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 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 Numerical Solutions of Fractional Fokker-Planck Equations Using Iterative Laplace Transform Method

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Limei Yan
Keyword(s):  
Laplace Transform ◽  
Numerical Solutions ◽  
Convergent Series ◽  
Space Time ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Promising Tool ◽  
The Laplace Transform ◽  
Fokker Planck Equations ◽  
Fokker Planck

A relatively new iterative Laplace transform method, which combines two methods; the iterative method and the Laplace transform method, is applied to obtain the numerical solutions of fractional Fokker-Planck equations. The method gives numerical solutions in the form of convergent series with easily computable components, requiring no linearization or small perturbation. The numerical results show that the approach is easy to implement and straightforward when applied to space-time fractional Fokker-Planck equations. The method provides a promising tool for solving space-time fractional partial differential equations.

The Axially Loaded Circular Cylinder

1962 ◽  
Vol 29 (2) ◽  
pp. 318-320
Author(s):  
H. D. Conway
Keyword(s):  
Exact Solution ◽  
Fourier Transform ◽  
Circular Cylinder ◽  
Closed Form ◽  
Cross Sections ◽  
Closed Form Solution ◽  
Form Solution ◽  
Concentrated Force ◽  
Transform Method ◽  
Fourier Transform Method

Commencing with Kelvin’s closed-form solution to the problem of a concentrated force acting at a given point in an indefinitely extended solid, a Fourier transform method is used to obtain an exact solution for the case when the force acts along the axis of a circular cylinder. Numerical values are obtained for the maximum direct stress on cross sections at various distances from the force. These are then compared with the corresponding stresses from the solution for an infinitely long strip, and in both cases it is observed that the stresses are practically uniform on cross sections greater than a diameter or width from the point of application of the load.

APPLICATION OF AN EFFECTIVE METHOD ON THE SYSTEM OF NONLINEAR FUZZY INTEGRO-DIFFERENTIAL EQUATIONS

2021 ◽  
Vol 21 (2) ◽  
pp. 407-422
Author(s):  
ANGBEEN IQBAL ◽  
JAMSHAD AHMAD ◽  
QAZI MAHMOOD UL HASSAN
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Graphical Representation ◽  
Numerical Approach ◽  
Fuzzy Arithmetic ◽  
Transform Method ◽  
Numerical Examples ◽  
Homotopy Perturbation ◽  
Sumudu Transform ◽  
Fuzzy Parameter

In real world physical applications purpose, it is complicated to acquire an exact solution of fuzzy differential equations due to complexities in fuzzy arithmetic and therefore creating the need for the use of reliable and efficient techniques in the solution of fuzzy differential equations. The purpose of this research paper is to utilize the reliable analytic approach of homotopy perturbation Sumudu transform method for better understanding of systems of non-linear fuzzy integro-differential equations, while using the concept of fuzzy parameter in certain dynamical problems to remove the hurdles faced in numerical approach. These mathematical models are of great interest in engineering and physics. Some numerical examples are also given to demonstrate the efficiency and superiority of the method, followed by graphical representation of the comparison of exact and approximated solution by using Maple 2017

On the Soliton Solution and Jacobi Doubly Periodic Solution of the Fractional Coupled Schrödinger–KdV Equation by a Novel Approach

2015 ◽  
Vol 16 (2) ◽  
pp. 79-95 ◽  
Author(s):  
S. Saha Ray
Keyword(s):  
Present Method ◽  
Numerical Solutions ◽  
Kdv Equation ◽  
Adomian Decomposition Method ◽  
Approximate Solutions ◽  
New Novel ◽  
Novel Approach ◽  
Adomian Decomposition ◽  
Novel Method ◽  
Doubly Periodic Solution

AbstractIn this paper, fractional coupled Schrödinger–Korteweg–de Vries equation (or Sch–KdV) equation with appropriate initial values has been solved by using a new novel method. The fractional derivatives are described in the Caputo sense. By using the present method, we can solve many linear and nonlinear coupled fractional differential equations. Basically, the present method originated from generalized Taylor’s formula [1]. The results reveal that the proposed method is very effective and simple for obtaining approximate solutions of fractional coupled Schrödinger–KdV equation. Numerical solutions are presented graphically to show the reliability and efficiency of the method. The method does not need linearization, weak nonlinearity assumptions or perturbation theory. The convergence of the method as applied to Sch–KdV is illustrated numerically as well as derived analytically. Moreover, the results are compared with those obtained by the Adomian decomposition method (ADM).

scholarly journals The solution of local fractional diffusion equation involving Hilfer fractional derivative

2019 ◽  
Vol 23 (Suppl. 3) ◽  
pp. 809-814
Author(s):  
Yun Qiao ◽  
Quan-Xi Qiao
Keyword(s):  
Exact Solution ◽  
Fractional Derivative ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Fractional Operator ◽  
Laplace Transform Method ◽  
Hilfer Fractional Derivative ◽  
Transform Method ◽  
Fractional Diffusion Equations ◽  
The Laplace Transform

In this present work the Yang-Fourier transform method incorporating the Laplace transform method is used to solve fractional diffusion equations involving the Hilfer fractional derivative and local fractional operator. The exact solution is obtained.

The Impact of Arbitrary Oriented Ellipsoidal Shell With a Barrier: Analytical Study

2015 ◽  
Vol 82 (4) ◽  
Author(s):  
Shahab Mansoor-Baghaei ◽  
Ali M. Sadegh
Keyword(s):  
Closed Form ◽  
Numerical Solutions ◽  
Closed Form Solution ◽  
Linearization Method ◽  
Form Solution ◽  
Ellipsoidal Shell ◽  
Explicit Finite Element ◽  
Closed Form Solutions ◽  
Transmitted Force ◽  
The Impact

In this paper, a closed form solution of an arbitrary oriented hollow elastic ellipsoidal shell impacting with an elastic flat barrier is presented. It is assumed that the shell is thin under the low speed impact. Due to the arbitrary orientation of the shell, while the pre-impact having a linear speed, the postimpact involves rotational and translational speed. Analytical solution for this problem is based on Hertzian theory (Johnson, W., 1972, Impact Strength of Materials, University of Manchester Institute of Science and Technology, Edward Arnold Publication, London) and the Vella’s analysis (Vella et al., 2012, “Indentation of Ellipsoidal and Cylindrical Elastic Shells,” Phys. Rev. Lett., 109, p. 144302) in conjunction with Newtonian method. Due to the nonlinearity and complexity of the impact equation, classical numerical solutions cannot be employed. Therefore, a linearization method is proposed and a closed form solution for this problem is accomplished. The closed form solution facilitates a parametric study of this type of problems. The closed form solution was validated by an explicit finite element method (FEM). Good agreement between the closed form solution and the FE results is observed. Based on the analytical method the maximum total deformation of the shell, the maximum transmitted force, the duration of the contact, and the rotation of the shell after the impact were determined. Finally, it was concluded that the closed form solutions were trustworthy and appropriate to investigate the impact of inclined elastic ellipsoidal shells with an elastic barrier.

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.

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