Nonstandard finite difference method for solving complex-order fractional Burgers' equations

An Analytical Approach for Fractional Hyperbolic Telegraph Equation Using Shehu Transform in One, Two and Three Dimensions

In the present research paper, an iterative approach named the iterative Shehu transform method is implemented to solve time-fractional hyperbolic telegraph equations in one, two, and three dimensions, respectively. These equations are the prominent ones in the field of physics and in some other significant problems. The efficacy and authenticity of the proposed method are tested using a comparison of approximated and exact results in graphical form. Both 2D and 3D plots are provided to affirm the compatibility of approximated-exact results. The iterative Shehu transform method is a reliable and efficient tool to provide approximated and exact results to a vast class of ODEs, PDEs, and fractional PDEs in a simplified way, without any discretization or linearization, and is free of errors. A convergence analysis is also provided in this research.

A new general integral transform for solving integral equations

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A new general integral transform which is covered all class of integral transform in the class of Laplace transform.

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We investigated the application of this new transform for solving ODE with constant and variable coefficient.

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This new transform can handle easily for fractional order integral equations and fractional order differential equations.

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We have discussed the advantage and disadvantage of other integral transformed which is defined during last 2 decades.

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We proved the related theorems for this new transform.

Introduction

Integral transforms are important to solve real problems. Appropriate choice of integral transforms helps to convert differential equations as well as integral equations into terms of an algebraic equation that can be solved easily.

During last two decades many integral transforms in the class of Laplace transform are introduced such as Sumudu, Elzaki, Natural, Aboodh, Pourreza, Mohand, G_transform, Sawi and Kamal transforms.

Objectives

In this paper, we introduce a general integral transform in the class of Laplace transform. We study the properties of this transform. Then we compare it with few exiting integral transforms in the Laplace family such as Laplace, Sumudu, Elzaki and G\_transforms, Pourreza, Aboodh and etc.

Methods

A new integral transform is introduced. Then some properties of this integral transform are discussed. This integral transform is used to solve this new transform is used for solving higher order initial value problems, integral equations and fractional order integral equation.

Results

It is proved that those new transforms in the class of Laplace transform which are introduced during last few decades are a special case of this general transform. It is shown that there is no advantage between theses transforms unless for special problems.

Conclusion

It has shown that this new integral transform covers those exiting transforms such as Laplace, Elzaki and Sumudu transforms for different value of p(s) and q(s). We used this new transform for solving ODE, integral equations and fractional integral equations. Also, we can introduce new integral transforms by using this new general integral transform.

Dynamics of fractional order nonlinear system: A realistic perception with neutrosophic fuzzy number and Allee effect

Introduction

The fusion of fractional order differential equations and fuzzy numbers has been widely used in modelling different engineering and applied sciences problems. In addition to these, the Allee effect, which is of high importance in field of biology and ecology, has also shown great contribution among other fields of sciences to study the correlation between density and the mean fitness of the subject.

Objectives

The present paper is intended to measure uncertain dynamics of an economy by restructuring the Cobb-Douglas paradigm of the renowned Solow-Swan model. The purpose of study is further boosted innovatively by subsuming the perception of logistic growth with Allee effect in the dynamics of physical capital and labor force.

Methods

Fractional order derivative and neutrosophic fuzzy (NF) theory are applied on the parameters of the Cobb-Douglas equation. Distinctively, cogitating fractional order derivative to study the change at each fractional stage; single-valued triangular neutrosophic fuzzy numbers (SVTNFN) to cope the uncertain situations; logistic growth function with Allee effect to analyze the factors in natural way, are the significant and novel features of this endeavor.

Results

The incorporation of the aforementioned theories and effects in the Cobb-Douglas equation, resulted in producing maximum sustainable capital investment and maximum capacity of labor force. The solutions in intervals located different possible solutions for different membership degrees, which accumulated the uncertain circumstances of a country.

Conclusion

Explicitly, these notions add new facts and figures not only in the dynamical study of capital and labor, which has been overlooked in classical models, but also left the door open for discussion and implementation on classical models of different fields.

A local stabilized approach for approximating the modified time-fractional diffusion problem arising in heat and mass transfer

INTRODUCTION

During the last years the modeling of dynamical phenomena has been advanced by including concepts borrowed from fractional order differential equations. The diffusion process plays an important role not only in heat transfer and fluid flow problems, but also in the modelling of pattern formation that arises in porous media. The modified time-fractional diffusion equation provides a deeper understanding of several dynamic phenomena.

OBJECTIVES

The purpose of the paper is to develop an efficient meshless technique for approximating the modified time-fractional diffusion problem formulated in the Riemann-Liouville sense.

METHODS

The temporal discretization is performed by integrating both sides of the modified time-fractional diffusion model. The unconditional stability of the time discretization scheme and the optimal convergence rate are obtained. Then, the spatial derivatives are discretized through a local hybridization of the cubic and Gaussian radial basis function. This hybrid kernel improves the condition of the system matrix. Therefore, the solution of the linear system can be obtained using direct solvers that reduce significantly computational cost. The main idea of the method is to consider the distribution of data points over the local support domain where the number of points is almost constant.

RESULTS

Three examples show that the numerical procedure has good accuracy and applicable over complex domains with various node distributions. Numerical results on regular and irregular domains illustrate the accuracy, efficiency and validity of the technique.

CONCLUSION

This paper adopts a local hybrid kernel meshless approach to solve the modified time-fractional diffusion problem. The main results of the research is the numerical technique with non-uniform distribution in irregular grids.

Conserved quantities, optimal system and explicit solutions of a (1 + 1)-dimensional generalised coupled mKdV-type system

Introduction

The purpose of this paper is to study, a (1 + 1)-dimensional generalised coupled modified Korteweg-de Vries-type system from Lie group analysis point of view. This system is studied in the literature for the first time. The authors found this system to be interesting since it is non-decouplable and possesses higher generalised symmetries.

Objectives

We look for the closed-form solutions and conservation laws of the system.

Methods

Optimal system of one-dimensional subalgebras for the system was obtained and then used to perform symmetry reductions and construct group invariant solutions. Power series solutions for the system were also obtained. The system has no variational principle and as such, we employed the multiplier method and used a homotopy integral formula to derive the conserved quantities.

Results

Group invariant solutions and power series solutions were constructed and three conserved vectors for the system were derived.

Conclusion

The paper studies the (1 + 1)-dimensional generalised coupled modified Korteweg-de Vries-type system for the first time and constructs its exact solutions and conservation laws.