergence of the fractional Féjer means and Bochner–Riesz means. Finally, we demonstrate the practical applications of the fractional Fourier series, particularly in the context of solving fractional partial differential equations with periodic boundary conditions, and showcase the utility of approximation methods on the fractional torus for recovering non-stationary signals. [ABSTRACT FROM AUTHOR]

In several recent studies, many researchers have shown the advantage of fractional calculus in the production of particular solutions of a huge number of linear and nonlinear partial differential equations. In this research work, different theorems related to the G-double Laplace transform (DGLT) are proved. The solution of the system of time-fractional partial differential equations is addressed using a new analytical method. This technique is a combination of the multi-G-Laplace transform and decomposition methods (MGLTDM). Moreover, we discuss the convergence of this method. Two examples are provided to check the applicability and efficiency of our technique. [ABSTRACT FROM AUTHOR]

Nonexistence theorems constitute an important part of the theory of differential and partial differential equations. Motivated by the numerous applications of fractional differential equations in diverse fields, in this paper, we studied sufficient conditions for the nonexistence of solutions (or, equivalently, necessary conditions for the existence of solutions) for nonlinear fractional differential inequalities and systems in the sense of Erdélyi-Kober. Our approach is based on nonlinear capacity estimates specifically adapted to the Erdélyi-Kober fractional operators and some integral inequalities. [ABSTRACT FROM AUTHOR]

This study aims to employ the extended direct algebraic method (EDAM) to generate and evaluate soliton solutions to the nonlinear, space-time conformable Estevez Mansfield-Clarkson equation (CEMCE), which is utilized to simulate shallow water waves. The proposed method entails transforming nonlinear fractional partial differential equations (NFPDEs) into nonlinear ordinary differential equations (NODEs) under the assumption of a finite series solution by utilizing Riccati ordinary differential equations. Various mathematical structures/solutions for the current model are derived in the form of rational, exponential, trigonometric, and hyperbolic functions. The wide range of obtained solutions allows for a thorough analysis of their actual wave characteristics. The 3D and 2D graphs are used to illustrate that these behaviors consistently manifest as periodic, dark, and bright kink solitons. Notably, the produced soliton solutions offer new and critical insights into the intricate behaviors of the CEMCE by illuminating the basic mechanics of the wave's interaction and propagation. By analyzing these solutions, academics can better understand the model's behavior in various settings. These solutions shed light on complicated issues such as configuration dispersion in liquid drops and wave behavior in shallow water. [ABSTRACT FROM AUTHOR]

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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. [ABSTRACT FROM AUTHOR]

This research offers a precise analytical solution for initial value problems of both linear and nonlinear fractional order partial differential equations. A new approach called the limit residual function method is developed and utilized to generate a series solution for the equations. The key tools of this method are the concepts of power series, residual function, and the limit at zero. The new method is characterized by the speed of determining the series solution coefficients and the limited mathematical operations used. The study examines five significant applications of fractional partial differential equations using this new method. To validate the accuracy of the results, they are compared with exact solutions in the classical case for the discussed applications. We conclude that the proposed approach is straightforward, effective, and successful in solving linear and nonlinear fractional differential equations. [ABSTRACT FROM AUTHOR]

This article introduces a new numerical algorithm dedicated to solving the most general form of variable-order fractional partial differential models. Both the time and spatial order of derivatives are considered as non-constant values. A combination of the shifted Chebyshev polynomials is used to approximate the solution of such equations. The coefficients of this combination are considered a function of time, and they are obtained using the collocation method. The theoretical aspects of the method are investigated, and then by solving some problems, the efficiency of the method is presented. [ABSTRACT FROM AUTHOR]

In this paper, we present a highly efficient analytical method that combines the Laplace transform and the residual power series approach to approximate solutions of nonlinear time-fractional partial differential equations (PDEs). First, we derive the analytical method for a general form of fractional partial differential equations. Then, we apply the proposed method to find approximate solutions to the time-fractional coupled Berger equations, the time-fractional coupled Korteweg–de Vries equations and time-fractional Whitham–Broer–Kaup equations. Secondly, we extend the proposed method to solve the two-dimensional time-fractional coupled Navier–Stokes equations. The proposed method is validated through various test problems, measuring quality and efficiency using error norms E 2 and E ∞ , and compared to existing methods. [ABSTRACT FROM AUTHOR]

