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In this study, time-fractional coupled Korteweg–de Vries (cKdV) equations are solved using an efficient and reliable numerical technique. The classical cKdV system has been generalized into the time-fractional cKdV system. We employ the local fractional homotopy analysis method (LFHAM) and the Adomian decomposition method (ADM) to propose an approximate solution for fractional cKdV equations. Both approaches determined findings are compared together. The findings clearly demonstrate that the suggested methods are appropriate and efficient for handling both linear and nonlinear issues in engineering and sciences. To demonstrate the suggested approaches' competencies, examples are provided. Convergent series form has been used to make the solutions. The relevance of the techniques is illustrated through graphic representations of the solution. [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]

The method of solving the vibration frequency of local fractional nonlinear oscillators under general initial conditions is studied by coupling the energy balance method with the Galerkin method. One test case of fractional vibration is given to prove the feasibility of this method. [ABSTRACT FROM AUTHOR]

In this paper, the solution and behaviour of local fractional Fokker–Planck equation (LFFPE) is investigated in fractal media. For this purpose, the local fractional homotopy perturbation Elzaki transform method (LFHPETM) is proposed and utilized to explore the solution of LFFPE. The proposed scheme is a merger of well known local fractional homotopy perturbation technique and recently introduced local fractional Elzaki transform (LFET). The convergence and uniqueness analyses for LFHPETM solution of the general partial differential equation is also presented along with the computational procedure of this new hybrid combination. Three examples of LFFPE are illustrated to depict the applicability of the employed method with graphical simulations on Cantor set. The copulation of LFET with local fractional homotopy perturbation method (LFHPM) efficiently provides the faster solution for LFFPE in a fractal domain as compared to the LFHPM. Furthermore, the achieved solutions are also in a good match with existing solutions. The 3D behavior of solutions of LFFPEs are presented for fractal order ln 2 / ln 3 . Figures illustrate the 3D surface graphics of solutions with respect to input variables. [ABSTRACT FROM AUTHOR]

Under the current research, the local fractional (3+1)-dimensional modified Zakharov–Kuznetsov equation (MZKE) is explored. With the Mittag–Leffler function (MLF) defined on the Cantor sets (CS), four special functions, namely the SH χ (ℑ χ) , CH χ (ℑ χ) , SE χ (ℑ χ) and CS χ (ℑ χ) are extracted to construct an auxiliary function. Then the auxiliary function, along with Yang's non-differentiable (ND) transformation, is manipulated to explore the ND exact solutions (ESs). By means of the proposed method, four different sets of the ND ESs are found in just one step. The nonlinear dynamics of the ND exact solutions on the CS are illustrated graphically. Furthermore, the ND exact solutions for χ = ln 2 / ln 3 and the classic exact solutions for χ = 1 are also compared and discussed in detail via the 2-D curves. The attained results reveal that the method is a simple but effective tool to deal with local fractional PDEs arising in physics and maths. [ABSTRACT FROM AUTHOR]

In recent years, the theory of local fractional calculus has been widely used in the description of the fractional circuits. This paper presents a fractal RLC-parallel resonant circuit (FRLC-PRC) using the local fractional derivative (LFD). The FRLC-PRC is modeled by studying the non-differentiable (ND) lumped elements, then the ND conductance is obtained with the help of the local fractional Laplace transform (LFLT) and the ND parallel-resonant angular frequency (ND PRAF) is analyzed. It is found that the FRLC-PRC becomes the ordinary one when the fractional order S = 1. The obtained results show that the LFD is a powerful tool in the description of fractal circuit systems. [ABSTRACT FROM AUTHOR]

Aided by the local fractional derivative, we present a new local fractional (3+1)-dimensional Kadomstev-Petviashvili equation for describing the fractal water wave in this work. The non-differentiable transform is utilized to convert the local fractional equation into a local fractional ODE. On defining the Mittag-Leffler function on the Cantor sets, then a trialfunction based on the Mittag-Leffler function is proposed to seek for the non-differentiable exact solutions. The results reveal that the proposed method is a promising way to study the local fractional PDE arising in engineering and physics. [ABSTRACT FROM AUTHOR]

