229 research outputs found
Conformable Derivative Operator in Modelling Neuronal Dynamics
This study presents two new numerical techniques for solving time-fractional one-dimensional cable differential equation (FCE) modeling neuronal dynamics. We have introduced new formulations for the approximate-analytical solution of the FCE by using modified homotopy perturbation method defined with conformable operator (MHPMC) and reduced differential transform method defined with conformable operator (RDTMC), which are derived the solutions for linear-nonlinear fractional PDEs. In order to show the efficiencies of these methods, we have compared the numerical and exact solutions of fractional neuronal dynamics problem. Moreover, we have declared that the proposed models are very accurate and illustrative techniques in determining to approximate-analytical solutions for the PDEs of fractional order in conformable sense
NOVEL METHODS FOR SOLVING THE CONFORMABLE WAVE EQUATION
In this paper, a two-dimensional conformable fractional wave equation describing a circular membrane undergoing axisymmetric vibrations is formulated. It was found that the analytical solutions of the fractional wave equation using the conformable fractional formulation can be easily and efficiently obtained using separation of variables and double Laplace transform methods. These solutions are compared with the approximate solution obtained using the differential transform method for certain cases
Solution of Multi-order Fractional Differential Equation Based on Conformable Derivative by Shifted Legendre Polynomial
The aim of this article is the way for finding approximation solution of multi-order fractional differential equation with conformable sense with use approximated function by shifted Legendre polynomial, the method is easy and powerful for get our results of the linear and non-linear equation, the background idea behind this method is finding system of algebra after achieving messing variable is that mean obtain approximate solution, a few examples illustrates for presented how much our method is capable
Approximate and Exact Solutions in the Sense of Conformable Derivatives of Quantum Mechanics Models Using a Novel Algorithm
The entirety of the information regarding a subatomic particle is encoded in a wave function. Solving quantum mechanical models (QMMs) means finding the quantum mechanical wave function. Therefore, great attention has been paid to finding solutions for QMMs. In this study, a novel algorithm that combines the conformable Shehu transform and the Adomian decomposition method is presented that establishes approximate and exact solutions to QMMs in the sense of conformable derivatives with zero and nonzero trapping potentials. This solution algorithm is known as the conformable Shehu transform decomposition method (CSTDM). To evaluate the efficiency of this algorithm, the numerical results in terms of absolute and relative errors were compared with the reduced differential transform and the two-dimensional differential transform methods. The comparison showed excellent agreement with these methods, which means that the CSTDM is a suitable alternative tool to the methods based on the Caputo derivative for the solutions of time-fractional QMMs. The advantage of employing this approach is that, due to the use of the conformable Shehu transform, the pattern between the coefficients of the series solutions makes it simple to obtain the exact solution of both linear and nonlinear problems. Consequently, our approach is quick, accurate, and easy to implement. The convergence, uniqueness, and error analysis of the solution were examined using Banach’s fixed point theory
A New Approach in Solving Regular and Singular Conformable Fractional Coupled Burger's Equations
The conformable double ARA decomposition approach is presented in this
current study to solve one-dimensional regular and singular conformable
functional Burger's equations. We investigate the conformable double ARA
transform's definition, existence requirements, and some basic properties. In
this study, we introduce a novel interesting method that combines the double
ARA transform with Adomian's decomposition method, in order to find the precise
solutions of some nonlinear fractional problems. Moreover, we use the new
approach to solve Burgers' equations for both regular and singular conformable
fractional coupled systems. We also provide several instances to demonstrate
the usefulness of the current study. Mathematica software has been used to get
numerical results
On the modulation instability analysis and deeper properties of the cubic nonlinear Schr¨odinger’s equation with repulsive δ-potential
This projected work applies the generalized exponential rational function method to extract the complex, trigonometric, hyperbolic, dark bright soliton solutions of the cubic nonlinear Schrödinger’s equation. Moreover, trigonometric, complex, strain conditions and dark-bright soliton wave distributions are also reported. Furthermore, the modulation instability analysis is also studied in detail. To better understand the dynamic behavior of some of the obtained solutions, several numerical simulations are presented in the paper. According to the obtained results, it is clear that the method has less limitations than other methods in determining the exact solutions of the equations. Despite the simplicity and ease of use of this method, it has a very powerful performance and is able to introduce a wide range of different types of solutions to such equations. The idea used in this paper is readily applicable to solving other partial differential equations in mathematical physics.Fundación Séneca (Spain), grant 20783/PI/18., and Ministry of Science, Innovation and Universities (Spain), grant PGC2018-097198-B- 100. Moreoer, this projected work was partially (not financial) supported by Harran University with the project HUBAP ID:20124
