New variational theory for coupled nonlinear fractal Schrödinger system

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
KangLe Wang
Keyword(s):  
Inverse Method ◽  
Equation System ◽  
Differential Equation System ◽  
Variational Theory ◽  
Transform Method ◽  
Content Type ◽  
Variational Iteration ◽  
Fractal Derivative ◽  
Fractal Differential Equation ◽  
Schrödinger System

Purpose The purpose of this paper is the coupled nonlinear fractal Schrödinger system is defined by using fractal derivative, and its variational principle is constructed by the fractal semi-inverse method. The approximate analytical solution of the coupled nonlinear fractal Schrödinger system is obtained by the fractal variational iteration transform method based on the proposed variational theory and fractal two-scales transform method. Finally, an example illustrates the proposed method is efficient to deal with complex nonlinear fractal systems. Design/methodology/approach The coupled nonlinear fractal Schrödinger system is described by using the fractal derivative, and its fractal variational principle is obtained by the fractal semi-inverse method. A novel approach is proposed to solve the fractal model based on the variational theory. Findings The fractal variational iteration transform method is an excellent method to solve the fractal differential equation system. Originality/value The author first presents the fractal variational iteration transform method to find the approximate analytical solution for fractal differential equation system. The example illustrates the accuracy and efficiency of the proposed approach.

An exact GITT solution for static bending of clamped parallelogram plate resting on an elastic foundation

2019 ◽  
Vol 36 (6) ◽  
pp. 2034-2047 ◽  
Author(s):  
Guangming Fu ◽  
Yudan Peng ◽  
Baojiang Sun ◽  
Chen An ◽  
Jian Su
Keyword(s):  
Boundary Conditions ◽  
Elastic Foundation ◽  
Design Methodology ◽  
Significant Contribution ◽  
Integral Transform ◽  
Auxiliary Problem ◽  
Equation System ◽  
Differential Equation System ◽  
Content Type ◽  
Integral Transform Technique

Purpose The purpose of this study is to propose generalised integral transform technique (GITT) to obtain the exact solutions for bending of clamped parallelogram plate resting on elastic foundation. Design/methodology/approach The GITT is used to solve the bending problem of the full clamped parallelogram plate under an elastic foundation. The auxiliary problem was developed and the corresponding eigenfunction and eigenvalue were calculated simultaneously. The original partial differential governed equation has been represented by the transformed ordinary differential equation system and solved by the subroutine DBVPFD from International Mathematics and Statistics Library. Findings The GITT has been proven to be an efficient approach to solve the bending problem of the plate with different loads, boundary conditions and elastic foundations. The parametric study indicates that the elastic foundation modulus has significant contribution in reducing the vertical deflections and moments for both rectangular and parallelogram plates. With the increasing of aspect ratio (a/b) and the elastic foundation modulus, the trends of the deflection and moment reduction decreased significantly. Originality/value The present hybrid analytical-numerical methodology was first used to solve the mechanics problem of the clamped parallelogram plate resting on elastic foundation. Excellent convergence and high accuracy was observed by comparing with the published results. It exhibits potential application to investigate the mechanics problem of the composite plate with different boundary conditions in the shipbuilding and civil engineering.

Variational principle and its fractal approximate solution for fractal Lane-Emden equation

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
KangLe Wang
Keyword(s):  
Differential Equation ◽  
Variational Principle ◽  
Iteration Method ◽  
Approximate Analytical Solution ◽  
Variational Iteration Method ◽  
Content Type ◽  
Emden Equation ◽  
Variational Iteration ◽  
Fractal Models ◽  
Fractal Differential Equation

Purpose The purpose of this paper is to describe the Lane–Emden equation by the fractal derivative and establish its variational principle by using the semi-inverse method. The variational principle is helpful to research the structure of the solution. The approximate analytical solution of the fractal Lane–Emden equation is obtained by the variational iteration method. The example illustrates that the suggested scheme is efficient and accurate for fractal models. Design/methodology/approach The author establishes the variational principle for fractal Lane–Emden equation, and its approximate analytical solution is obtained by the variational iteration method. Findings The variational iteration method is very fascinating in solving fractal differential equation. Originality/value The author first proposes the variational iteration method for solving fractal differential equation. The example shows the efficiency and accuracy of the proposed method. The variational iteration method is valid for other nonlinear fractal models as well.

A new fractal model for the soliton motion in a microgravity space

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
KangLe Wang
Keyword(s):  
Variational Principle ◽  
Analytical Solution ◽  
Conservation Laws ◽  
Inverse Method ◽  
Approximate Analytical Solution ◽  
Microgravity Condition ◽  
Fractal Model ◽  
Content Type ◽  
Scale Method ◽  
Fractal Derivative

Purpose On a microgravity condition, a motion of soliton might be subject to a microgravity-induced motion. There is no theory so far to study the effect of air density and gravity on the motion property. Here, the author considers the air as discrete molecules and a motion of a soliton is modeled based on He’s fractal derivative in a microgravity space. The variational principle of the alternative model is constructed by semi-inverse method. The variational principle can be used to establish the conservation laws and reveal the structure of the solution. Finally, its approximate analytical solution is found by using two-scale method and homotopy perturbation method (HPM). Design/methodology/approach The author establishes a new fractal model based on He’s fractal derivative in a microgravity space and its variational principle is obtained via the semi-inverse method. The approximate analytical solution of the fractal model is obtained by using two-scale method and HPM. Findings He’s fractal derivative is a powerful tool to establish a mathematical model in microgravity space. The variational principle of the fractal model can be used to establish the conservation laws and reveal the structure of the solution. Originality/value The author proposes the first fractal model for the soliton motion in a microgravtity space and obtains its variational principle and approximate solution.

