Numerical study of variational problems of moving or fixed boundary conditions by Muntz wavelets

2020 ◽  
pp. 107754632097479
Author(s):  
Ashish Rayal ◽  
Sag R Verma
Keyword(s):  
Exact Solution ◽  
Boundary Conditions ◽  
Approximation Method ◽  
Computational Algorithm ◽  
Numerical Study ◽  
Operational Matrix ◽  
Variational Problems ◽  
Algebraic Equations ◽  
Suggested Approach ◽  
Fixed Boundary

In this study, an approximation method with an integral operational matrix based on the Muntz wavelets basis is presented to solve the variational problems of moving or fixed boundary conditions and a computational algorithm is given for the suggested approach. First, the integral operational matrix is created through the Muntz wavelets. Then, by using this integral operational matrix with Lagrange multipliers, the present approach reduces the variational problem into the system of algebraic equations. This approach is examined by some illustrative examples, and the acquired results prove that the suggested approach can solve the variational problems effectively with higher accuracy. The proposed approach yields better and comparable results with some other existing schemes given in the literature. The approximate wavelet solutions derived by the suggested approach are very identical to the corresponding exact solution.

A numerical study of the nonlinear fractional mathematical model of tumor cells in presence of chemotherapeutic treatment

2020 ◽  
Vol 13 (03) ◽  
pp. 2050021 ◽  
Author(s):  
Sachin Kumar ◽  
Abdon Atangana
Keyword(s):  
Tumor Cells ◽  
Immune Cells ◽  
Initial Conditions ◽  
Numerical Study ◽  
Operational Matrix ◽  
Medical Science ◽  
Algebraic Equations ◽  
Normal Cells ◽  
Operational Matrix Method ◽  
Chemotherapeutic Treatment

Cancer belongs to the class of diseases which is symbolized by out of control cells growth. These cells affect DNAs and damage them. There exist many treatments available in medical science as radiation therapy, targeted therapy, surgery, palliative care and chemotherapy. Chemotherapy is one of the most popular treatments which depends on the type, location and grade of cancer. In this paper, we are working on modeling and prediction of the effect of chemotherapy on cancer cells using a fractional differential equation by using the differential operator in Caputo’s sense. The presented model depicts the interaction between tumor, normal and immune cells in a tumor by using a system of four coupled fractional partial differential equations (PDEs). For this system, initial conditions of tumor cells and dimensions are taken in such a way that tumor is spread out enough in size and can be detected easily with the clinical machines. An operational matrix method with Genocchi polynomials is applied to study this system of fractional PDEs (FPDEs). An operational matrix for fractional differentiation is derived. Applying the collocation method and using this matrix, the nonlinear system is reduced to a system of algebraic equations, which can be solved using Newton iteration method. The salient features of this paper are the pictorial presentations of the numerical solution of the concerned equation for different particular cases to show the effect of fractional exponent on diffusive nature of immune cells, tumor cells, normal cells and chemotherapeutic drug and depict the interaction among immune cells, normal cells and tumor cells in a tumor site.

Application of Bernstein Polynomial Multiwavelets for Solving Non Linear Variational Problems with Moving and Fixed Boundaries

2020 ◽  
Vol 13 ◽  
Author(s):  
Sandeep Dixit ◽  
Shweta Pandey ◽  
S.R. Verma
Keyword(s):  
Variational Problem ◽  
Direct Method ◽  
Bernstein Polynomials ◽  
Operational Matrix ◽  
Variational Problems ◽  
Equation System ◽  
Primary Objective ◽  
Necessary Condition ◽  
Algebraic Equations ◽  
Operational Matrix Of Integration

Background: In this article, an efficient direct method has been proposed in order to solve physically significant variational problems. The proposed technique finds its basis in Bernstein polynomials multiwavelets (BPMWs). The mechanism of the proposed method is to transform the variational problem into an algebraic equation system through the use of BPMWs. Objective: Since the necessary condition of extremization consists of a differential equation that cannot be easily integrated in complex cases, an approximated numerical solution becomes a necessity. Our primary objective is to establish a wavelet based method for solving variational problems of physical interest. Besides being computationally more effective, the proposed approach yields relatively more accurate results than other comparable methods. The approach employs fewer basis elements, which in turn increases the simplicity, decreases the calculation time, and furnishes better results. Methods: An operational matrix of integration, which is based on the BPMWs, is presented. We substitute the approximated values of , unknown function and their derivative functions with BPMWs operational matrix of integration and BPMWs. On substituting the respective values in the given variational problem, it gets converted into a system of algebraic equations. The obtained system is further solved using the Lagrange multiplier. Results: The results obtained yield a greater degree of convergence as compared to other existing numerical methods. Numerical illustrations based on physical variational problems and the comparisons of outcomes with exact solutions demonstrate that the proposed method yields better efficiency, applicability, and accuracy. Conclusion: The proposed method gives better results than other comparable methods, even with the use of a fewer number of basis elements. The large order of matrices, such as 32, 64, and 512, obtained by using other available methods is far too high to achieve accuracy in results in comparison to the ones we obtain by using matrices of relatively lower orders, such as 7, 8 and 13, in the proposed method. This method can also be used for extremization functional occurring in electrical circuits and mechanical physical problems.

