Sense Of Conformable Derivative Research Articles

Nonlinear dynamic wave characteristics of optical soliton solutions in ion-acoustic wave

Traveling wave solutions are utilized to depict reaction-diffusion and analyze electrical signal transmission and propagation in space-time nonlinear fractional order partial differential equations like the space-time fractional Telegraph and Kolmogorov-Petrovsky Piskunov equations, and that are used in physical science to model combustion, biological research to model nerve impulse propagation, chemical dynamics to model concentration in order wave propagation, and plasma to model the progression of a set of duffing oscillators. In this study, the new generalized (G′/G)−expansion technique was employed to construct some novel and more universal closed-form traveling wave solutions in the sense of conformable derivatives which explain the above-stated phenomena properly. By utilizing complex fractional transformation, the ordinary differential equations are generated from fractional order differential equations. The recommended technique allowed us to produce some dynamical wave patterns of kink, single soliton, compacton, periodic shape, multiple periodic waves, anti-kink, and other structures are developed, which are shown using 3D plots and contour plots to illustrate the physical layout clearly. The traveling waveform responses can be defined in terms of functions based on trigonometry, hyperbolic operations, and rational functions and that are quick, flexible, and simple to reproduce. Furthermore, the obtained closed-form solutions for nonlinear fractional evolution equations make stability analysis and accuracy comparison amongst numerical solvers easier, which asserts that the new generalized (G′/G)−expansion technique is one of the most proficient and effective approaches.

On Partial Exact Controllability of Fractional Control Systems in Conformable Sense

In this work, we investigate the partial exact controllability of fractional semilinear control systems in the sense of conformable derivatives. Initially, we establish the existence and uniqueness of the mild solution for this type of fractional control systems. Then, by employing a contraction mapping principle, we obtain sufficient conditions for the conformable fractional semilinear system to be partially exactly controllable, assuming that its associated linear part is partially exactly controllable. To demonstrate the efficacy of the theoretical findings, a typical example is provided at the end.

Well-posedness and Ulam-Hyers stability results of solutions to pantograph fractional stochastic differential equations in the sense of conformable derivatives

<abstract><p>One kind of stochastic delay differential equation in which the delay term is dependent on a proportion of the current time is the pantograph stochastic differential equation. Electric current collection, nonlinear dynamics, quantum mechanics, and electrodynamics are among the phenomena modeled using this equation. A key idea in physics and mathematics is the well-posedness of a differential equation, which guarantees that the solution to the problem exists and is a unique and meaningful solution that relies continuously on the initial condition and the value of the fractional derivative. Ulam-Hyers stability is a property of equations that states that if a function is approximately satisfying the equation, then there exists an exact solution that is close to the function. Inspired by these findings, in this research work, we established the Ulam-Hyers stability and well-posedness of solutions of pantograph fractional stochastic differential equations (PFSDEs) in the framework of conformable derivatives. In addition, we provided examples to analyze the theoretical results.</p></abstract>

On fractional order computational solutions of low-pass electrical transmission line model with the sense of conformable derivative

In this study, we solve fractional order PDEs with free parameters that regulate wave motion in a low-pass electrical transmission line equation (LPET) using the unified technique. To convert the governing equation into an ordinary differential equation (ODE), we have used a simple linear fractional transformation. The unified technique can therefore be used to find various single and extracting wave solutions to the equation of attention. Using Maple-18 software, all of the established solutions are validated and verified. By selecting appropriate values for the fractional order and free parameters, the two dimensional (2D) and three dimensional (3D) profiles of some of the assimilation solutions demonstrating kink, singular kink, cross kink, bright-dark kink, lump type kink, anti-kink, dark-anti kink, singular periodic, periodic, singular soliton solutions are displayed. The effect of free parameters and fractionality on the dynamical properties of the precise and exploited solutions are graphically illustrated and discussed in detail. We have also compared our results to those found in the literature review. Wherever possible, the judgment demonstrates no significant differences between our realized and published solutions. The results of this investigation show that the adopting technique is both productive and powerful in extracting wave solutions to nonlinear evolution problems that arise in the areas of applied mathematics, mathematical physics, and engineering.

On Survey of the Some Wave Solutions of the Non-Linear Schrödinger Equation (NLSE) in Infinite Water Depth

In this work, we use two different analytic schemes which are the Sine-Gordon expansion technique and the modified exp -expansion function technique to construct novel exact solutions of the non-linear Schrödinger equation, describing gravity waves in infinite deep water, in the sense of conformable derivative. After getting various travelling wave solutions, we plot 3D, 2D and contour surfaces to present behaviours obtained exact solutions.