Based on the local fractional derivative, a fractal non-linear heat conduction equation, which can model the behavior of the heat transfer in the fractal medium, is extracted in this work. On defining the Mittag-Leffler function on the Cantor sets, two special functions namely the THv (µv) function and CHv (µv) function are constructed, and then are employed along with Yang's non-differentiable transformation seek for the non-differentiable exact solutions. The obtained results confirm that the proposed method iseffective and powerful, and can provide a promising way to find the exact solutions of the fractal PDE. [ABSTRACT FROM AUTHOR]

In this paper, a new fractal nerve impulses modeling is successfully described via the Yang's local fractional derivative in a microgravity space, and its approximate analytical solution is obtained by a new Adomian decomposition method. The efficiency and accuracy analysis of the proposed method is elucidated according to the graphs. The result shows that our method is excellent and accurate in dealing with Local fractional differential equations. [ABSTRACT FROM AUTHOR]

In this paper, the Local Fractional Laplace Decomposition Method (LFLDM) is used for solving a type of Two-Dimensional Fractional Diffusion Equation (TDFDE). In this method, first we apply the Laplace transform and its inverse to the main equation, and then the Adomian decomposition is used to obtain approximate/analytical solution. The accuracy and applicability of the LFLDM is shown through two examples. The LFLDM results are in good agreement with the exact solution of the problems. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new frequency amplitude formula for the local fractional nonlinear oscillation via local fractional calculus. It is more general than the He's frequency amplitude formula. Several test cases of local fractional nonlinear oscillations are given to prove the feasibility of the improved formula. [ABSTRACT FROM AUTHOR]

Fractal geometry plays an important role in the description of the characteristics of nature. Local fractional calculus, a new branch of mathematics, is used to handle the non-differentiable problems in mathematical physics and engineering sciences. The local fractional inequalities, local fractional ODEs and local fractional PDEs via local fractional calculus are studied. Fractional calculus is also considered to express the fractal behaviors of the functions, which have fractal dimensions. The interesting problems from fractional calculus and fractals are reported. With the scaling law, the scaling-law vector calculus via scaling-law calculus is suggested in detail. Some special functions related to the classical, fractional, and power-law calculus are also presented to express the Kohlrausch–Williams–Watts function, Mittag-Leffler function and Weierstrass–Mandelbrot function. They have a relation to the ODEs, PDEs, fractional ODEs and fractional PDEs in real-world problems. Theory of the scaling-law series via Kohlrausch–Williams–Watts function is suggested to handle real-world problems. The hypothesis for the tempered Xi function is proposed as the Fractals Challenge, which is a new challenge in the field of mathematics. The typical applications of fractal geometry are proposed in real-world problems. [ABSTRACT FROM AUTHOR]

In this work, the (2 + 1) -dimensional Kadomtsev–Petviashvili model is investigated. A novel variational scheme, namely, the variational transform wave method (VTWM), is successfully established to seek the solitary wave solution of the Kadomtsev–Petviashvili model. Furthermore, the fractal solitary solution of fractal Kadomtsev–Petviashvili model is also studied based on the local fractional derivative. Numerical examples are given to fully demonstrate that the VTWM is straightforward, efficient and attractive. Finally, the physical properties of solitary wave solutions are demonstrated by some 3D graphs. [ABSTRACT FROM AUTHOR]

In this paper, a local fractional Riccati differential equation method is applied. A new travelling wave solution to the nonlinear local fractional (2 + 1) -dimensional dispersive long water wave system is investigated. After travelling wave transformation, the governing model studied is converted into nonlinear ordinary differential equation. Some properties with the strain conditions are also reported. [ABSTRACT FROM AUTHOR]

In this paper, we establish a Lyapunov-type inequality for the half-linear local fractional ordinary differential equation based on the formulation of the local fractional derivative. In addition, we apply the inequality to investigate the non-existence and uniqueness of solutions for related homogeneous and non-homogeneous local fractional boundary value problems. [ABSTRACT FROM AUTHOR]