A Numerical Approach for Fractional Order Riccati Differential Equation Using B-Spline Operational Matrix

2015 ◽  
Vol 18 (2) ◽  
Author(s):  
Hossein Jafari ◽  
Haleh Tajadodi ◽  
Dumitru Baleanu
Keyword(s):  
Differential Equation ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Numerical Approach ◽  
Algebraic Equations ◽  
Riccati Differential Equation ◽  
Suggested Approach ◽  
B Spline ◽  
Riccati Differential Equations ◽  
The Given

AbstractIn this article, we develop an effective numerical method to achieve the numerical solutions of nonlinear fractional Riccati differential equations. We found the operational matrix within the linear B-spline functions. By this technique, the given problem converts to a system of algebraic equations. This technique is used to solve fractional Riccati differential equation. The obtained results are illustrated both applicability and validity of the suggested approach.

scholarly journals An Efficient Approach for Solving Nonlinear Troesch’s and Bratu’s Problems by Wavelet Analysis Method

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
A. Kazemi Nasab ◽  
Z. Pashazadeh Atabakan ◽  
A. Kılıçman
Keyword(s):  
Exact Solution ◽  
Wavelet Analysis ◽  
Operational Matrix ◽  
Nonlinear Problems ◽  
Basis Functions ◽  
Wavelet Basis ◽  
Algebraic Equations ◽  
Analysis Method ◽  
Chebyshev Wavelets ◽  
Chebyshev Wavelet

We introduce Chebyshev wavelet analysis method to solve the nonlinear Troesch and Bratu problems. Chebyshev wavelets expansions together with operational matrix of derivative are employed to reduce the computation of nonlinear problems to a system of algebraic equations. Several examples are given to validate the efficiency and accuracy of the proposed technique. We compare the results with those ones reported in the literature in order to demonstrate that the method converges rapidly and approximates the exact solution very accurately by using only a small number of Chebyshev wavelet basis functions. Convergence analysis is also included.

An Efficient Legendre Spectral Tau Matrix Formulation for Solving Fractional Subdiffusion and Reaction Subdiffusion Equations

2015 ◽  
Vol 10 (2) ◽  
Author(s):  
E. H. Doha ◽  
A. H. Bhrawy ◽  
S. S. Ezz-Eldien
Keyword(s):  
Boundary Conditions ◽  
Spectral Method ◽  
Operational Matrix ◽  
Neumann Boundary ◽  
Variable Coefficients ◽  
Dirichlet Boundary Conditions ◽  
Fractional Integrals ◽  
Algebraic Equations ◽  
Matrix Formulation ◽  
Subdiffusion Equation

In this work, we discuss an operational matrix approach for introducing an approximate solution of the fractional subdiffusion equation (FSDE) with both Dirichlet boundary conditions (DBCs) and Neumann boundary conditions (NBCs). We propose a spectral method in both temporal and spatial discretizations for this equation. Our approach is based on the space-time shifted Legendre tau-spectral method combined with the operational matrix of fractional integrals, described in the Riemann–Liouville sense. The main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations in the unknown expansion coefficients of the sought-for spectral approximations. In addition, this approach is also investigated for solving the FSDE with the variable coefficients and the fractional reaction subdiffusion equation (FRSDE). For conforming the validity and accuracy of the numerical scheme proposed, four numerical examples with their approximate solutions are presented. Also, comparisons between our numerical results and those obtained by compact finite difference method (CFDM), Box-type scheme (B-TS), and FDM with Fourier analysis (FA) are introduced.