Fractal soliton solutions for the fractal-fractional shallow water wave equation arising in ocean engineering

The generalized shallow water wave equation is an important mathematical model that is used to elaborate ocean engineering, weather simulations, tsunami prediction and tidal currents. In this work, the generalized (3 + 1)-dimensional fractal-fractional shallow water wave equation (FFSWWE) is investigated where fractal-fractional derivative is taken in the conformable derivative sense. Some new fractal soliton solutions of FFSWWE are successfully derived by the fractal-fractional variational wave method (FFVWM), which is a new mathematical technology. This new method has the advantages of being simple, efficient, and direct. The 3D graphics that describe these new fractal soliton solutions that were obtained are tremendously important for improving our understanding of physical oceanography.

The Fractional Series Solutions for the Conformable Time-Fractional Swift-Hohenberg Equation through the Conformable Shehu Daftardar-Jafari Approach with Comparative Analysis

The major objective of this study is to derive fractional series solutions of the time-fractional Swift-Hohenberg equations (TFSHEs) in the sense of conformable derivative using the conformable Shehu transform (CST) and the Daftardar-Jafari approach (DJA). We call it the conformable Shehu Daftardar-Jafari approach (CSDJA). One of the universal equations used in the description of pattern formation in spatially extended dissipative systems is the Swift-Hohenberg equation. To assess the effectiveness and consistency of the suggested approach, the numerical results are compared with those obtained by the Elzaki decomposition method (EDM) in the sense of relative and absolute error functions, proving that the CSDJA is an effective substitute for techniques that use He’s or Adomian polynomials to solve TFSHEs. The transition from the imprecise solution to the precise solution at various values of fractional-order derivatives is shown using the recurrence error function. Furthermore, the exact and approximative solutions are compared using 2D and 3D graphics and also numerically in the form of relative and absolute error functions. The results show that the procedure is quick, precise, and easy to implement, and it yields outstanding results. The recommended approach’s strength, which gives it an advantage over the Adomian decomposition and homotopy perturbation methods, is its algorithm for dealing with nonlinear problems without the use of Adomian polynomials or He’s polynomials. The advantage of this method is that it does not make any assumptions about physical parameters. As a result, it can be used to solve both weakly and strongly nonlinear problems and circumvent some of the drawbacks of perturbation techniques.

Dynamical Analysis of Nonlocalized Wave Solutions of 2 + 1 -Dimensional CBS and RLW Equation with the Impact of Fractionality and Free Parameters

This study retrieves some new exact solutions to the 2 + 1 -dimensional Calogero-Bogoyavlenskii-Schilf (CSB) equation and regularized long wave (RLW) equation in the context of nonlinear traveling wave phenomena. In this regard, the advanced exp − φ ξ -expansion method is imposed to the 2 + 1 -dimensional CBS and RLW equation, and consequently, rogue, kink, singular kink, periodic, singular, and multiple soliton solutions are exhibited in terms of trigonometric, hyperbolic, and rational function solutions. To enucleate the underlying nonlocalized traveling wave features, accomplished exact solutions are established by making their dynamic behavior of the exact solutions exhibited in three-dimensional (3D) and two-dimensional (2D) combined chart with the help of computational software Maple 18. All of our accomplished solutions are claimed to be new in the sense of conformable derivative, chosen a unique fractional type wave transformation, dynamical behavior of fractionally and free variable, and the imposed method on our preferred equations.

Analysis of some new wave solutions of DNA-Peyrard–Bishop equation via mathematical method

The main goal of this paper is to extract the closed-form wave solutions by means of [Formula: see text]-expansion method to the DNA flow known as the Peyrard–Bishop model which describes the nonlinear interaction between neighboring displacements and hydrogen bonds. The aforementioned model is investigated for the time-fractional order and the fractional derivatives are used in the sense of conformable derivatives. The projected method provides additional generic and comprehensive wave solutions incorporated with physical parameters whose specific values reveal different shapes of the waveform such as periodic soliton, singular kink soliton, singular anti-kink soliton, and other different types of solitons. The obtained soliton solutions are novel, advanced, distinct, and relevant based on the available information in the literature which may be applied to more complex phenomena. The three-dimensional, two-dimensional, and contour plots of some of the attained results are sketched by assigning individual values of the parameters to scrutinize the dynamic behavior of the waves.

