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  • Open Access


    Optimal Shape Factor and Fictitious Radius in the MQ-RBF: Solving Ill-Posed Laplacian Problems

    Chein-Shan Liu1, Chung-Lun Kuo1, Chih-Wen Chang2,*

    CMES-Computer Modeling in Engineering & Sciences, Vol.139, No.3, pp. 3189-3208, 2024, DOI:10.32604/cmes.2023.046002

    Abstract To solve the Laplacian problems, we adopt a meshless method with the multiquadric radial basis function (MQ-RBF) as a basis whose center is distributed inside a circle with a fictitious radius. A maximal projection technique is developed to identify the optimal shape factor and fictitious radius by minimizing a merit function. A sample function is interpolated by the MQ-RBF to provide a trial coefficient vector to compute the merit function. We can quickly determine the optimal values of the parameters within a preferred rage using the golden section search algorithm. The novel method provides the More >

  • Open Access


    Wavelet Multi-Resolution Interpolation Galerkin Method for Linear Singularly Perturbed Boundary Value Problems

    Jiaqun Wang1,2, Guanxu Pan2, Youhe Zhou2, Xiaojing Liu2,*

    CMES-Computer Modeling in Engineering & Sciences, Vol.139, No.1, pp. 297-318, 2024, DOI:10.32604/cmes.2023.030622

    Abstract In this study, a wavelet multi-resolution interpolation Galerkin method (WMIGM) is proposed to solve linear singularly perturbed boundary value problems. Unlike conventional wavelet schemes, the proposed algorithm can be readily extended to special node generation techniques, such as the Shishkin node. Such a wavelet method allows a high degree of local refinement of the nodal distribution to efficiently capture localized steep gradients. All the shape functions possess the Kronecker delta property, making the imposition of boundary conditions as easy as that in the finite element method. Four numerical examples are studied to demonstrate the validity More >

  • Open Access



    Harinakshi Karkeraa , Nagaraj N. Katagia,† , Ramesh B. Kudenattib

    Frontiers in Heat and Mass Transfer, Vol.18, pp. 1-10, 2022, DOI:10.5098/hmt.18.41

    Abstract In this paper, we study the characteristics of laminar boundary-layer flow of a viscous incompressible fluid over a moving wedge. The transformed boundary-layer equation given by the Falkner-Skan equation is solved by an efficient easy-to-use approximate method based on uniform Haar wavelets in conjunction with quasilinearization and collocation approach. The residual and error estimates are computed to confirm the validity of the obtained results. A meaningful comparison between the present solutions with existing numerical results in the literature is carried out to highlight the benefits and efficiency of proposed method. Furthermore, the influence of variable More >

  • Open Access


    Efficient Numerical Scheme for Solving Large System of Nonlinear Equations

    Mudassir Shams1, Nasreen Kausar2,*, Shams Forruque Ahmed3, Irfan Anjum Badruddin4, Syed Javed4

    CMC-Computers, Materials & Continua, Vol.74, No.3, pp. 5331-5347, 2023, DOI:10.32604/cmc.2023.033528

    Abstract A fifth-order family of an iterative method for solving systems of nonlinear equations and highly nonlinear boundary value problems has been developed in this paper. Convergence analysis demonstrates that the local order of convergence of the numerical method is five. The computer algebra system CAS-Maple, Mathematica, or MATLAB was the primary tool for dealing with difficult problems since it allows for the handling and manipulation of complex mathematical equations and other mathematical objects. Several numerical examples are provided to demonstrate the properties of the proposed rapidly convergent algorithms. A dynamic evaluation of the presented methods More >

  • Open Access


    A Pseudo-Spectral Scheme for Systems of Two-Point Boundary Value Problems with Left and Right Sided Fractional Derivatives and Related Integral Equations

    I. G. Ameen1, N. A. Elkot2, M. A. Zaky3,*, A. S. Hendy4,5, E. H. Doha2

    CMES-Computer Modeling in Engineering & Sciences, Vol.128, No.1, pp. 21-41, 2021, DOI:10.32604/cmes.2021.015310

    Abstract We target here to solve numerically a class of nonlinear fractional two-point boundary value problems involving left- and right-sided fractional derivatives. The main ingredient of the proposed method is to recast the problem into an equivalent system of weakly singular integral equations. Then, a Legendre-based spectral collocation method is developed for solving the transformed system. Therefore, we can make good use of the advantages of the Gauss quadrature rule. We present the construction and analysis of the collocation method. These results can be indirectly applied to solve fractional optimal control problems by considering the corresponding More >

