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• Open Access

ARTICLE

### The Second Kind Chebyshev Wavelet Method for Fractional Differential Equations with Variable Coefficients

CMES-Computer Modeling in Engineering & Sciences, Vol.93, No.3, pp. 187-202, 2013, DOI:10.3970/cmes.2013.093.187

Abstract In this article, the second kind Chebyshev wavelet method is presented for solving a class of multi-order fractional differential equations (FDEs) with variable coefficients. We first construct the second kind Chebyshev wavelet, prove its convergence and then derive the operational matrix of fractional integration of the second kind Chebyshev wavelet. The operational matrix of fractional integration is utilized to reduce the fractional differential equations to a system of algebraic equations. In addition, illustrative examples are presented to demonstrate the efficiency and accuracy of the proposed method. More >

• Open Access

ARTICLE

### Wavelet operational matrix method for solving fractional integral and differential equations of Bratu-type

CMES-Computer Modeling in Engineering & Sciences, Vol.92, No.4, pp. 353-368, 2013, DOI:10.3970/cmes.2013.092.353

Abstract In this paper, a wavelet operational matrix method based on the second kind Chebyshev wavelet is proposed to solve the fractional integral and differential equations of Bratu-type. The second kind Chebyshev wavelet operational matrix of fractional order integration is derived. A truncated second kind Chebyshev wavelet series together with the wavelet operational matrix is utilized to reduce the fractional integral and differential equations of Bratu-type to a system of nonlinear algebraic equations. The convergence and the error analysis of the method are also given. Two examples are included to verify the validity and applicability of More >

• Open Access

ARTICLE

### Numerical solution of fractional partial differential equations using Haar wavelets

CMES-Computer Modeling in Engineering & Sciences, Vol.91, No.4, pp. 269-287, 2013, DOI:10.3970/cmes.2013.091.269

Abstract In this paper, we present a computational method for solving a class of fractional partial differential equations which is based on Haar wavelets operational matrix of fractional order integration. We derive the Haar wavelets operational matrix of fractional order integration. Haar wavelets method is used because its computation is sample as it converts the original problem into Sylvester equation. Finally, some examples are included to show the implementation and accuracy of the approach. More >

• Open Access

ARTICLE

### Numerical solution of nonlinear fractional integral differential equations by using the second kind Chebyshev wavelets

CMES-Computer Modeling in Engineering & Sciences, Vol.90, No.5, pp. 359-378, 2013, DOI:10.3970/cmes.2013.090.359

Abstract By using the differential operator matrix and the product operation matrix of the second kind Chebyshev wavelets, a class of nonlinear fractional integral-differential equations is transformed into nonlinear algebraic equations, which makes the solution process and calculation more simple. At the same time, the maximum absolute error is obtained through error analysis. It also can be used under the condition that no exact solution exists. Numerical examples verify the validity of the proposed method. More >

• Open Access

ARTICLE

### Numerical Solution of Space-Time Fractional Convection-Diffusion Equations with Variable Coefficients Using Haar Wavelets

CMES-Computer Modeling in Engineering & Sciences, Vol.89, No.6, pp. 481-495, 2012, DOI:10.3970/cmes.2012.089.481

Abstract In this paper, we present a computational method for solving a class of space-time fractional convection-diffusion equations with variable coefficients which is based on the Haar wavelets operational matrix of fractional order differentiation. Haar wavelets method is used because its computation is sample as it converts the original problem into Sylvester equation. Error analysis is given that shows efficiency of the method. Finally, a numerical example shows the implementation and accuracy of the approach. More >

• Open Access

ARTICLE

### Haar Wavelet Operational Matrix Method for Solving Fractional Partial Differential Equations

CMES-Computer Modeling in Engineering & Sciences, Vol.88, No.3, pp. 229-244, 2012, DOI:10.3970/cmes.2012.088.229

Abstract In this paper, Haar wavelet operational matrix method is proposed to solve a class of fractional partial differential equations. We derive the Haar wavelet operational matrix of fractional order integration. Meanwhile, the Haar wavelet operational matrix of fractional order differentiation is obtained. The operational matrix of fractional order differentiation is utilized to reduce the initial equation to a Sylvester equation. Some numerical examples are included to demonstrate the validity and applicability of the approach. More >

• Open Access

ARTICLE

### Application of Geometric Approach for Fuzzy Linear Systems to a Fuzzy Input-Output Analysis

CMES-Computer Modeling in Engineering & Sciences, Vol.88, No.2, pp. 93-106, 2012, DOI:10.3970/cmes.2012.088.093

Abstract Uncertainties in some parameters of problems of Leontief input-output analysis lead naturally to fuzzy linear systems. In this work, we consider input-output model, where the technology matrix is crisp and the vector of final outputs is fuzzy. The model is expressed by a fuzzy linear system with crisp matrix and with fuzzy right-hand side vector. We apply a geometric method for solving the system. The method finds the solution in the form of a fuzzy set of vectors. The solution set is shown to be a parallelepiped in coordinate space and is expressed by an More >

• Open Access

ARTICLE

### Strong Solutions of the Fuzzy Linear Systems

CMES-Computer Modeling in Engineering & Sciences, Vol.76, No.3&4, pp. 207-216, 2011, DOI:10.3970/cmes.2011.076.207

Abstract We consider a fuzzy linear system with crisp coefficient matrix and with an arbitrary fuzzy number in parametric form on the right-hand side. It is known that the well-known existence and uniqueness theorem of a strong fuzzy solution is equivalent to the following: The coefficient matrix is the product of a permutation matrix and a diagonal matrix. This means that this theorem can be applicable only for a special form of linear systems, namely, only when the system consists of equations, each of which has exactly one variable. We prove an existence and uniqueness theorem, More >

• Open Access

ARTICLE

### A Geometric Approach to Solve Fuzzy Linear Systems

CMES-Computer Modeling in Engineering & Sciences, Vol.75, No.3&4, pp. 189-204, 2011, DOI:10.3970/cmes.2011.075.189

Abstract In this paper, linear systems with a crisp real coefficient matrix and with a vector of fuzzy triangular numbers on the right-hand side are studied. A new method, which is based on the geometric representations of linear transformations, is proposed to find solutions. The method uses the fact that a vector of fuzzy triangular numbers forms a rectangular prism in n-dimensional space and that the image of a parallelepiped is also a parallelepiped under a linear transformation. The suggested method clarifies why in general case different approaches do not generate solutions as fuzzy numbers. It… More >

• Open Access

ARTICLE

### Inverse Sensitivity Analysis of Singular Solutions of FRF matrix in Structural System Identification

CMES-Computer Modeling in Engineering & Sciences, Vol.37, No.2, pp. 113-152, 2008, DOI:10.3970/cmes.2008.037.113

Abstract The problem of structural damage detection based on measured frequency response functions of the structure in its damaged and undamaged states is considered. A novel procedure that is based on inverse sensitivity of the singular solutions of the system FRF matrix is proposed. The treatment of possibly ill-conditioned set of equations via regularization scheme and questions on spatial incompleteness of measurements are considered. The application of the method in dealing with systems with repeated natural frequencies and (or) packets of closely spaced modes is demonstrated. The relationship between the proposed method and the methods based More >

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