scholarly journals An Analytical Investigation of Fractional-Order Biological Model Using an Innovative Technique

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Hassan Khan ◽  
Adnan Khan ◽  
Maysaa Al Qurashi ◽  
Dumitru Baleanu ◽  
Rasool Shah
Keyword(s):  
Fractional Order ◽  
Present Method ◽  
Iterative Procedure ◽  
Laplace Transformation ◽  
Small Volume ◽  
Analytical Investigation ◽  
Form Solution ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Closed Contact

In this paper, a new so-called iterative Laplace transform method is implemented to investigate the solution of certain important population models of noninteger order. The iterative procedure is combined effectively with Laplace transformation to develop the suggested methodology. The Caputo operator is applied to express the noninteger derivative of fractional order. The series form solution is obtained having components of convergent behavior toward the exact solution. For justification and verification of the present method, some illustrative examples are discussed. The closed contact is observed between the obtained and exact solutions. Moreover, the suggested method has a small volume of calculations; therefore, it can be applied to handle the solutions of various problems with fractional-order derivatives.

scholarly journals Modified Modelling for Heat Like Equations within Caputo Operator

Energies ◽  
2020 ◽  
Vol 13 (8) ◽  
pp. 2002 ◽  
Author(s):  
Hassan Khan ◽  
Adnan Khan ◽  
Maysaa Al-Qurashi ◽  
Rasool Shah ◽  
Dumitru Baleanu
Keyword(s):  
Exact Solution ◽  
Present Method ◽  
Numerical Solutions ◽  
Graphical Representation ◽  
Close Contact ◽  
Form Solution ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Closed Form Solutions ◽  
Novel Approach

The present paper is related to the analytical solutions of some heat like equations, using a novel approach with Caputo operator. The work is carried out mainly with the use of an effective and straight procedure of the Iterative Laplace transform method. The proposed method provides the series form solution that has the desired rate of convergence towards the exact solution of the problems. It is observed that the suggested method provides closed-form solutions. The reliability of the method is confirmed with the help of some illustrative examples. The graphical representation has been made for both fractional and integer-order solutions. Numerical solutions that are in close contact with the exact solutions to the problems are investigated. Moreover, the sample implementation of the present method supports the importance of the method to solve other fractional-order problems in sciences and engineering.

scholarly journals A Note on Double Conformable Laplace Transform Method and Singular One Dimensional Conformable Pseudohyperbolic Equations

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 949 ◽  
Author(s):  
Hassan Eltayeb ◽  
Said Mesloub ◽  
Yahya T. Abdalla ◽  
Adem Kılıçman
Keyword(s):  
Present Method ◽  
Decomposition Method ◽  
Numerical Solutions ◽  
High Accuracy ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Numerical Examples ◽  
One Dimensional ◽  
Linear And Nonlinear ◽  
Laplace Decomposition Method

The purpose of this article is to obtain the exact and approximate numerical solutions of linear and nonlinear singular conformable pseudohyperbolic equations and conformable coupled pseudohyperbolic equations through the conformable double Laplace decomposition method. Further, the numerical examples were provided in order to demonstrate the efficiency, high accuracy, and the simplicity of present method.

scholarly journals Analysis of Optical Solitons for Nonlinear Schrödinger Equation with Detuning Term by Iterative Transform Method

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1850
Author(s):  
Nehad Ali Shah ◽  
Praveen Agarwal ◽  
Jae Dong Chung ◽  
Essam R. El-Zahar ◽  
Y. S. Hamed
Keyword(s):  
Fractional Order ◽  
Present Method ◽  
Nonlinear Partial Differential Equations ◽  
Optical Solitons ◽  
Optical Soliton ◽  
Transform Method ◽  
Computational Costs ◽  
Nonlinear Schrödinger ◽  
Linear And Nonlinear ◽  
Integral Order

In this article, the iteration transform method is used to evaluate the solution of a fractional-order dark optical soliton, bright optical soliton, and periodic solution of the nonlinear Schrödinger equations. The Caputo operator describes the fractional-order derivatives. The solutions of some illustrative examples are presented to show the validity of the proposed technique without using any polynomials. The proposed method provides the series form solutions, which converge to the exact results. Using the present methodology, the solutions of fractional-order problems as well as integral-order problems are calculated. The present method has less computational costs and a higher rate of convergence. Therefore, the suggested algorithm is constructive to solve other fractional-order linear and nonlinear partial differential equations.

scholarly journals Analytical Solutions of (2+Time Fractional Order) Dimensional Physical Models, Using Modified Decomposition Method

2019 ◽  
Vol 10 (1) ◽  
pp. 122 ◽  
Author(s):  
Hassan Khan ◽  
Umar Farooq ◽  
Rasool Shah ◽  
Dumitru Baleanu ◽  
Poom Kumam ◽  
...  
Keyword(s):  
Fractional Order ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Analytical Techniques ◽  
Form Solution ◽  
Physical Models ◽  
Fractional Partial Differential Equations ◽  
Analytical Technique ◽  
Adomian Decomposition ◽  
Closed Contact