Gronwall–Bellman-type inequalities provide a very effective way to investigate the qualitative and quantitative properties of solutions of nonlinear integral and differential equations. In recent years, local fractional calculus has attracted the attention of many researchers. In this paper, based on the basic knowledge of local fractional calculus and the method of proving inequality on the set of real numbers, some new Gronwall–Bellman inequalities are discussed and established on the fractal space. Our results are a powerful supplement to the existing literature. [ABSTRACT FROM AUTHOR]

In this paper, we first created a fractal Date–Jimbo–Kashiwara–Miwa (FDJKM) long ripple wave model in a non-smooth boundary or microgravity space recorded. Using fractal semi-inverse skill (FSIS) and fractal traveling wave transformation (FTWT), the fractal variational principle (FVP) was derived, and the strong minimum necessary circumstance was attested with the He Wierstrass function. We have discovered two distinct solitary wave solutions, the square form of the hyperbolic secant function and the hyperbolic secant function form. Then, soliton solutions are cultivated through FVP and the minimum steady state condition. Finally, the influences of non-smooth boundaries on solitons were tackled, and the properties of the solution were demonstrated through three-dimensional contour lines. Fractal dimension can impact waveforms, but cannot affect their vertex values. The presentation of soliton solutions (SWS) using techniques is not only laudable but also noteworthy. The technique employed can also be used to investigate solitary wave solutions of other local fractional calculus partial differential equations. [ABSTRACT FROM AUTHOR]

This paper studies a local fractional vibration system with a damped non-linear term to reveal its frequency property and its energy consumption. A modification of He's frequency formulation is recommended for this purpose. Some examples are given to illustrate the solving process and the reliability of the method. Additionally, the effect of the initial conditions on the vibrating properties is elucidated. This paper offers a new window for fast and effective insight into local fractional vibration systems. [ABSTRACT FROM AUTHOR]

In this paper, we give a main example indicating the ineffectiveness of the local fractional derivatives on the Riemann curvature tensor that is a common tool in calculating curvature of a Riemannian manifold. For this, first we introduce a general local fractional derivative operator that involves the mostly used ones in the literature as conformable, alternative, truncated M− and V−fractional derivatives. Then, according to this general operator, a particular Riemannian metric on the real affine space Rn that is different from the Euclidean one is defined. In conclusion, our main example states that the Riemann curvature tensor of Rn endowed with this particular metric is identically 0, that is, one is locally isometric to Euclidean space.

The fractional derivatives have been widely applied in many fields and has attracted widespread attention. This paper extracts a new fractional (3+1)-dimensional modified Zakharov–Kuznetsov equation (MZKe) with the local fractional derivative (LFD) for the first time. Two special functions, namely, the LT δ (Ξ δ) and LC δ (Ξ δ) functions that are derived on the basis of the Mittag-Leffler function (MLF) defined on the Cantor set (CS), are employed to construct the auxiliary trial function to look into the exact solutions (ESs). Aided by Yang's non-differentiable (ND) transformation, six groups of the ND ESs are found. The ND ESs on the CS for δ = ln 2 / ln 3 are depicted graphically. Additionally, as a comparison, the ESs of the classic (3+1)-dimensional MZKe for δ = 1 are also illustrated. The outcomes reveal that the derived method is powerful and effective, and can be used to deal with the other local fractional PDEs. [ABSTRACT FROM AUTHOR]

In the context of fractal space, this study presents a higher-order nonlinear local fractional Bratu-type equation and thoroughly examines this generalized nonlinear equation. Additional analysis and identification of particular special situations of the generalized local fractional Bratu equation is performed. Finally, the Adomian decomposition method is utilized to derive that solution for the generalized Bratu equation of local fractional type. This study contributes to a deeper understanding of these equations and provides a practical computational approach to their solutions. [ABSTRACT FROM AUTHOR]

