A numerical solution for heat transfer past a stretching sheet with ohmic dissipation and suction or injection problem using Haar wavelet quasilinearization method

This paper represents a numerical analysis for heat transfer of a Jeffrey fluid flow past a stretching sheet with ohmic dissipation and suction/injection. The partial differential equations are reduced into a set of convenient nonlinear ordinary differential equations with the boundary conditions. Haar wavelet quasilinearization method (HWQM) is used to solve ordinary differential equations. The effect of various related parameters on velocity and temperature profiles are computed and analyzed. Then, comparison is made between the numerical results of proposed method with existing numerical solutions found in the literature, and reasonable agreement is noted

A numerical solution for nonlinear heat transfer of fin problems using the Haar wavelet quasilinearization method

The aim of this paper is to study the new application of Haar wavelet quasilinearization method (HWQM) to solve one-dimensional nonlinear heat transfer of fin problems. Three different types of nonlinear problems are numerically treated and the HWQM solutions are compared with those of the other method. The effects of temperature distribution of a straight fin with temperature-dependent thermal conductivity in the presence of various parameters related to nonlinear boundary value problems are analyzed and discussed. Numerical results of HWQM gives excellent numerical results in terms of competitiveness and accuracy compared to other numerical methods. This method was proven to be stable, convergent and, easily coded

Numerical solution of elliptic partial differential equations by haar wavelet operational matrix method / Nor Artisham Che Ghani

The purpose of this study is to establish a simple numerical method based on the Haar wavelet operational matrix of integration for solving two dimensional elliptic partial differential equations of the form, Ñ2u(x, y) + ku(x, y) = f (x, y) with the Dirichlet boundary conditions. To achieve the target, the Haar wavelet series were studied, which came from the expansion for any two dimensional functions g(x, y) defined on L2 ([0,1)´ [0,1)), i.e. g(x, y) =Σc h (x)h ( y) ij i j or compactly written as HT (x)CH( y) , where C is the coefficient matrix and H(x) or H( y) is a Haar function vector. Wu (2009) had previously used this expansion to solve first order partial differential equations. In this work, we extend this method to the solution of second order partial differential equations. The main idea behind the Haar operational matrix for solving the second order partial differential equations is the determination of the coefficient matrix, C. If the function f (x, y) is known, then C can be easily computed as H × F × HT , where F is the discrete form for f (x, y) . However, if the function u(x, y) appears as the dependent variable in the elliptic equation, the highest partial derivatives are first expanded as Haar wavelet series, i.e. u HT (x)CH( y) xx = and u HT (x)DH( y) yy = , and the coefficient matrices C and D usually can be solved by using Lyapunov or Sylvester type equation. Then, the solution u(x, y) can easily be obtained through Haar operational matrix. The key to this is the identification for the form of coefficient matrix when the function is separable. Three types of elliptic equations solved by the new method are demonstrated and the results are then compared with exact solution given. For the beginning, the computation iii was carried out for lower resolution. As expected, the more accurate results can be obtained by increasing the resolution and the convergence are faster at collocation points. This research is preliminary work on two dimensional space elliptic equation via Haar wavelet operational matrix method

Numerical solution of elliptic partial differential equations by Haar wavelet operational matrix method / Nor Artisham Che Ghani

The purpose of this study is to establish a simple numerical method based on the Haar wavelet operational matrix of integration for solving two dimensional elliptic partial differential equations of the form, Ñ2u(x, y) + ku(x, y) = f (x, y) with the Dirichlet boundary conditions. To achieve the target, the Haar wavelet series were studied, which came from the expansion for any two dimensional functions g(x, y) defined on L2 ([0,1)´ [0,1)), i.e. g(x, y) =Σc h (x)h ( y) ij i j or compactly written as HT (x)CH( y) , where C is the coefficient matrix and H(x) or H( y) is a Haar function vector. Wu (2009) had previously used this expansion to solve first order partial differential equations. In this work, we extend this method to the solution of second order partial differential equations. The main idea behind the Haar operational matrix for solving the second order partial differential equations is the determination of the coefficient matrix, C. If the function f (x, y) is known, then C can be easily computed as H × F × HT , where F is the discrete form for f (x, y) . However, if the function u(x, y) appears as the dependent variable in the elliptic equation, the highest partial derivatives are first expanded as Haar wavelet series, i.e. u HT (x)CH( y) xx = and u HT (x)DH( y) yy = , and the coefficient matrices C and D usually can be solved by using Lyapunov or Sylvester type equation. Then, the solution u(x, y) can easily be obtained through Haar operational matrix. The key to this is the identification for the form of coefficient matrix when the function is separable. Three types of elliptic equations solved by the new method are demonstrated and the results are then compared with exact solution given. For the beginning, the computation was carried out for lower resolution. As expected, the more accurate results can be obtained by increasing the resolution and the convergence are faster at collocation points. This research is preliminary work on two dimensional space elliptic equation via Haar wavelet operational matrix method. We hope to extend this method for solving diffusion equation, k u t u = Ñ2 ¶ ¶ and wave equation, c u t u 2 2 2 2 = Ñ ¶ ¶ in a plane

