Combining the reproducing kernel method with a practical technique to solve the system of nonlinear singularly perturbed boundary value problems

We introduce a new method for solving Riccati differential equations, which is based on reproducing kernel method and quasilinearization technique. The quasilinearization technique is used to reduce the Riccati differential equation to a sequence of linear problems. The resulting sets of differential equations are treated by using reproducing kernel method. The solutions of Riccati differential equations obtained using many existing methods give good approximations only in the neighborhood of the initial position. However, the solutions obtained using the present method give good approximations in a larger interval, rather than a local vicinity of the initial position. Numerical results compared with other methods show that the method is simple and effective.

This article proposes a new approach for solving linear Volterra integral equations with variable coefficients using the Reproducing Kernel Method (RKM). This method eliminates the need for the Gram-Schmidt process. However, the accuracy of RKM is influenced by various factors, including the selection of points, bases, space, and implementation method. The main objective of this article is to introduce a generalized method based on the Reproducing Kernel, which is successful in solving a special type of singular weakly nonlinear boundary value problems (BVPs). The easy implementation, elimination of the Gram-Schmidt process, fewer calculations, and high accuracy of the present method are interesting. The conformity of numerical results, including tables and figures, with theorems related to error analysis and convergence order, confirms the practicality of the present method.

Numerical investigations for systems of second-order periodic boundary value problems using reproducing kernel method

Two-dimensional orthogonal polynomials for vibration of rectangular composite plates

In this study, an iterative approximation is proposed by using the reproducing kernel method (RKM) for the nonlinear advection equation. To apply the iterative RKM, specific reproducing kernel spaces are defined and their kernel functions are presented. The proposed method requires homogenising the initial or boundary conditions of the problem under consideration. After homogenising the initial condition of the advection equation, a linear operator selection is made, and then the approximate solution is constructed using orthonormal basis functions in serial form. Convergence analysis of the approximate solution is demonstrated through the lemma and theorem. Numerical outcomes are provided in the form of graphics and tables to show the efficiency and accuracy of the presented method.

The loss of orthogonality in the Gram-Schmidt orthogonalization process

Purpose: In this paper, a new algorithm is presented for solving one-dimensional parabolic partial differential equations subject to integral conditions. Methods: The algorithm is based on the transverse method of lines and the reproducing kernel method. The transverse method of lines can reduce a one-dimensional parabolic partial differential equation subject to integral conditions to a series of ordinary differential equations(ODEs) with integral boundary conditions. The reproducing kernel method is a relative new analytical technique, which can solve successfully ODEs with integral boundary conditions.

Study on flexural vibration of triangular composite plates influenced by fibre orientation

Two data analytic research areas—penalized splines and reproducing kernel methods—have become very vibrant since the mid-1990s. This article shows how the former can be embedded in the latter via theory for reproducing kernel Hilbert spaces. This connection facilitates cross-fertilization between the two bodies of research. In particular, connections between support vector machines and penalized splines are established. These allow for significant reductions in computational complexity, and easier incorporation of special structure such as additivity.

The traditional reproducing kernel method (TRKM) cannot obtain satisfactory numerical results for solving the partial differential equation (PDE). In this study, for the first time, the abovementioned problems are solved by adaptive piecewise interpolation reproducing kernel method (APIRKM) to obtain the exact and approximate solutions of partial differential equations by means of series expansion using reconstructed kernel function. The highlight of this paper is to obtain more accurate approximate solution and save more time through adaptive discovery. Numerical solutions of the three examples show that the present method is more advantageous than TRKM.

Generalized multiple scale reproducing kernel particle methods

An efficient model correction method is proposed by using the modal measurement from a structural system. The method corrects/updates the mass and stiffness matrix without imposing any parameterization. It considers the information from both the nominal finite-element model and the measurement of modal frequencies and mode shapes. The method is computationally very efficient and it does not require computation of the complete set of eigenvalues and eigenvectors of the nominal model. Instead, only the nominal eigenvalues and eigenvectors of the modes to be corrected are needed. The Gram-Schmidt orthogonalization process is used to construct a basis that satisfies the mass orthogonality condition. This basis is used to transform the eigenvectors of the nominal model so that the corrected model is compatible with the measurement. A thousand-degree-of-freedom chainlike system and a 1,440-degree-of-freedom structural frame are used to illustrate the proposed method.

Chapter 5 - Vector Spaces

In the previous chapter, the theory and methods of elementary linear algebra were extended to linear spaces, in that finite-dimensional vector spaces were extended to include infinite-dimensional sequence and function spaces, that the inner product was applied to derive such mathematical tools as orthogonal projection and the Gram-Schmidt orthogonalization process, and that the concept of completeness was studied to introduced Hilbert and Banach spaces.

An alternative approach to orthogonal graph theoretical invariants