The objective of this paper is investigating the numerical solution and optical phenomena of highly nonlinear fractional-order partial differential equation namely fractional order Lakshmanan–Porsezian–Daniel model, incorporating spatio-temporal dispersion effects. The fractional derivative is considered in Caputo sense because this is the most crucial tool for working with integer order models in a fractional sense under the right subsidiary conditions, it has many benefits. To achieve the objective of this study, the Improved Adomian decomposition method has been utilized. This method is known for effectively handling nonlinearities and fractional derivatives iteratively. The obtained soliton solutions are presented graphically. We conduct a comparative analysis at ρ = 1 , focusing on the bright optical soliton solutions. To the best of our knowledge, there have not been previously reported identical or related findings. By comparing the obtained results with the exact solution, the accuracy of the considered method is examined. The applied method is explicit, efficient, and user-friendly for managing a wider range of fractional order nonlinear models, as demonstrated by the numerical outcomes. [ABSTRACT FROM AUTHOR]

In this paper, we apply the fractional homotopy perturbation transform method (FHPTM) to deliver an effective semi-analytical technique for determining fractional-order Kuramoto–Sivashinsky equations. The project technique combines the Laplace transform with the Caputo–Fabrizio fractional derivative of order α where α ∈ (0 , 1 ] . Fractional-order Kuramoto–Sivashinsky equation is indeed important in the field of nonlinear physics and mathematics. It is a fractional partial differential equation that describes the behaviour of waves in certain dissipative media, such as flames and chemicals. The FHPTM is described to be fast and accurate. Illustrative examples are included to demonstrate the efficiency and reliability of the presented techniques. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of at least two weak solutions to a class of singular two-phase problems with variable exponents involving a ψ-Hilfer fractional operator and Dirichlet-type boundary conditions when the term source is dependent on one parameter. Here, we use the fiber method and the Nehari manifold to prove our results. [ABSTRACT FROM AUTHOR]

Hyperbolic linear theory of heat propagation has been established in the framework of a Caputo time fractional order derivative. The solution of a system of integer and fractional order initial value problems is achieved by employing the Adomian decomposition approach. The obtained solution is in convergent infinite series form, demonstrating the method’s strengths in solving fractional differential equations. Moreover, the double Laplace transform method is employed to acquire the solution of a system of integer and fractional order boundary conditions in the Laplace domain. An inversion of double Laplace transforms has been achieved numerically by employing the Xiao algorithm in the space-time domain. Considering the non-Fourier effect of heat conduction, the finite speed of thermal wave propagation has been attained. The role of the fractional order parameter has been examined scientifically. The results obtained by considering the fractional order theory and the integer order theory perfectly coincide as a limiting case of fractional order parameter approaches one. [ABSTRACT FROM AUTHOR]

This research paper introduces a novel approach to deriving traveling wave solutions (TWSs) for the Caputo-fractional Klein–Gordon equations. This research presents a distinct methodological advancement by introducing TWSs of a particular time-fractional partial differential equation, utilizing a non-local fractional operator, specifically the Caputo derivative. To achieve our goal, a novel transformation is considered, that converts a time-fractional partial differential equation into fractional ordinary differential equations, enabling analytical solutions through various analytical methods. This paper employs the homotopy analysis method to achieve the target objectives. To demonstrate the efficiency and applicability of the proposed transform and method, two examples are discussed and analyzed in figures. [ABSTRACT FROM AUTHOR]

This study introduces the generalized Khater Method (gKM) as a tool for investigating the existence and dynamics of soliton solutions within the realm of the Fractional Coupled Higgs System (FCHS) with conformable fractional derivatives. The methodology involves obtaining Nonlinear Ordinary Differential Equations (NODEs) from FCHS via a fractional complex transformation. The reduced NODEs are further used to uncover various families of soliton solutions, elucidating their intricate interrelationships and dynamics. Using some 3D graphics, we visually depict the behavior of some soliton solutions, including shock, lump-type shock, periodic, multiple periodic, rogue, and kink solitons. These findings contribute to a better understanding of the dynamics inherent in the FCHS and have potential applications in domains dealing with nonlinear fractional partial differential equations. Arguably, the study also highlights the importance of incorporating the conformable fractional derivative into the FCHS to provide it with more degrees of freedom, which allows for a deeper exploration of its behaviors and could have significant implications for particle physics, particularly for phenomena such as mass origin and electroweak violation of symmetry. [ABSTRACT FROM AUTHOR]