This study proposes a new fractal modified equal width-Burgers equation (MEWBE) with the local fractional derivative (LFD) for the first time. By defining the Mittag-Leffler function (MLF) on the Cantor set (CS), two special functions, namely, the T H υ (μ υ) and C H υ (μ υ) functions, are derived for constructing the auxiliary function to seek the non-differentiable (ND) exact solutions. And 16 groups of the ND exact solutions are successfully established. The solutions on the CS are depicted graphically to interpret the nonlinear dynamic behaviors. Furthermore, the comparative results of the fractal MEWBE and the classical MEWBE are also discussed. The obtained results confirm that the proposed method is effective and powerful, and can provide a promising way to find the ND exact solutions of the local fractional PDEs. [ABSTRACT FROM AUTHOR]

In this work, the local fractional Camassa-Holm-Kadomtsev–Petviashvili equation (LFCHKPE) is defined on Cantor sets by using the local fractional derivative for the first time. The exact travelling-wave solution of the non-differential type for the LFCHKPE is obtained by using the local fractional wave method (LFWM). The LFWM is very simple and effective and then gets good results. The obtained non-differential travelling-wave solutions are elaborated by some 3D figures. A comparison with classical Camassa-Holm-Kadomtsev–Petviashvili equation (CCHKPE) is also presented. The LFWM sheds a new light on the exact travelling wave solution of local fractional wave equations.

In this paper, an analog of the conformable fractional derivative is defined in an arbitrary finite-dimensional commutative associative algebra. Functions taking values in the indicated algebras and having derivatives in the sense of a conformable fractional derivative are called φ-monogenic. A relation between the concepts of φ-monogenic and monogenic functions in such algebras has been established. Two new definitions have been proposed for the fractional derivative of the functions with values in finite-dimensional commutative associative algebras. [ABSTRACT FROM AUTHOR]

In this study, the local fractional derivative is employed to build the fractional Bogoyavlensky–Konopelchenko model, which is then used to develop the interaction between long wave propagation and Riemann wave propagating under particular conditions. The major goal of this study is to obtain some new solitary wave solutions of the local fractional Bogoyavlensky–Konopelchenko model using two effective methods, the Yang–Machado–Baleanu–Cattain wave method (YMBCWM) and fractional sech function method (FSFM). These obtained solitary wave solutions are unique from those found in the literature. Several 3D simulation figures show the dynamic behavior of these new solitary wave solutions. The two novel approaches bring new perspectives for resolving the same class of fractional wave equations. [ABSTRACT FROM AUTHOR]

In this paper, we mainly consider the local fractional Vakhnenko-Parkes equation with the local fractional derivative for the first time. Some new soliton solutions of local fractional Vakhnenko-Parkes equation are derived by using local fractional wave method. These obtained soliton solutions suggest that this proposed approach is effective, simple and reliable. Finally, the physical characteristics of these new soliton solutions are described through 3-D figures. [ABSTRACT FROM AUTHOR]

In this study, a new fractional Zakharov–Kuznetsov equation (ZKE) within the local fractional derivative (LFD) is derived. Yang's non-differentiable (ND) traveling wave transform is introduced, then two novel techniques namely the Mittag-Leffler function-based method (MLFBM) and Yang's special function method (Y-SFM) are adopted to seek for the ND exact solutions for the first time. With the aid of the Mathematica software, the dynamic behaviors of the different solutions on the Cantor sets are illustrated via the 3D plots by assigning the appropriate parameters. The attained results confirm that the mentioned methods are effective and straightforward, which can be used to study the ND exact solutions of the local fractional partial differential equations (PDEs). [ABSTRACT FROM AUTHOR]

Based on the local fractional derivative, a new local fractional Klein-Fock-Gordon equation is derived in this paper for the first time. A simple method namely Yang's special function method is used to seek for the non-differentiable exact solutions. The whole calculation process strongly shows that the proposed method is simple and effective, and can be applied to investigate the non-differentiable exact solutions of the other local fractional PDE. [ABSTRACT FROM AUTHOR]

In this article, a new (2 + 1)‐dimensional local fractional breaking soliton equation is derived with the local fractional derivative. Applying the traveling wave transform of the non‐differentiable type, the (2 + 1)‐dimensional local fractional breaking soliton equation is converted into a nonlinear local fractional ordinary differential equation. By defining a set of elementary functions on Cantor sets, a novel analytical technique namely the Mittag–Leffler function‐based method is employed for the first time ever to construct the exact solutions. The solutions on the Cantor sets are presented via the 3‐D contours. It reveals that the proposed method is effective and powerful and is expected to give some inspiration for the study of the local fractional PDEs. [ABSTRACT FROM AUTHOR]

