Efficient spectral-Galerkin algorithms for direct solution of fourth-order differential equations using Jacobi polynomials

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A Jacobi–Gauss collocation method for solving nonlinear Lane–Emden type equations (2012, methodology): “The use of general Jacobi polynomials has the advantage of obtaining the equations in terms of the Jacobi parameters a and b (see [14,15]).”
Efficient spectral ultraspherical-Galerkin algorithms for the direct solution of 2nth-order linear differential equations (2009, methodology): “From the numerical point of view, Doha and Abd-Elhameed [17,18], Doha and Bhrawy [19–21], Doha [22] and Shen [23,24] have constructed efficient spectral-Galerkin algorithms using compact combinations of orthogonal polynomials for solving elliptic equations of the second-and fourth-order in various s”
On shifted Jacobi spectral method for high-order multi-point boundary value problems (2012, methodology): “In the present paper we intend to extend the application of Jacobi polynomials from Galerkin method for solving two-point linear problems (see, [10,11]) to tau and collocation methods to solve the linear and nonlinear multi-point BVPs.”
A Jacobi spectral Galerkin method for the integrated forms of fourth‐order elliptic differential equations (2009, unknown): “27) resulting from the integrated forms are sparse and are therefore cheaper to solve than those obtained from the differentiated forms (see, [47]).”