This research employs advanced computational methodologies to analyze solitary wave solutions associated with the fractional nonlinear Chaffee–Infante ( C I ) equation, extending classical diffusion models with broad applications in materials science, fluid dynamics, and signal processing. The study makes notable contributions to the modeling of anomalous diffusion in porous materials, the comprehension of nonlinear dynamics, and the analysis of wave behavior incorporating memory and non-local effects. The research enhances our understanding of practical applications and provides valuable insights into complex wave dynamics within fluid dynamics, nonlinear optics, and plasma physics. The computational techniques utilized in this investigation, specifically the extended unified ( E U ) and trigonometric–quantic–B-spline (TQBS ) approaches, demonstrate superior effectiveness in comparison to existing methods, promising heightened accuracy and efficiency in solving fractional partial differential equations. The numerical scheme, Trigonometric–Quantic–B–spline, in tandem with analytical solutions, serves to validate the model and adapt it to real-world complexities. This integrated approach not only quantifies accuracy but also supports parameter estimation, ensuring the model's applicability in diverse engineering and scientific scenarios. Graphical representations of both analytical and numerical solutions are presented, offering a visual elucidation of the model's characteristics and validating solution accuracy. The incorporation of the Trigonometric–Quantic–B–spline scheme provides a robust foundation for further exploration and applications in varied scientific and engineering domains. [ABSTRACT FROM AUTHOR]

In recent years, various complex systems and real-world phenomena have been shown to include memory and hereditary properties that change with respect to time, space, or other variables. Consequently, fractional partial differential equations containing variable-order fractional operators have been extensively resorted for modeling such phenomena accurately. In this paper, we consider the two-dimensional fractional cable equation with the Caputo variable-order fractional derivative in the time direction, which is preferable for describing neuronal dynamics in biological systems. A point-wise scheme, namely, the Crank–Nicolson finite difference method, along with a group-wise scheme referred to as the explicit decoupled group method are proposed to solve the problem under consideration. The stability and convergence analyses of the numerical schemes are provided with complete details. To demonstrate the validity of the proposed methods, numerical simulations with results represented in tabular and graphical forms are given. A quantitative analysis based on the CPU timing, iteration counting, and maximum absolute error indicates that the explicit decoupled group method is more efficient than the Crank–Nicolson finite difference scheme for solving the variable-order fractional equation. [ABSTRACT FROM AUTHOR]

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). [ABSTRACT FROM AUTHOR]

Purpose: Fractional Fokker-Planck equation (FFPE) and time fractional coupled Boussinesq-Burger equations (TFCBBEs) play important roles in the fields of solute transport, fluid dynamics, respectively. Although there are many methods for solving the approximate solution, simple and effective methods are more preferred. This paper aims to utilize Laplace Adomian decomposition method (LADM) to construct approximate solutions for these two types of equations and gives some examples of numerical calculations, which can prove the validity of LADM by comparing the error between the calculated results and the exact solution. Design/methodology/approach: This paper analyzes and investigates the time-space fractional partial differential equations based on the LADM method in the sense of Caputo fractional derivative, which is a combination of the Laplace transform and the Adomian decomposition method. LADM method was first proposed by Khuri in 2001. Many partial differential equations which can describe the physical phenomena are solved by applying LADM and it has been used extensively to solve approximate solutions of partial differential and fractional partial differential equations. Findings: This paper obtained an approximate solution to the FFPE and TFCBBEs by using the LADM. A number of numerical examples and graphs are used to compare the errors between the results and the exact solutions. The results show that LADM is a simple and effective mathematical technique to construct the approximate solutions of nonlinear time-space fractional equations in this work. Originality/value: This paper verifies the effectiveness of this method by using the LADM to solve the FFPE and TFCBBEs. In addition, these two equations are very meaningful, and this paper will be helpful in the study of atmospheric diffusion, shallow water waves and other areas. And this paper also generalizes the drift and diffusion terms of the FFPE equation to the general form, which provides a great convenience for our future studies. [ABSTRACT FROM AUTHOR]

Recently, fractional derivatives have become increasingly important for describing phenomena occurring in science and engineering fields. In this paper, we consider a numerical method for solving the fractional Burgers' equations (FBEs), a vital topic in fractional partial differential equations. Due to the difficulty of the fractional derivatives, the nonlinear FBEs are linearized through the Rubin–Graves linearization scheme combined with the implicit the third-order Adams–Moulton scheme. Additionally, in the spatial direction of the FBEs, the fourth-order central finite difference scheme is used to obtain more accurate solutions. The convergence of the proposed scheme is theoretically and numerically analyzed. Also, the efficiency is demonstrated through several numerical experiments and compared with that of existing methods. [ABSTRACT FROM AUTHOR]