  • Open Access


    Spectral Solutions of Linear and Nonlinear BVPs Using Certain Jacobi Polynomials Generalizing Third- and Fourth-Kinds of Chebyshev Polynomials

    W. M. Abd-Elhameed1,2,*, Asmaa M. Alkenedri2

    CMES-Computer Modeling in Engineering & Sciences, Vol.126, No.3, pp. 955-989, 2021, DOI:10.32604/cmes.2021.013603

    Abstract This paper is dedicated to implementing and presenting numerical algorithms for solving some linear and nonlinear even-order two-point boundary value problems. For this purpose, we establish new explicit formulas for the high-order derivatives of certain two classes of Jacobi polynomials in terms of their corresponding Jacobi polynomials. These two classes generalize the two celebrated non-symmetric classes of polynomials, namely, Chebyshev polynomials of third- and fourth-kinds. The idea of the derivation of such formulas is essentially based on making use of the power series representations and inversion formulas of these classes of polynomials. The derived formulas More >

  • Open Access


    Numerical Solving of a Boundary Value Problem for Fuzzy Differential Equations

    Afet Golayoğlu Fatullayev1, Canan Köroğlu2

    CMES-Computer Modeling in Engineering & Sciences, Vol.86, No.1, pp. 39-52, 2012, DOI:10.3970/cmes.2012.086.039

    Abstract In this work we solve numerically a boundary value problem for second order fuzzy differential equations under generalized differentiability in the form y''(t) = p(t)y'(t) + q(t)y(t) + F(t) y(0) = γ, y(l) = λ where t ∈T = [0,l], p(t)≥0, q(t)≥0 are continuous functions on [0,l] and [γ]α = [γ_αα], [λ]α = [λ_α¯α] are fuzzy numbers. There are four different solutions of the problem (0.1) when the fuzzy derivative is considered as generalization of the H-derivative. An algorithm is presented and the finite difference method is used for solving obtained problems. The applicability More >

  • Open Access


    The coupling FEM and NBEM with non-matching grids for a class of nonlinear boundary value problems

    Ju E Yang, Qiya Hu, Dehao Yu

    The International Conference on Computational & Experimental Engineering and Sciences, Vol.16, No.3, pp. 71-72, 2011, DOI:10.3970/icces.2011.016.071

    Abstract In this paper, based on the natural boundary reduction method advanced bu Feng and Yu, we are concerned with a domain decomposition method with nonmatching grids for a certain nonlinear interface problem in unbounded domains. We first discuss a new coupling of finite element and boundary element by adding an auxiliary circle. Then we use a dual basis multipier on the interface to provide the numerical analysis with nonmatching grids. Finally, we give some numerical examples further to confirm our theoretical results. More >

  • Open Access


    General ray method for solution of the Dirichlet boundary value problems for elliptic partial differential equations in domains with complicated geometry

    A. Grebennikov1

    The International Conference on Computational & Experimental Engineering and Sciences, Vol.15, No.3, pp. 85-90, 2010, DOI:10.3970/icces.2010.015.085

    Abstract New General Ray (GR) method for solution of the Dirichlet boundary value problem for the class of elliptic Partial Differential Equations (PDE) is proposed. GR-method consists in application of the Radon transform directly to the PDE and in reduction PDE to assemblage of Ordinary Differential Equations (ODE). The class of the PDE includes the Laplace, Poisson and Helmgoltz equations. GR-method presents the solution of the Dirichlet boundary value problem for this type of equations by explicit analytical formulas that use the direct and inverse Radon transform. Proposed version of GR-method justified theoretically, realized by fast algorithms and More >

  • Open Access


    The Lie-Group Shooting Method for Nonlinear Two-Point Boundary Value Problems Exhibiting Multiple Solutions

    Chein-Shan Liu1

    The International Conference on Computational & Experimental Engineering and Sciences, Vol.5, No.2, pp. 55-84, 2008, DOI:10.3970/icces.2008.005.055

    Abstract The present paper provides a Lie-group shooting method for the numerical solutions of second-order nonlinear boundary value problems exhibiting multiple solutions. It aims to find all solutions as easy as possible. The boundary conditions considered are classified into four types, namely the Dirichlet, the first Robin, the second Robin and the Neumann. The two Robin type problems are transformed into a canonical one by using the technique of symmetric extension of the governing equations. The Lie-group shooting method is very effective to search unknown initial condition through a weighting factor r(0,1). Furthermore, the closed-form solutions are More >

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