A New Numerical Technique for Solving Fractional Partial Differential Equations
We propose conformable Adomian decomposition method (CADM) for fractional
partial differential equations (FPDEs). This method is a new Adomian
decomposition method (ADM) based on conformable derivative operator (CDO) to
solve FPDEs. At the same time, conformable reduced differential transform
method (CRDTM) for FPDEs is briely given and a numerical comparison is made
between this method and the newly introduced CADM. In applied science, CADM can
be used as an alternative method to obtain approximate and analytical solutions
for FPDEs as CRDTM. In this study, linear and non-linear three problems are
solved by these two methods. In these methods, the obtained solutions take the
form of a convergent series with easily computable algorithms. For the
applications, the obtained results by these methods are compared to each other
and with the exact solutions. When applied to FPDEs, it is seem that CADM
approach produces easy, fast and reliable solutions as CRDTM
Abundant optical soliton solutions for an integrable (2+1)-dimensional nonlinear conformable Schrödinger system
Abstract The analytical solutions of the integrable generalized ( 2 + 1 ) -dimensional nonlinear conformable Schrodinger (NLCS) system of equations was explored in this paper with the aid of three novel techniques which consist of ( G ′ / G ) -expansion method, generalized Riccati equation mapping method and the Kudryashov method in the conformable sense. We have discovered a new and more general variety of exact traveling wave solutions by using the proposed methods with a variety of soliton solutions of several structures. With several plots illustrating the behavior of dynamic shapes of the solutions, the findings are highly applicable and detailed the physical dynamic of the considered nonlinear system
Applied mathematical modelling with new parameters and applications to some real life problems
Some Epidemic models with fractional derivatives were proved to be well-defined, well-posed and more accurate [34, 51, 116], compared to models with the conventional derivative. An Ebola epidemic model with non-linear transmission is fully analyzed. The model is expressed with the conventional time derivative with a new parameter included,
which happens to be fractional (that derivative is called the derivative). We proved that the model is well-de ned and well-posed. Moreover, conditions for boundedness and dissipativity of the trajectories are established. Exploiting the generalized Routh-Hurwitz Criteria, existence and stability analysis of equilibrium points for the
Ebola model are performed to show that they are strongly dependent on the non-linear transmission. In particular, conditions for existence and stability of a unique endemic equilibrium to the Ebola system are given. Numerical simulations are provided for particular expressions of the non-linear transmission, with model's parameters taking di erent values. The resulting simulations are in concordance with the usual threshold
behavior. The results obtained here may be signi cant for the ght and prevention
against Ebola haemorrhagic fever that has so far exterminated hundreds of families and
is still a ecting many people in West-Africa and other parts of the world.
The full comprehension and handling of the phenomenon of shattering, sometime happening
during the process of polymer chain degradation [129, 142], remains unsolved
when using the traditional evolution equations describing the degradation. This traditional
model has been proved to be very hard to handle as it involves evolution of
two intertwined quantities. Moreover, the explicit form of its solution is, in general,
impossible to obtain. We explore the possibility of generalizing evolution equation modeling
the polymer chain degradation and analyze the model with the conventional time
derivative with a new parameter. We consider the general case where the breakup rate
depends on the size of the chain breaking up. In the process, the alternative version of
Sumudu integral transform is used to provide an explicit form of the general solution
representing the evolution of polymer sizes distribution. In particular, we show that
this evolution exhibits existence of complex periodic properties due to the presence of
cosine and sine functions governing the solutions. Numerical simulations are performed
for some particular cases and prove that the system describing the polymer chain degradation
contains complex and simple harmonic poles whose e ects are given by these
functions or a combination of them. This result may be crucial in the ongoing research
to better handle and explain the phenomenon of shattering. Lastly, it has become a conjecture that power series like Mittag-Le er functions and
their variants naturally govern solutions to most of generalized fractional evolution models
such as kinetic, di usion or relaxation equations. The question is to say whether or
not this is always true! Whence, three generalized evolution equations with an additional
fractional parameter are solved analytically with conventional techniques. These
are processes related to stationary state system, relaxation and di usion. In the analysis,
we exploit the Sumudu transform to show that investigation on the stationary
state system leads to results of invariability. However, unlike other models, the generalized
di usion and relaxation models are proven not to be governed by Mittag-Le er
functions or any of their variants, but rather by a parameterized exponential function,
new in the literature, more accurate and easier to handle. Graphical representations
are performed and also show how that parameter, called ; can be used to control the
stationarity of such generalized models.Mathematical SciencesPh. D. (Applied Mathematics
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