In this paper, we study traveling wavefronts in an anomalous diffusion predator–prey model with the modified Leslie–Gower and Holling-type II schemes. We perform a traveling wave analysis to show that the model has heteroclinic trajectories connecting two steady state solutions of the resulting system of fractional partial differential equations and corresponding to traveling wavefronts. This also includes numerical results to show the existence of traveling wavefronts. Furthermore, we obtain the numerical time-dependent solutions in order to show the evolution of wavefronts. We find that wavefronts exist that travel faster in the anomalous subdiffusive regime than in the normal diffusive one. Our results emphasize that the main properties of traveling waves and invasions are altered by anomalous subdiffusion in this model. [ABSTRACT FROM AUTHOR]

In this paper, we use the local fractional variational iteration transform method LFVITM to solve a class of linear and nonlinear partial differential equations (PDEs), as well as a system of PDEs which are involving local fractional differential operators (LFDOs). The technique combines the variational iteration transform approach and the Yang–Laplace transform. To show how effective and precise the LFVITM four examples are given. [ABSTRACT FROM AUTHOR]

In this work, travelling wave solutions of a nonlinear system of fractional Schrödinger equations (FSEs) with conformable fractional derivatives are studied. We examine the fractional generalization of the Schrödinger equation, a topic of great importance in quantum physics, using the analytic approach known as the modified extended direct algebraic method. Our approach involves the use of a fractional complex transformation to produce nonlinear ordinary differential equations, which are then solved to reveal travelling wave solutions. The two- and three-dimensional graphs that provide visual representations of the system's behaviour present a variety of wave profiles, including periodic, kink, anti-kink, shocks, lumps, and other soliton waves. The study sheds light on the dynamics of FSEs by revealing multiple families of travelling wave solutions and their complex relationships. These results provide insight into nonlinear fractional partial differential equations and a greater understanding of the dynamics of FSEs than previous attempts in the literature. [ABSTRACT FROM AUTHOR]

In the past, the existence and uniqueness of the solutions of fractional differential equations have been investigated by many researchers theoretically in various approaches in the literature. In this paper, there is no discussion of the existence of solutions for the nonlinear differential equations with fractal fractional operators. The objective of this work is to present novel contraction approaches, notably the α-φ-contraction α-type of the F-contraction, within the context of F-metric and orbital metric spaces. The aim of this study is to illustrate certain fixed point theorems that offer a new and direct approach to establish the existence and uniqueness of the solution to the general partial differential equations by employing the fractal fractional operators. [ABSTRACT FROM AUTHOR]

In this study, an accurate analytical solution is presented for fuzzy FPDEs. It is done by using a novel method called the Laplace-residual power series (LRPSM) to build a series solution to the given problems. The fundamental instruments of the employed method are the Laplace transform, fractional Laurent, and fractional power series. Using the idea of a limit at infinity, we provide a series solution to a fuzzy FPDE with quick convergence and simple coefficient finding. We analyze three cases to obtain approximate and exact solutions to show the effectiveness and reliability of the Laplace- residual power series approach. To demonstrate the accuracy of the suggested procedure, we compare the findings to the real data. [ABSTRACT FROM AUTHOR]

It is mentioned that understanding linear and non-linear thermo-elasticity systems is important for understanding temperature, elasticity, stresses, and thermal conductivity. One of the most crucial aspects of the current research is the solution to these systems. The fractional form of several thermo-elastic systems is explored, and elegant solutions are provided. The solutions of fractional and integer thermo-elastic systems are further discussed using tables and diagrams. The closed contact between the LRPSM and exact solutions is displayed in the graphs. Plotting fractional problem solutions demonstrates their convergence towards integer-order problem solutions for suitable modeling. The tables confirm that greater precision is rapidly attained as the terms of the derived series solution increase. The faster convergence and stability of the suggested method support its modification for other fractional non-linear complex systems in nature. [ABSTRACT FROM AUTHOR]

This paper aims to establish the existence and uniqueness of mild solutions to abstract sequential fractional differential equations. The approach employed involves the utilization of resolvent operators and the fixed-point theorem. Additionally, we investigate a specific example concerning a partial differential equation incorporating the Caputo fractional derivative. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]