In this article, a new analytical technique based on an innovative transformation is used to solve (2+time fractional-order) dimensional physical models. The proposed method is the hybrid methodology of Shehu transformation along with Adomian decomposition method. The series form solution is obtained by using the suggested method which provides the desired rate of convergence. Some numerical examples are solved by using the proposed method. The solutions of the targeted problems are represented by graphs which have confirmed closed contact between the exact and obtained solutions of the problems. Based on the novelty and straightforward implementation of the method, it is considered to be one of the best analytical techniques to solve linear and non-linear fractional partial differential equations.

scholarly journals Fractional-Order Investigation of Diffusion Equations via Analytical Approach

2021 ◽  
Vol 8 ◽  
Author(s):  
Haobin Liu ◽  
Hassan Khan ◽  
Saima Mustafa ◽  
Lianming Mou ◽  
Dumitru Baleanu
Keyword(s):  
Applied Sciences ◽  
Fractional Order ◽  
Closed Form Solution ◽  
Approximate Solutions ◽  
Form Solution ◽  
Diffusion Equations ◽  
Physical Problems ◽  
Research Article ◽  
Closed Contact ◽  
Caputo Fractional Order Derivative

This research article is mainly concerned with the analytical solution of diffusion equations within a Caputo fractional-order derivative. The motivation and novelty behind the present work are the application of a sophisticated and straight forward procedure to solve diffusion equations containing a derivative of a fractional-order. The solutions of some illustrative examples are calculated to confirm the closed contact between the actual and the approximate solutions of the targeted problems. Through analysis it is shown that the proposed solution has a higher rate of convergence and provides a closed-form solution. The small number of calculations is the main advantage of the proposed method. Due to a comfortable and straight forward implementation, the suggested method can be utilized to nonlinear fractional-order problems in various applied science branches. It can be extended to solve other physical problems of fractional-order in multiple areas of applied sciences.

scholarly journals The Analysis of Fractional-Order Helmholtz Equations via a Novel Approach

2021 ◽  
Author(s):  
Pongsakorn Sunthrayuth ◽  
Zeyad Al-Zhour ◽  
Yu-Ming Chu
Keyword(s):  
Fractional Order ◽  
Present Method ◽  
Computational Cost ◽  
Analytical Techniques ◽  
Form Solution ◽  
Effective Technique ◽  
Helmholtz Equations ◽  
Inverse Transform ◽  
Novel Approach ◽  
Fractional Analysis

This paper is related to the fractional view analysis of Helmholtz equations, using innovative analytical techniques. The fractional analysis of the proposed problems has been done in terms of Caputo-operator sense. In the current methodology, first, we applied the r-Laplace transform to the targeted problem. The iterative method is then implemented to obtain the series form solution. After using the inverse transform of the r-Laplace, the desire analytical solution is achieved. The suggested procedure is verified through specific examples of the fractional Helmholtz equations. The present method is found to be an effective technique having a closed resemblance with the actual solutions. The proposed technique has less computational cost and a higher rate of convergence. The suggested methods are therefore very useful to solve other systems of fractional order problems.

Homotopy analysis transform algorithm to solve time-fractional foam drainage equation

2016 ◽  
Vol 5 (3) ◽  
Author(s):  
Mukesh Singh ◽  
Mohd Naseem ◽  
Amit Kumar ◽  
Sunil Kumar
Keyword(s):  
Laplace Transform ◽  
Fractional Derivative ◽  
Homotopy Analysis Method ◽  
Present Method ◽  
Analysis Method ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Homotopy Analysis ◽  
Foam Drainage Equation ◽  
Foam Drainage

AbstractThis paper emphasizes on finding the solution for a foam drainageequation using the technique of modified homotopy analysis transform method (MHATM). MHATM is a new amalgamation of the homotopy analysis method and Laplace transform method with homotopy polynomial. Comparisons are made between the results of the proposed method for different values of fractional derivative α and exact solutions. Then, we analyze the results by numerical simulations, which demonstrate the simplicity and effectiveness of the present method.

scholarly journals Exact Evaluation of Infinite Series Using Double Laplace Transform Technique

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Hassan Eltayeb ◽  
Adem Kılıçman ◽  
Said Mesloub
Keyword(s):  
Laplace Transform ◽  
Infinite Series ◽  
Present Method ◽  
Laplace Transform Method ◽  
General Technique ◽  
Transform Method ◽  
Double Laplace Transform ◽  
Laplace Transform Technique ◽  
Exact Evaluation

Double Laplace transform method was applied to evaluate the exact value of double infinite series. Further we generalize the current existing methods and provide some examples to illustrate and verify that the present method is a more general technique.

Dynamic Rate Effects on Timoshenko Beam Response

1985 ◽  
Vol 52 (2) ◽  
pp. 439-445 ◽  
Author(s):  
T. J. Ross
Keyword(s):  
Timoshenko Beam ◽  
Bending Moment ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Rate Effects ◽  
The Laplace Transform ◽  
Step Loading ◽  
Fixed End ◽  
Constitutive Formulation ◽  
Viscoelastic Timoshenko Beam

The problem of a viscoelastic Timoshenko beam subjected to a transversely applied step-loading is solved using the Laplace transform method. It is established that the support shear force is amplified more than the support bending moment for a fixed-end beam when strain rate influences are accounted for implicitly in the viscoelastic constitutive formulation.

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