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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 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]

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.

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]

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, 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]

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 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]

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, the initial value problem is discussed for the local fractional Caudrey-Dodd-Gibbon-Kaeada equation. The fractional complex transform and the new iterative method are used to solve the problem, and the approximate analytical solutions are obtained. [ABSTRACT FROM AUTHOR]

The local fractional variational iteration method is applied to a modified Fisher's equation defined on Cantor sets with the fractal conditions. The solution process is simple, and the accuracy of the approximate solution is high. The method provides an unrivaled tool for local differential equations. [ABSTRACT FROM AUTHOR]

In order to solve the local fractional differential equations, we couple the fractional residual method with the Adomian decomposition method via the local fractional calculus operator. Several examples are given to illustrate the solution process and the reliability of the method. [ABSTRACT FROM AUTHOR]

This article investigates the local fractional generalized Kadomtsev–Petviashvili equation and the local fractional Kadomtsev–Petviashvili-modified equal width equation. It presents traveling-wave transformation in a nondifferentiable type for the governing equations, which translate them into local fractional ordinary differential equations. It also investigates nondifferentiable traveling-wave solutions for certain proposed models, using an ansatz method based on some generalized functions defined on fractal sets. Several interesting graphical representations as 2D, 3D, and contour plots at some selected parameters are presented, by considering the integer and fractional derivative orders to illustrate the physical naturality of the inferred solutions. Further results are also introduced in some details. [ABSTRACT FROM AUTHOR]

In this paper, we study the space-time Hölder continuity of the solution to the following nonlinear time-fractional slow and fast diffusion equation: ∂ β + ν 2 (- Δ) α / 2 u (t , x) = I t γ σ u (t , x) W ˙ (t , x) , t > 0 , x ∈ R d , where W ˙ is the space-time white noise, α ∈ (0 , 2 ] , β ∈ (0 , 2) , γ ≥ 0 and ν > 0 . The existence/uniqueness of a random field solution has been obtained in [9] under the condition that 2 (β + γ) - 1 - d β / α > 0 . The Hölder regularity of the solution has been obtained in the same reference, but only for the case β + γ ≤ 1 . In this paper, we use the idea from the local fractional derivative to establish the Hölder regularity of the solution for all possible cases – β ∈ (0 , 2) , which in particular recovers the special case in [9] when β ∈ (0 , 1 - γ ] . As a rather surprising consequence, when γ = 0 , α = 2 and β is close to 2, the space and time Hölder exponents are both to 1 - , which is different from the known Hölder exponents for the stochastic wave equation which are (1 / 2) - . [ABSTRACT FROM AUTHOR]

In this paper, the nonlinear system of local fractional partial differential equations is solved via local fractional Elzaki transform decomposition method. The local fractional Elzaki decomposition transform method combines a local fractional Elzaki transform and the Adomian decomposition method. Applications related to the nonlinear system of local fractional partial differential equations are presented. [ABSTRACT FROM AUTHOR]

Local fractional calculus has gained wide attention in the field of circuit design. In this paper, we propose the zero-input response(ZIR) of fractal RC circuit modeled by local fractional derivative(LFD) for the first time. With help of the law of switch and the Kirchhoff Voltage Laws, the transient local fractional ordinary differential equation is established, and the corresponding exact solution behavior defined on Cantor sets is presented. What we found especially interesting was that the fractal RC becomes the ordinary one in the particular case κ = 1. The results obtained in this paper reveal that the local fractional calculus is a powerful tool to analyze the fractal circuit systems.

As an important electronic device, filter is applied to all kinds of electronic products. In this paper, a new ℘-order R-L High-pass filter (HPF) modeled by the local fractional derivative (LFD) is proposed for the first time. With the help of the local fractional Laplace transform (LFLT), we obtain the non-differentiable(ND) transfer function, and present the expressions of ND amplitude-frequency characteristic (AFC) and ND phase-frequency characteristics (PFC). The corresponding parameters and properties of the ℘-order R-L HPF are also studied. What’s interesting is that the ℘-order R-L HPF becomesthe ordinary one in the exceptional case at ℘ = 1. The obtained results in this paper reveal the sufficiency of the local fractional derivative for analyzing the circuit systems in fractal space.