Extended Haar Wavelet Quasilinearization method for solving boundary value problems / Nor Artisham Che Ghani

Several computational methods have been proposed to solve single nonlinear ordinary differential equations. In spite of the enormous numerical effort, however yet numerically accurate and robust algorithm is still missing. Moreover, to the best of our knowledge, only a few works are dedicated to the numerical solution of coupled nonlinear ordinary differential equations. Hence, a robust algorithm based on Haar wavelets and the quasilinearization process is provided in this study for solving both numerical solutions; single nonlinear ordinary differential equations and systems of coupled nonlinear ordinary differential equations, including two of them are the new problems with some additional related parameters. In this research, the generation of Haar wavelets function, its series expansion and one-dimensional matrix for a chosen interval 0, B is introduced in detail. We expand the usual defined interval 0, 1 to 0, B because the actual problem does not necessarily involve only limit B to one, especially in the case of coupled nonlinear ordinary differential equations. To achieve the target, quasilinearization technique is used to linearize the nonlinear ordinary differential equations, and then the Haar wavelet method is applied in the linearized problems. Quasilinearization technique provides a sequence of function which monotonic quadratically converges to the solution of the original equations. The highest derivatives appearing in the differential equations are first expanded into Haar series. The lower order derivatives and the solutions can then be obtained quite easily by using multiple integration of Haar wavelet. All the values of Haar wavelet functions are substituted into the quasilinearized problem. The wavelet coefficient can be calculated easily by using MATLAB software. The universal subprogram is introduced to calculate the integrals of Haar wavelets. This will provide small computational time. The initial approximation can be determined from mathematical or physical consideration. In the demonstration problem, the performance of Haar wavelet quasilinearization method (HWQM) is compared with the existing numerical solutions that showed the same basis found in the literature. For the beginning, the computation was carried out for lower resolution. As expected, the more accurate results can be obtained by increasing the resolution and the convergence are faster at collocation points. For systems of coupled nonlinear ordinary differential equations, the equations are obtained through the similarity transformations. The transformed equations are then solved numerically. This is contrary to Runge-Kutta method, where the boundary value problems of HWQM need not to be reduced into a system of first order ordinary differential equations. Besides in terms of accuracy, efficiency and applicability in solving nonlinear ordinary differential equations for a variety of boundary conditions, this method also allow simplicity, fast and small computation cost since most elements of the matrices of Haar wavelet and its integration are zeros, it were contributed to the speeding up of the computation. This method can therefore serve as very useful tool in many physical applications

A numerical solution for heat transfer past a stretching sheet with ohmic dissipation and suction or injection problem using Haar wavelet quasilinearization method

This paper represents a numerical analysis for heat transfer of a Jeffrey fluid flow past a stretching sheet with ohmic dissipation and suction/injection. The partial differential equations are reduced into a set of convenient nonlinear ordinary differential equations with the boundary conditions. Haar wavelet quasilinearization method (HWQM) is used to solve ordinary differential equations. The effect of various related parameters on velocity and temperature profiles are computed and analyzed. Then, comparison is made between the numerical results of proposed method with existing numerical solutions found in the literature, and reasonable agreement is noted

Heat transfer over a steady stretching surface in the presence of suction

The purpose of this paper is to present the Cattaneo–Christov heat flux model for Maxwell fluid past a stretching surface where the presence of suction/injection is taken into account. The governing system of equations is reduced to the ordinary differential equations with the boundary conditions by similarity transformation. These equations are then solved numerically by two approaches, Haar wavelet quasilinearization method (HWQM) and Runge–Kutta–Gill method (RK Gill). The behavior of various pertinent parameters on velocity and temperature distributions is analyzed and discussed. Comparison of the obtained numerical results is made between both methods and with the existing numerical solutions found in the literature, and reasonable agreement is noted

A numerical solution for nonlinear heat transfer of fin problems using the Haar wavelet quasilinearization method

The aim of this paper is to study the new application of Haar wavelet quasilinearization method (HWQM) to solve one-dimensional nonlinear heat transfer of fin problems. Three different types of nonlinear problems are numerically treated and the HWQM solutions are compared with those of the other method. The effects of temperature distribution of a straight fin with temperature-dependent thermal conductivity in the presence of various parameters related to nonlinear boundary value problems are analyzed and discussed. Numerical results of HWQM gives excellent numerical results in terms of competitiveness and accuracy compared to other numerical methods. This method was proven to be stable, convergent and, easily coded. © 2019 The Author