scholarly journals Quasilinearized Semi-Orthogonal B-Spline Wavelet Method for Solving Multi-Term Non-Linear Fractional Order Equations

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1549
Author(s):  
Can Liu ◽  
Xinming Zhang ◽  
Boying Wu
Keyword(s):  
Fractional Order ◽  
Operational Matrix ◽  
Order Equation ◽  
Quasilinearization Method ◽  
Wavelet Method ◽  
Spline Wavelet ◽  
B Spline ◽  
Spline Wavelets ◽  
Fractional Differential Operator ◽  
Non Linear

In the present article, we implement a new numerical scheme, the quasilinearized semi-orthogonal B-spline wavelet method, combining the semi-orthogonal B-spline wavelet collocation method with the quasilinearization method, for a class of multi-term non-linear fractional order equations that contain both the Riemann–Liouville fractional integral operator and the Caputo fractional differential operator. The quasilinearization method is utilized to convert the multi-term non-linear fractional order equation into a multi-term linear fractional order equation which, subsequently, is solved by means of semi-orthogonal B-spline wavelets. Herein, we investigate the operational matrix and the convergence of the proposed scheme. Several numerical results are delivered to confirm the accuracy and efficiency of our scheme.

scholarly journals Two-dimensional nonlinear time fractional reaction–diffusion equation in application to sub-diffusion process of the multicomponent fluid in porous media

Meccanica ◽  
2020 ◽  
Author(s):  
P. Pandey ◽  
S. Das ◽  
E-M. Craciun ◽  
T. Sadowski
Keyword(s):  
Diffusion Equation ◽  
Present Article ◽  
Fractional Order ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Two Dimensional ◽  
Linear Algebraic Equations ◽  
Non Linear

AbstractIn the present article, an efficient operational matrix based on the famous Laguerre polynomials is applied for the numerical solution of two-dimensional non-linear time fractional order reaction–diffusion equation. An operational matrix is constructed for fractional order differentiation and this operational matrix converts our proposed model into a system of non-linear algebraic equations through collocation which can be solved by using the Newton Iteration method. Assuming the surface layers are thermodynamically variant under some specified conditions, many insights and properties are deduced e.g., nonlocal diffusion equations and mass conservation of the binary species which are relevant to many engineering and physical problems. The salient features of present manuscript are finding the convergence analysis of the proposed scheme and also the validation and the exhibitions of effectiveness of the method using the order of convergence through the error analysis between the numerical solutions applying the proposed method and the analytical results for two existing problems. The prominent feature of the present article is the graphical presentations of the effect of reaction term on the behavior of solute profile of the considered model for different particular cases.

Third-Kind Chebyshev Wavelet Method for the Solution of Fractional Order Riccati Differential Equations

2019 ◽  
Vol 28 (14) ◽  
pp. 1950247 ◽  
Author(s):  
Sadiye Nergis Tural-Polat
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Algebraic Equations ◽  
Wavelet Method ◽  
Riccati Differential Equations ◽  
The Third ◽  
Chebyshev Wavelet ◽  
Fractional Orders

In this paper, we derive the numerical solutions of the various fractional-order Riccati type differential equations using the third-kind Chebyshev wavelet operational matrix of fractional order integration (C3WOMFI) method. The operational matrix of fractional order integration method converts the fractional differential equations to a system of algebraic equations. The third-kind Chebyshev wavelet method provides sparse coefficient matrices, therefore the computational load involved for this method is not as severe and also the resulting method is faster. The numerical solutions agree with the exact solutions for non-fractional orders, and also the solutions for the fractional orders approach those of the integer orders as the fractional order coefficient [Formula: see text] approaches to 1.

scholarly journals B-spline Wavelet Method for Solving Fredholm Hammerstein Integral Equation Arising from Chemical Reactor Theory

2018 ◽  
Vol 7 (3) ◽  
pp. 163-169 ◽  
Author(s):  
P. K. Sahu ◽  
A. K. Ranjan ◽  
S. Saha Ray
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Integral Equation ◽  
Chemical Reactor ◽  
Steady State Solution ◽  
Algebraic Equations ◽  
Hammerstein Integral Equation ◽  
Wavelet Method ◽  
Spline Wavelet ◽  
B Spline

Abstract Mathematical model for an adiabatic tubular chemical reactor which processes an irreversible exothermic chemical reaction has been considered. For steady state solution for an adiabatic tubular chemical reactor, the model can be reduced to ordinary differential equation with a parameter in the boundary conditions. Again the ordinary differential equation has been converted into a Hammerstein integral equation which can be solved numerically. B-spline wavelet method has been developed to approximate the solution of Hammerstein integral equation. This method reduces the integral equation to a system of algebraic equations. The numerical results obtained by the present method have been compared with the available results.

A Double Thick Beam FEM with Explosion Resistance Dynamics Analysis Based on Spline Wavelets in Underground Structure

2012 ◽  
Vol 479-481 ◽  
pp. 1247-1252
Author(s):  
Han Yi Liu ◽  
Gan Long Su ◽  
Qing Ou Yang ◽  
Qing Su
Keyword(s):  
Thick Plate ◽  
Underground Structure ◽  
Damage Mechanism ◽  
Displacement Function ◽  
Smooth Functions ◽  
Spline Wavelet ◽  
B Spline ◽  
Blast Damage ◽  
Spline Wavelets ◽  
Thick Beam

Isoparametric B-spline wavelet functions can provide excellent approximation for smooth functions. In this paper, the response of composite laminated plate subjected to explosions is analyzed by using a double thick beam FEM mode based on the spline wavelet function. This mode, which can be considered in the distribution of normal stress and transverse shear stress simultaneously, is very important in the analysis of thick plate or blast-damage mechanism. The mode in analyzing underground structure and earth medium respond has the advantage of high precision and fast constringency and is easy to establish a element displacement function. The comparison between simulative and experimental results shows the effectiveness and precision of this method.

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