Analytical behavior of soliton solutions to the couple type fractional-order nonlinear evolution equations utilizing a novel technique

Shallow water waves are one of the most prominent and widely used waves in coastal areas. The anomalous bi-directional transmission of long waves over shallow water is described by the nonlinear fractional partial differential equations (NLFPDEs), namely space-time-fractional coupled Whitham-Broer-Kaup (WBK) and coupled approximate long water (ALW) wave equations. To achieve new analytical solutions, the enhanced extended tanh-function technique for the mentioned fractional equations in the sense of conformable derivative is used. Some well-known wave shapes of solitons, namely, kink, bell-shape, and other types, through the suggested technique are achieved, and the 3D, contour, list point plots, and vector plots of the solutions are sketched to further illustrate the wave profile. The results achieved in this study have been validated using the computational software Maple and found to be accurate. It is observed that the method is realistic, and dependable to investigate more generalized analytical wave solutions.

An Effective New Iterative Method to Solve Conformable Cauchy Reaction-Diffusion Equation via the Shehu Transform

For the first time, we establish a new procedure by using the conformable Shehu transform (CST) and an iteration method for solving fractional-order Cauchy reaction-diffusion equations (CRDEs) in the sense of conformable derivative (CD). We call this recommended method the conformable Shehu transform iterative method (CSTIM). To evaluate the efficacy and consistency of CSTIM for conformable partial differential equations (PDEs), the absolute errors of four CRDEs are reviewed graphically and numerically. Furthermore, graphical significances are correspondingly predicted for several values of fractional-order derivatives. The results and examples establish that our new method is unpretentious, accurate, valid, and capable. CSTIM does not necessarily use He’s polynomials and Adomian polynomials when solving nonlinear problems, so it has a strong advantage over the homotopy analysis and Adomian decomposition methods. The convergence and absolute error analysis of the series solutions is also offered.

New soliton solutions for the space-time fractional modified third order Korteweg–de Vries equation

In present research work, the soliton solutions of the space-time fractional modified third order Korteweg–de Vries (KdV) equation are explored by Sardar-subequation method (SSM). The (KdV) equation has an established version based on the shallow water waves in the oceans, oceanography theory and the ion-acoustic waves in plasma. The fractional order partial differential equation (PDE) is transform to a non-linear ordinary differential equation (ODE) by using traveling wave transformation. It is noted that in this work the fractional differential equation is solved in the sense of conformable derivative. Then soliton solutions of reduced equation are established by SSM. The present technique provides bright, dark, singular, periodic singular, combined bright-dark and combined dark-singular soliton solutions.

Mathematical analysis and numerical simulation of the Ebola epidemic disease in the sense of conformable derivative

The aim of this paper is to construct and analyze a mathematical model for the response of T-cytotoxic lymphocytes to the Ebola virus using the Herz-tuckwill model with nonlinear Conformable order differential equations which derive from real biological data, then study the global stability of the equilibria using an appropriate Lyapunov function and the LaSalle invariance principle. Furthermore, we demonstrate the impact of the non-integer order of the model compared with the integral order. Finally consider a numerical simulation that justifies the biological hypotheses and the theory results. The numerical results provided that, the non-integer order has a great impact on the treatments of Ebola virus in this model.

Nonlinear dynamical study to time fractional Dullian–Gottwald–Holm model of shallow water waves

This paper studies the exact traveling wave solutions to the nonlinear Dullin–Gottwald–Holm model which has the application in shallow-water waves in which the fractional derivative is considered in the sense of conformable derivative. Diverse exact solutions in hyperbolic, trigonometric and plane wave forms are obtained using two integration norms. For this purpose [Formula: see text]-expansion method and reccati mapping techniques are used. The 3D plots and their corresponding contour graphs are also depicted. Being concise and straightforward, the calculations demonstrate the effectiveness and convenience of the method for solving other nonlinear partial differential equations.

Analytical Solutions for the Nonlinear Partial Differential Equations Using the Conformable Triple Laplace Transform Decomposition Method

In this paper, a novel analytical method for solving nonlinear partial differential equations is studied. This method is known as triple Laplace transform decomposition method. This method is generalized in the sense of conformable derivative. Important results and theorems concerning this method are discussed. A new algorithm is proposed to solve linear and nonlinear partial differential equations in three dimensions. Moreover, some examples are provided to verify the performance of the proposed algorithm. This method presents a wide applicability to solve nonlinear partial differential equations in the sense of conformable derivative.