In this work, the local fractional Zakharov–Kuznetsov‐modified equal width dynamical (LFZKMEWD) model is investigated on Cantor sets by using the local fractional derivative (LFD). The fractal variational wave method (FVWM) is employed to obtain the exact traveling wave solutions of the nondifferentiable type for the LFZKMEW model. The numerical example illustrates the FVWM is efficient and straightforward. The properties of exact traveling wave solutions are also elaborated by some figures. [ABSTRACT FROM AUTHOR]

In this work, the local fractional Camassa-Holm-Kadomtsev–Petviashvili equation (LFCHKPE) is defined on Cantor sets by using the local fractional derivative for the first time. The exact travelling-wave solution of the non-differential type for the LFCHKPE is obtained by using the local fractional wave method (LFWM). The LFWM is very simple and effective and then gets good results. The obtained non-differential travelling-wave solutions are elaborated by some 3D figures. A comparison with classical Camassa-Holm-Kadomtsev–Petviashvili equation (CCHKPE) is also presented. The LFWM sheds a new light on the exact travelling wave solution of local fractional wave equations. [ABSTRACT FROM AUTHOR]

A new local fractional modified Benjamin–Bona–Mahony equation is proposed within the local fractional derivative in this study for the first time. By defining some elementary functions via the Mittag–Leffler function (MLF) on the Cantor sets (CSs), a set of nonlinear local fractional ordinary differential equations (NLFODEs) is constructed. Then, a fast algorithm namely Yang's special function method is employed to find the non-differentiable (ND) exact solutions. By this method, we can extract abundant exact solutions in just one step. Finally, the obtained solutions on the CS are outlined in the form of the 3-D plot. The whole calculation process clearly shows that Yang's special function method is simple and effective, and can be applied to investigate the exact ND solutions of the other local fractional PDEs. [ABSTRACT FROM AUTHOR]

The Boussinesq–Kadomtsev–Petviashvili-like model is a famous wave equation which is used to describe the shallow water waves in ocean beaches and lakes. When shallow water waves propagate in microgravity or with unsmooth boundaries, the Boussinesq–Kadomtsev–Petviashvili-like model is modified into its fractal model by the local fractional derivative (LFD). In this paper, we mainly study the fractal Boussinesq–Kadomtsev–Petviashvili-like model (FBKPLM) based on the LFD on Cantor sets. Two efficient and reliable mathematical approaches are successfully implemented to obtain the different types of fractal traveling wave solutions of the FBKPLM, which are fractal variational method (FVM) and fractal Yang wave method (FYWM). Finally, some three-dimensional (3D) simulation graphs are employed to elaborate the properties of the fractal traveling wave solutions. [ABSTRACT FROM AUTHOR]

In this paper, the fractal Zakharov–Kuznetsov–Benjamin–Bona–Mahony model (FZKBBM) is studied based on the local fractional derivative sense on Cantor sets for the first time. The different types of traveling wave solutions of the FZKBBM are successfully obtained by using two reliable and efficient approaches, which are fractal Yang wave method (FYWM) and fractal variational method (FVM). The properties of the obtained traveling wave solutions of non-differential type are elaborated by using some three-dimensional simulation graphs. [ABSTRACT FROM AUTHOR]

In this paper, we aim to discuss a fractional complex Ginzburg–Landau equation by using the parabolic law and the law of weak non-local nonlinearity. Then, we derive dynamic behaviors of the given model under certain parameter regions by employing the planar dynamical system theory. Further, we apply the ansatz method to derive soliton, bright and kinked solitons and verify their existence by imposing certain conditions. In addition, we integrate our solutions in appropriate dimensions to explain their behavior at various groups of parameters. Moreover, we compare the graphical representations of the established solutions at different fractional derivatives and illustrate the impact of the fractional derivative on the investigated soliton solutions as well. [ABSTRACT FROM AUTHOR]

