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Journal articles on the topic 'Spline theory. Differential equations'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Tarang, M. "STABILITY OF THE SPLINE COLLOCATION METHOD FOR SECOND ORDER VOLTERRA INTEGRO‐DIFFERENTIAL EQUATIONS." Mathematical Modelling and Analysis 9, no. 1 (March 31, 2004): 79–90. http://dx.doi.org/10.3846/13926292.2004.9637243.

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Numerical stability of the spline collocation method for the 2nd order Volterra integro‐differential equation is investigated and connection between this theory and corresponding theory for the 1st order Volterra integro‐differential equation is established. Results of several numerical tests are presented. Straipsnyje nagrinejamas antros eiles Volteros integro‐diferencialiniu lygčiu splainu kolokaci‐jos metodo skaitinis stabilumas ir nustatytas ryšys tarp šios teorijos ir atitinkamos pirmos eiles Volterra integro‐diferencialiniu lygčiu teorijos. Pateikti keleto skaitiniu eksperimentu rezultatai.

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2

Ibrahim, M. A. K., A. El-Safty, and Shadia M. Abo-Hasha. "Application of spline functions to neutral delay-differential equations." International Journal of Computer Mathematics 62, no. 3-4 (January 1996): 233–39. http://dx.doi.org/10.1080/00207169608804540.

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3

Ayad, A. "Spline approximation for second order fredholm integro-differential equations." International Journal of Computer Mathematics 66, no. 1-2 (January 1998): 79–91. http://dx.doi.org/10.1080/00207169808804626.

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4

Zhao, J., M. S. Cheung, and S. F. Ng. "Spline Kantorovich method and analysis of general slab bridge deck." Canadian Journal of Civil Engineering 25, no. 5 (October 1, 1998): 935–42. http://dx.doi.org/10.1139/l98-030.

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In this paper, the spline Kantorovich method is developed and applied to the analysis and design of bridge decks. First, the bridge deck is mapped into a unit square in the Xi - eta plane. The governing partial differential equation of the plate is reduced to the ordinary differential equation in the longitudinal direction of the bridge by the routine Kantorovich method. Spline point collocation method is then used to solve the derived ordinary differential equation to obtain the displacements and internal forces of the bridge deck. Mindlin plate theory is incorporated into the differential equation and, as a result, the effect of shear deformation of the plate is also considered. Possible shear locking is avoided by the reduced integration technique. Numerical examples show that the proposed new numerical model is versatile, efficient, and reliable.Key words: Kantorovich method, spline function, partial differential equations, ordinary differential equations, point collocation method, bridge deck.

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5

El-Safty, A., and Shadia M. Abo-Hasha. "Stability of 2h-step spline method for delay differential equations." International Journal of Computer Mathematics 74, no. 3 (January 2000): 315–24. http://dx.doi.org/10.1080/00207160008804945.

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6

Ayad, A. "Spline approximation for first Order fredholm delay integro-differential equations." International Journal of Computer Mathematics 70, no. 3 (January 1999): 467–76. http://dx.doi.org/10.1080/00207169908804768.

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7

Srivastava, Hari Mohan, Pshtiwan Othman Mohammed, Juan L. G. Guirao, and Y. S. Hamed. "Some Higher-Degree Lacunary Fractional Splines in the Approximation of Fractional Differential Equations." Symmetry 13, no. 3 (March 5, 2021): 422. http://dx.doi.org/10.3390/sym13030422.

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In this article, we begin by introducing two classes of lacunary fractional spline functions by using the Liouville–Caputo fractional Taylor expansion. We then introduce a new higher-order lacunary fractional spline method. We not only derive the existence and uniqueness of the method, but we also provide the error bounds for approximating the unique positive solution. As applications of our fundamental findings, we offer some Liouville–Caputo fractional differential equations (FDEs) to illustrate the practicability and effectiveness of the proposed method. Several recent developments on the the theory and applications of FDEs in (for example) real-life situations are also indicated.

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8

Mittal, R. C., and Amit Tripathi. "Numerical solutions of two-dimensional Burgers’ equations using modified Bi-cubic B-spline finite elements." Engineering Computations 32, no. 5 (July 6, 2015): 1275–306. http://dx.doi.org/10.1108/ec-04-2014-0067.