New Exact Soliton Solutions of the ( 3 + 1 )-Dimensional Conformable Wazwaz–Benjamin–Bona–Mahony Equation via Two Novel Techniques

In this work, the ( 3 + 1 )-dimensional Wazwaz–Benjamin–Bona–Mahony equation is formulated in the sense of conformable derivative. Two novel methods of generalized Kudryashov and exp − φ ℵ are investigated to obtain various exact soliton solutions. All algebraic computations are done with the help of the Maple software. Graphical representations are provided in 3D and 2D profiles to show the behavior and dynamics of all obtained solutions at various parameters’ values and conformable orders using Wolfram Mathematica.

Solitary and periodic wave solutions to the family of new 3D fractional WBBM equations in mathematical physics

For the newly implemented 3D fractional Wazwaz-Benjamin-Bona-Mahony (WBBM) equation family, the present study explores the exact singular, solitary, and periodic singular wave solutions via the (G′/G2)-expansion process. In the sense of conformable derivatives, the equations considered are translated into ordinary differential equations. In spite with many trigonometric, complex hyperbolic, and rational functions, some fresh exact singular, solitary, and periodic wave solutions to the deliberated equations in fractional systems are attained by the implementation of the (G′/G2)-expansion technique through the computational software Mathematica. The unique solutions derived by the process defined are articulated with the arrangement of the functions tanh, sech; tan, sec; coth, cosech, and cot, cosec. With three-dimensional (3D), two dimensional (2D) and contour graphics, some of the latest solutions created have been envisaged, selecting appropriate arbitrary constraints to illustrate their physical representation. The outcomes were obtained to determine the power of the completed technique to calculate the exact solutions of the equations of the WBBM that can be used to apply the nonlinear water model in the ocean and coastal engineering. All the solutions given have been certified by replacing their corresponding equations with the computational software Mathematica.

Exact and solitary wave solutions of conformable time fractional Clannish Random Walker’s Parabolic and Ablowitz-Kaup-Newell-Segur equations via modified mathematical methods

The nonlinear fractional differential equations (FDEs) composed by mathematical modeling through nonlinear corporeal structures. The study of these kind models has an energetic position in different field of applied sciences. The dominant intension of this article is to extract novel analytical wave solutions of the conformable time fractional Clannish Random Walker’s Parabolic (CRWP) and (2 + 1)-dimensional fractional Ablowitz-Kaup-Newell-Segur (AKNS) equations in the sense of conformable derivative with the aid of modified mathematical methods, called modified extended auxiliary equation mapping and modified F-expansion schemes. By the virtue of employed techniques, different types of solutions are obtained in the form of trigonometric, hyperbolic, exponential and rational functions respectively. To prompt the essential propagated features, some investigated solutions are exhibited in form of 3D and 2D is planned by passing on the precise values to the parameters under the constrain conditions. The accomplished solutions show that these presented schemes are reliable, applicable and efficient.

Periodic and solitary wave solutions to a family of new 3D fractional WBBM equations using the two-variable method

For the newly implemented 3D fractional Wazwaz–Benjamin–Bona–Mahony​ (WBBM) equation family, the present study explores exact singular, solitary, and periodic singular wave solutions via the (G′∕G,1∕G)-expansion process. In the sense of conformable derivatives, the equations considered are transformed into ordinary differential equations. In spite of many trigonometric, complex hyperbolic, and rational functions, some fresh exact singular, solitary, and periodic wave solutions to the deliberated equations in fractional systems are attained by the implementation of the (G′∕G,1∕G)-expansion technique through the computational software Mathematica. The unique solutions derived by the process defined are articulated with the arrangement of the functions tanh, sech; tan, sec; coth, csch, and cot, csc. With three-dimensional graphics, some of the latest solutions created have been envisaged, by selecting appropriate arbitrary constraints to illustrate their physical representation. The outcomes which are obtained to show the power of the computational technique for the WBBM equations can be applied to other nonlinear water model equations in ocean and coastal engineering. All the obtained solutions have been verified by the computational software Mathematica.

Comparative study of two techniques on some nonlinear problems based ussing conformable derivative

AbstractIn this paper, three eminent types of time-fractional nonlinear partial differential equations are considered, which are the fractional foam drainage equation, fractional Gardner equation, and fractional Fornberg–Whitham equation in the sense of conformable fractional derivative. The approximate solutions of these considered problems are constructed and discussed using the conformable fractional variational iteration method and conformable fractional reduced differential transform method. The conformable derivative is one of the admirable choices to handle nonlinear physical problems of different fields of interest. Comparisons of approximate solution obtained by two techniques, to each other and with the exact solutions are also presented and affirm that the considered methods are efficient and reliable techniques to study other nonlinear fractional equations and models in the sense of conformable derivative. To explain the effects of several parameters and variables on the movement, the approximate results are shown in tables and two-and three-dimensional surface graphs.