Direct numerical method for isoperimetric fractional variational problems based on operational matrix

2017 ◽  
Vol 24 (14) ◽  
pp. 3063-3076 ◽  
Author(s):  
Samer S Ezz–Eldien ◽  
Ali H Bhrawy ◽  
Ahmed A El–Kalaawy
Keyword(s):  
Direct Method ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Variational Problems ◽  
Test Problems ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Direct Numerical Method ◽  
Fractional Variational Problems ◽  
A Direct Method

In this paper, we applied a direct method for a solution of isoperimetric fractional variational problems. We use shifted Legendre orthonormal polynomials as basis function of operational matrices of fractional differentiation and fractional integration in combination with the Lagrange multipliers technique for converting such isoperimetric fractional variational problems into solving a system of algebraic equations. Also, we show the convergence analysis of the presented technique and introduce some test problems with comparisons between our numerical results with those introduced using different methods.

scholarly journals A new approximation method to solve boundary value problems by using functional perturbation concepts

2018 ◽  
Vol 36 (3) ◽  
pp. 9-25
Author(s):  
Somayeh Pourghanbar ◽  
Mojtaba Ranjbar
Keyword(s):  
Exact Solution ◽  
Boundary Conditions ◽  
Perturbation Method ◽  
Boundary Value Problems ◽  
Approximation Method ◽  
Boundary Value

Functional perturbation method (FPM) is presented for the solution of dierential equations with boundary conditions. Some properties of FPM are utilized to reduce the dierential equation with variable coecients to the equations with constant coecients. The FPM can be applied directly for many types of dierential equations. The exact solution is obtained by only the rst term of the Frechet series for polynomial cases. Four examples are included to demonstrate the method.

scholarly journals Numerical study of n-pentane separation using adsorption column

2005 ◽  
Vol 48 (spe) ◽  
pp. 267-274 ◽  
Author(s):  
Adriano da Silva ◽  
Viviana Cocco Mariani ◽  
Antônio Augusto Ulson de Souza ◽  
Selene Maria de Arruda Guelli Ulson Souza
Keyword(s):  
Heat Transfer ◽  
Computational Algorithm ◽  
Numerical Study ◽  
Algebraic Equations ◽  
Adsorption Column ◽  
Zeolite 5A ◽  
Mass And Heat Transfer ◽  
Model Equations ◽  
The Mathematical Model ◽  
Finite Method

This work simulated numerically the n-pentane separation of a mixture of iso-pentane, n-pentane and nitrogen, using an adsorption column with zeolite 5A. The mathematical model equations of the mass and heat transfer in the adsorption column are presented, as well as the boundary and initials conditions, beyond some hypotheses and considerations. The Volume Finite Method was used in the discretization of the equations to get the system of algebraic equations and posterior development of the computational algorithm. The numerical results using the Differencing Central (CDS) and Upwind (UDS) interpolations were compared with experimental results found in the literature. The influence of the partial pressure in the adsorption column performance was also analyzed.

scholarly journals An Efficient Spectral Approximation for Solving Several Types of Parabolic PDEs with Nonlocal Boundary Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
E. Tohidi ◽  
A. Kılıçman
Keyword(s):  
Boundary Conditions ◽  
Operational Matrix ◽  
Nonlocal Boundary ◽  
Main Idea ◽  
Nonlocal Boundary Conditions ◽  
Algebraic Equations ◽  
Parabolic Pdes ◽  
Direct Collocation ◽  
Operational Matrix Of Differentiation ◽  
The Given

The problem of solving several types of one-dimensional parabolic partial differential equations (PDEs) subject to the given initial and nonlocal boundary conditions is considered. The main idea is based on direct collocation and transforming the considered PDEs into their associated algebraic equations. After approximating the solution in the Legendre matrix form, we use Legendre operational matrix of differentiation for representing the mentioned algebraic equations clearly. Three numerical illustrations are provided to show the accuracy of the presented scheme. High accurate results with respect to the Bernstein Tau technique and Sinc collocation method confirm this accuracy.

scholarly journals A Collocation Method Based on the Bernoulli Operational Matrix for Solving Nonlinear BVPs Which Arise from the Problems in Calculus of Variation

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Emran Tohidi ◽  
Adem Kılıçman
Keyword(s):  
Collocation Method ◽  
Necessary Conditions ◽  
Operational Matrix ◽  
Bernoulli Polynomials ◽  
Variational Problems ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Lagrange Equations ◽  
Calculus Of Variation ◽  
Operational Matrix Of Differentiation

A new collocation method is developed for solving BVPs which arise from the problems in calculus of variation. These BVPs result from the Euler-Lagrange equations, which are the necessary conditions of the extremums of problems in calculus of variation. The proposed method is based upon the Bernoulli polynomials approximation together with their operational matrix of differentiation. After imposing the collocation nodes to the main BVPs, we reduce the variational problems to the solution of algebraic equations. It should be noted that the robustness of operational matrices of differentiation with respect to the integration ones is shown through illustrative examples. Complete comparisons with other methods and superior results confirm the validity and applicability of the presented method.

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