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. [ABSTRACT FROM AUTHOR]

In this research, we discuss the construction of the analytic solution of the homogenous initial boundary value problem including partial differential equations of fractional order. Since the homogenous initial boundary value problem involves a local fractional order derivative, it has classical initial and boundary conditions. By means of separation of the variables method and the inner product defined on L2 [0, l], the solution is constructed in the form of a Fourier series including the exponential function. The illustrative examples present the applicability and influence of the separation of variables method on time fractional diffusion problems. Moreover, as the fractional order α tends to 1, the solution of the fractional diffusion problem tends to the solution of the diffusion problem which proves the accuracy of the solution. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we investigate a 3-D diffusion equation within the scope of the local fractional derivative. For this model, we establish local fractional vector operators and a local fractional Laplace operator defined on Cantor-type cylindrical coordinate and Cantor-type spherical coordinate, respectively. With the help of the spherical symmetry method based on those operators, we obtain exact traveling wave solutions of the 3-D diffusion equation. The results reveal that the proposed schemes are very effective for obtaining nondifferentiable solutions of fractional diffusion problems.

The local fractional derivative (LFD) has attracted wide attention in the field of engineering application. In this paper, the LFD is used to model the fractional Sallen-Key filter for the first time. The non-differentiable(ND) transfer function is obtained by using the local fractional Laplace transform(LFLT). And the amplitude frequency response is analyzed in detail for different fractional order $\varsigma $ . It is found that the fractional Sallen-Key filter becomes the ordinary one in the special case $\varsigma =1$ . The obtained results of this paper show the powerful ability of local fractional calculus in the analysis of complex problems arising in engineering fields.

The local fractional derivative has gained more and more attention in the field of fractal electrical circuits. In this paper, we propose a new ζ-order RLC** resonant circuit described by the local fractional derivative for the first time. By studying the non-differentiable lumped elements, the non-differentiable equivalent impedance is obtained with the help of the local fractional Laplace transform. Then the non-differentiable resonant angular frequency is studied and the non-differentiable resonant characteristic is analyzed with different input signals and parameters, where it is observed that the ζ-order RLC resonant circuit becomes the ordinary one for the special case when the fractional order ζ = 1. The obtained results show that the local fractional derivative is a powerful tool in the description of fractal circuit systems. [ABSTRACT FROM AUTHOR]

The linear and non-linear Klein-Gordon equations are considered. The fractional complex transform is used to convert the equations on a continuous space/time to fractals ones on Cantor sets, the resultant equations are solved by local fractional reduced differential transform method. Three examples are given to show the effectiveness of the technology. [ABSTRACT FROM AUTHOR]

In present work, nonlinear fractional partial differential equations namely transport equation and Fokker-Planck equation involving local fractional differential operators, are investigated by means of the local fractional homotopy perturbation Sumudu transform method. The proposed method is a coupling of homotopy perturbation method with local fractional Sumudu transform and is used to describe the non-differentiable problems. Numerical simulation results are projected to show the efficiency of the proposed technique.

Local fractional calculus has gained wide attention in the field of circuit design. In this paper, we propose the zero-input response(ZIR) of fractal RC circuit modeled by local fractional derivative(LFD) for the first time. With help of the law of switch and the Kirchhoff Voltage Laws, the transient local fractional ordinary differential equation is established, and the corresponding exact solution behavior defined on Cantor sets is presented. What we found especially interesting was that the fractal RC becomes the ordinary one in the particular case κ = 1. The results obtained in this paper reveal that the local fractional calculus is a powerful tool to analyze the fractal circuit systems. [ABSTRACT FROM AUTHOR]

In this paper, the non-linear local fractional Bratu-type equation is described by the local fractional derivative in a fractal space, and its variational formulation is successfully established according to semi-inverse transform method. Finally, we find the approximate analytical solution of the local fractional Bratu-type equation by using Adomina decomposition method. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the approximate analytical solution of variable coefficients modified Korteweg-de Vries equation with local fractional derivative by using new iterative method. [ABSTRACT FROM AUTHOR]

As an important electronic device, filter is applied to all kinds of electronic products. In this paper, a new ℘ -order R-L High-pass filter (HPF) modeled by the local fractional derivative (LFD) is proposed for the first time. With the help of the local fractional Laplace transform (LFLT), we obtain the non-differentiable(ND) transfer function, and present the expressions of ND amplitude-frequency characteristic (AFC) and ND phase-frequency characteristics (PFC). The corresponding parameters and properties of the ℘ -order R-L HPF are also studied. What's interesting is that the ℘ -order R-L HPF becomesthe ordinary one in the exceptional case at ℘ = 1. The obtained results in this paper reveal the sufficiency of the local fractional derivative for analyzing the circuit systems in fractal space. [ABSTRACT FROM AUTHOR]

In order to make the local variational iteration algorithm converge faster and more effective, the Sumudu transform is adopted and a proper initial solution is chosen. Some examples are given to show that the presented method is reliable, efficient and easy to implement from a computational viewpoint. [ABSTRACT FROM AUTHOR]