scholarly journals Numerical Analysis of Time-Fractional Diffusion Equations via a Novel Approach

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Nehad Ali Shah ◽  
S. Saleem ◽  
Ali Akgül ◽  
Kamsing Nonlaopon ◽  
Jae Dong Chung
Keyword(s):  
Nonlinear Problems ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Variational Theory ◽  
Transform Method ◽  
Fractional Diffusion Equations ◽  
Variational Iteration ◽  
Novel Approach ◽  
Linear And Nonlinear ◽  
Iteration Technique

The aim of this paper is a new semianalytical technique called the variational iteration transform method for solving fractional-order diffusion equations. In the variational iteration technique, identifying of the Lagrange multiplier is an essential rule, and variational theory is commonly used for this purpose. The current technique has the edge over other methods as it does not need extra parameters and polynomials. The validity of the proposed method is verified by considering some numerical problems. The solution achieved has shown that the better accuracy of the proposed technique. This paper proposes a simpler method to calculate the multiplier using the Shehu transformation, making a valuable technique to researchers dealing with various linear and nonlinear problems.

scholarly journals Transient Dynamics in the Random Growth and Reset Model

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 306
Author(s):  
Tamás S. Biró ◽  
Lehel Csillag ◽  
Zoltán Néda
Keyword(s):  
Differential Equation ◽  
Mean Field ◽  
Equation System ◽  
Transient Dynamics ◽  
Type Model ◽  
Discrete Version ◽  
Differential Equation System ◽  
Practical Applications ◽  
Random Growth ◽  
Recursive Integration

A mean-field type model with random growth and reset terms is considered. The stationary distributions resulting from the corresponding master equation are relatively easy to obtain; however, for practical applications one also needs to know the convergence to stationarity. The present work contributes to this direction, studying the transient dynamics in the discrete version of the model by two different approaches. The first method is based on mathematical induction by the recursive integration of the coupled differential equations for the discrete states. The second method transforms the coupled ordinary differential equation system into a partial differential equation for the generating function. We derive analytical results for some important, practically interesting cases and discuss the obtained results for the transient dynamics.

Identification of three-dimensional transient temperature fields in thick-walled elements using the inverse method

2018 ◽  
Vol 28 (1) ◽  
pp. 138-150 ◽  
Author(s):  
Magdalena Jaremkiewicz
Keyword(s):  
Heat Transfer ◽  
Heat Transfer Coefficient ◽  
Transfer Coefficient ◽  
Inverse Method ◽  
Thermal Stresses ◽  
Computing Time ◽  
Transient Temperature ◽  
Content Type ◽  
Inner Surface ◽  
The Heat Transfer Coefficient

Purpose The purpose of this paper is to propose a method of determining the transient temperature of the inner surface of thick-walled elements. The method can be used to determine thermal stresses in pressure elements. Design/methodology/approach An inverse marching method is proposed to determine the transient temperature of the thick-walled element inner surface with high accuracy. Findings Initially, the inverse method was validated computationally. The comparison between the temperatures obtained from the solution for the direct heat conduction problem and the results obtained by means of the proposed inverse method is very satisfactory. Subsequently, the presented method was validated using experimental data. The results obtained from the inverse calculations also gave good results. Originality/value The advantage of the method is the possibility of determining the heat transfer coefficient at a point on the exposed surface based on the local temperature distribution measured on the insulated outer surface. The heat transfer coefficient determined experimentally can be used to calculate thermal stresses in elements with a complex shape. The proposed method can be used in online computer systems to monitor temperature and thermal stresses in thick-walled pressure components because the computing time is very short.

A nonlinear magnetic circuit model for periodic eddy current problems using T,ϕ-ϕ formulation

2017 ◽  
Vol 36 (3) ◽  
pp. 649-664 ◽  
Author(s):  
Rene Plasser ◽  
Gergely Koczka ◽  
Oszkár Bíró
Keyword(s):  
Eddy Current ◽  
Magnetic Circuit ◽  
Iteration Process ◽  
Equation System ◽  
Circuit Model ◽  
Computational Time ◽  
Combined Method ◽  
3D Fem ◽  
Content Type ◽  
Transformer Model

Purpose A transformer model is used as a benchmark for testing various methods to solve 3D nonlinear periodic eddy current problems. This paper aims to set up a nonlinear magnetic circuit problem to assess the solving procedure of the nonlinear equation system for determining the influence of various special techniques on the convergence of nonlinear iterations and hence the computational time. Design/methodology/approach Using the T,ϕ-ϕ formulation and the harmonic balance fixed-point approach, two techniques are investigated: the so-called “separate method” and the “combined method” for solving the equation system. When using the finite element method (FEM), the elapsed time for solving a problem is dominated by the conjugate gradient (CG) iteration process. The motivation for treating the equations of the voltage excitations separately from the rest of the equation system is to achieve a better-conditioned matrix system to determine the field quantities and hence a faster convergence of the CG process. Findings In fact, both methods are suitable for nonlinear computation, and for comparing the final results, the methods are equally good. Applying the combined method, the number of iterations to be executed to achieve a meaningful result is considerably less than using the separated method. Originality/value To facilitate a quick analysis, a simplified magnetic circuit model of the 3D problem was generated to assess how the different ways of solutions will affect the full 3D solving process. This investigation of a simple magnetic circuit problem to evaluate the benefits of computational methods provides the basis for considering this formulation in a 3D-FEM code for further investigation.

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