In this paper, we present a computational algorithm, namely, local fractional natural homotopy analysis method (LFNHAM) to explore the solutions of local fractional coupled Helmholtz and local fractional coupled Burgers' equations (LFCHEs and LFCBEs). This work also investigates the uniqueness and convergence of the solution of a general local fractional partial differential equation (LFPDE) obtained by the suggested method in view of theory of fixed point and Banach spaces. Furthermore, the error analysis of the LFNHAM solution is also discussed. Moreover, the numerical simulations are presented for each of the local fractional coupled equations on the Cantor set. The computational procedure clearly illustrates the validity and reliability of the proposed method for achieving the solutions of local fractional coupled Helmholtz and coupled Burgers' equations. The proposed method also minimizes the computational work unlike other conventional methods while still giving extremely precise results. The implemented combination supplies a more general solution as compared to other methods and assimilates their consequences as a special case. In addition, the acquired solutions are also in excellent match with previously determined solutions.

Abstract In this paper, we present a generalized Jensen-type inequality for generalized harmonically convex function on the fractal sets, and a generalized Jensen–Mercer inequality involving local fractional integrals is obtained. Moreover, we establish some generalized Jensen–Mercer-type local fractional integral inequalities for harmonically convex function. Also, we obtain some generalized related results using these inequalities on the fractal space. Finally, we give applications of generalized means and probability density function.

In this paper, we investigate the Riemann wave model (RWM) on Cantor sets by using the local fractional derivative (LFD). A novel computational approach is provided to seek the exact traveling-wave solution of the non-differential type for the local fractional Riemann wave model (LFRWM). The proposed scheme is called local fractional traveling-wave method (LFTWM). An example is given to illustrate that the LFTWM is simple and excellent. The properties of the obtained traveling-wave solutions are elaborated by some 3D graphs. The LFTWM sheds a new light on solving the local fractional wave equations (LFWE) in physics and engineering. [ABSTRACT FROM AUTHOR]

The various aspects of differential calculus are always on the path to progress and excellence, and these trends have been more highlighted in recent decades. More specifically, tremendous advances have been made in the field of fractional calculus. One of the main branches in this field is the local fractional derivative, which has been used successfully to describe many real‐world phenomena in science, and engineering. The present contribution aims to adopt a newly proposed analytical technique to construct exact solutions to local fractional Schamel's equation defined on Cantor sets. To this end, a set of elementary functions are constructed on Cantor sets is defined. Furthermore, the combination of these modified functions is utilized to constitute a formal representation of the searched exact solution for the equation. Numerical simulations related to some of the obtained solutions are also included. The acquired results confirm that the method used is not only very straightforward but also efficient in terms of application. In order to perform complicated and tedious calculations while solving this problem, the use of a symbolic computational packages in Maple or Mathematica is inevitable. The work emphasizes the power of the employed method in providing various exact solutions to different physical problems involving local fractional derivatives. [ABSTRACT FROM AUTHOR]

This paper derives a new fractional Fokas system with the aid of the local fractional derivative. A novel technology named the extended rational fractal sine–cosine method is presented for the first time ever to develop the abundant traveling wave solutions. Two families (six sets) of the traveling wave solutions on Cantor set are successfully constructed. The dynamic behaviors of the solutions on Cantor sets are provided via numerical simulations. The obtained results in this work strongly prove that the proposed approach is simple but effective, which is expected to shed a new light on the study of the traveling wave theory of the local fractional equations. [ABSTRACT FROM AUTHOR]

In this work, the Kadomtsov–Petviashvili–Benjamin–Bona–Mahony model is described by the local fractional derivative (LFD) on Cantor sets. A novel algorithm is presented to seek the exact traveling wave solution of the nondifferentiable type for the local fractional Kadomtsov–Petviashvili–Benjamin–Bona–Mahony model based on the variational theory, which is called variational wave transform method (VWTM). This new algorithm provides a new idea for seeking the exact traveling wave solutions in fractal space with simplicity and efficiency. The physical properties of traveling wave solutions are described by some 3D simulation figures. [ABSTRACT FROM AUTHOR]