In this paper, a formula of the local fractional partial derivative of inverse matrix is presented and proved. With the help of the derived formula, two new non-linear PDE are derived including the local fractional non-isospectral self-dual Yang- Mills equation and the local fractional principal chiral field equation. It is shown that the formula of the local fractional partial derivative of inverse matrix can be used to derive some other local fractional non-linear PDE in soliton theory. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a local fractional PDE with Laguerre type derivative. The considered equation represents a general extension of the classical heat and wave equations. The method of separation of variables is used to solve the differential equation defined in a bounded domain. [ABSTRACT FROM AUTHOR]

This paper proposes a new method to solve local fractional differential equation. The method divides the studied equation into a system, where the initial solution is obtained from a residual equation. The new method is therefore named as the fractional residual method. Examples are given to elucidate its efficiency and reliability. [ABSTRACT FROM AUTHOR]

In this paper, we define a new type of string stability based on Mittag-Leffler function so-called Mittag-Leffler (p\alpha)-string stability. This kind of stability for a class of singularly perturbed stochastic systems of fractional order will be considered. The fractional derivative in these systems is in the local sense. String stability indicates uniform boundedness of the interconnected system if the initial cases of interconnected system be uniformly bounded. The deduction of the sufficient conditions of stability is based on a mixture of the concept of the Mittag-Leffler stability with the notion of p-mean string stability of singularly perturbed stochastic systems. In the sequel, our purpose is to investigate the full order system in their lower order subsystems, i.e., the reduced order system and the boundary layer correction.

In this paper, the periodic solution for the local fractional Boussinesq equation can be obtained in the sense of the local fractional derivative. It is given by applying direct integration with symmetry condition. Meanwhile, the periodic solution of the non-differentiable type with generalized functions defined on Cantor sets is analyzed. As a result, we have a new point to look the local fractional Boussinesq equation through the local fractional derivative theory. [ABSTRACT FROM AUTHOR]

The present paper tends to define a new type of string stability based on Mittag-Leer function that is called (pff)-string stability. This kind of stability will be considered for a class of singularly perturbed stochastic systems of fractional order. The fractional derivative in these systems is situated in the local sense. String stability indicates uniform boundedness of the interconnected system, if the initial cases of interconnected system be uniformly bounded. The deduction of the sufficient conditions of stability is based on a mixture of the concept of the Mittag-Leer stability with the notion of p-mean string stability of singularly perturbed stochastic systems. In this sense the objective, it is argued, is to investigate the full order system in their lower order subsystems, i.e., the reduced order system and the boundary layer correction. [ABSTRACT FROM AUTHOR]

In this paper, the modified Fornberg-Whitham equation is described by the local fractional derivative for the first time. The fractal complex transform and the modified reduced differential transform method are successfully adopted to solve the modified local Fornberg-Whitham equation defined on fractal sets. [ABSTRACT FROM AUTHOR]

In this article, the local fractional Homotopy perturbation method is utilized to solve the non-homogeneous heat conduction equations. The operator is considered in the sense of the local fractional differential operator. Comparative results between non-homogeneous and homogeneous heat conduction equations are presented. The obtained result shows the non-differentiable behavior of heat conduction of the fractal temperature field in homogeneous media.

This paper proposes a new method called the local fractional Fourier sine transform to solve fractional differential equations on a fractal space. The method takes full advantages of the Yang-Fourier transform, the local fractional Fourier cosine, and sine transforms. A 1-D local fractional heat transfer equation is used as an example to reveal the merits of the new technology, and the example can be used as a paradigm for other applications. [ABSTRACT FROM AUTHOR]

In this paper, local fractional cylindrical wave solutions on Signorini hyperelastic materials are studied. In particular, we focus on the so-called Signorini potential. Cantor-type cylindrical coordinates are used to analyze, both from dynamical and geometrical point of view, wave solutions, so that the nonlinear fundamental equations of the fractional model are explicitly given. In the special case of linear approximation we explicitly compute the fractional wave profile.

In this short note we present a new general definition of local fractional derivative, that depends on an unknown kernel. For some appropriate choices of the kernel we obtain some known cases. We establish a relation between this new concept and ordinary differentiation. Using such formula, most of the fundamental properties of the fractional derivative can be derived directly.

The variational iteration method was originally proposed to solve non-linear problems of differential equations, this paper shows that it is also a powerful mathematical tool to local fractional differential equations. Two local fractional modified Korteweg-de Vries equations are used as examples to reveal the simple solution process. [ABSTRACT FROM AUTHOR]

We apply the new local fractional reduced differential transform method to obtain the solutions of some linear and nonlinear partial differential equations on Cantor set. The reported results are compared with the related solutions presented in the literature and the graphs are plotted to show their behaviors. The results prove that the presented method is faster and easy to apply. [ABSTRACT FROM AUTHOR]