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Purpose – The purpose of this paper is to develop an efficient numerical scheme for non-linear two-dimensional (2D) parabolic partial differential equations using modified bi-cubic B-spline functions. As a test case, method has been applied successfully to 2D Burgers equations. Design/methodology/approach – The scheme is based on collocation of modified bi-cubic B-Spline functions. The authors used these functions for space variable and for its derivatives. Collocation form of the partial differential equation results into system of first-order ordinary differential equations (ODEs). The obtained system of ODEs has been solved by strong stability preserving Runge-Kutta method. The computational complexity of the method is O(p log(p)), where p denotes total number of mesh points. Findings – Obtained numerical solutions are better than those available in literature. Ease of implementation and very small size of computational work are two major advantages of the present method. Moreover, this method provides approximate solutions not only at the grid points but also at any point in the solution domain. Originality/value – First time, modified bi-cubic B-spline functions have been applied to non-linear 2D parabolic partial differential equations. Efficiency of the proposed method has been confirmed with numerical experiments. The authors conclude that the method provides convergent approximations and handles the equations very well in different cases.

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9

Ibrahim, M. A. K., A. El-Safty, and Shadia M. Abo-Hasha. "On the p-stability of quadratic spline for delay differential equations." International Journal of Computer Mathematics 52, no. 3-4 (January 1994): 219–23. http://dx.doi.org/10.1080/00207169408804306.

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10

WELLS, J. C., V. E. OBERACKER, M. R. STRAYER, and A. S. UMAR. "SPECTRAL PROPERTIES OF DERIVATIVE OPERATORS IN THE BASIS-SPLINE COLLOCATION METHOD." International Journal of Modern Physics C 06, no. 01 (February 1995): 143–67. http://dx.doi.org/10.1142/s0129183195000125.

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We discuss the basis-spline collocation method for the lattice solution of boundary-value differential equations, drawing particular attention to the difference between lattice and continuous collocation methods. Spectral properties of the basis-spline lattice representation of the first and second spatial derivatives are studied for the case of periodic boundary conditions with homogeneous lattice spacing and compared to spectra obtained using traditional finite-difference schemes. Basis-spline representations are shown to give excellent resolution on small-length scales and to satisfy the chain rule with good fidelity for the lattice-derivative operators using high-order splines. Application to the one-dimensional Dirac equation shows that very high-order spline representations of the Hamiltonian on odd lattices avoid the notorious spectral-doubling problem.

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11

Slassi, Mehdi. "A Milstein-based free knot spline approximation for stochastic differential equations." Journal of Complexity 28, no. 1 (February 2012): 37–47. http://dx.doi.org/10.1016/j.jco.2011.03.005.

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12

Sahu, P. K., A. K. Ranjan, and S. Saha Ray. "B-spline Wavelet Method for Solving Fredholm Hammerstein Integral Equation Arising from Chemical Reactor Theory." Nonlinear Engineering 7, no. 3 (September 25, 2018): 163–69. http://dx.doi.org/10.1515/nleng-2017-0116.

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Abstract Mathematical model for an adiabatic tubular chemical reactor which processes an irreversible exothermic chemical reaction has been considered. For steady state solution for an adiabatic tubular chemical reactor, the model can be reduced to ordinary differential equation with a parameter in the boundary conditions. Again the ordinary differential equation has been converted into a Hammerstein integral equation which can be solved numerically. B-spline wavelet method has been developed to approximate the solution of Hammerstein integral equation. This method reduces the integral equation to a system of algebraic equations. The numerical results obtained by the present method have been compared with the available results.

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13

Mahmoud, S. M. "A class of three-point spline collocation methods for solving delay differential equations." International Journal of Computer Mathematics 84, no. 10 (October 2007): 1495–508. http://dx.doi.org/10.1080/00207160701303409.

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14

Wang, Feng, and Song Xiang. "A nth-Order Shear Deformation Theory for the Free Vibration Analysis of the Isotropic Plates on Elastic Foundations." Advanced Materials Research 1033-1034 (October 2014): 860–63. http://dx.doi.org/10.4028/www.scientific.net/amr.1033-1034.860.

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A nth-order shear deformation theory for free vibration of the isotropic plates resting on a two-parameter Pasternak foundations is developed. The present theory does not require shear correction factor, and satisfies the zero transverse shear stress boundary conditions on the top and bottom surface of the plate. Governing equations are derived from the principle of virtual displacements. Meshless global collocation method based on the thin plate spline radial basis function is used to solve the governing differential equations. The accuracy of the present theory is demonstrated by comparing the present results with available published results.

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15

Alshomrani, Ali Saleh, Sapna Pandit, Abdullah K. Alzahrani, Metib Said Alghamdi, and Ram Jiwari. "A numerical algorithm based on modified cubic trigonometric B-spline functions for computational modelling of hyperbolic-type wave equations." Engineering Computations 34, no. 4 (June 12, 2017): 1257–76. http://dx.doi.org/10.1108/ec-05-2016-0179.

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Purpose The main purpose of this work is the development of a numerical algorithm based on modified cubic trigonometric B-spline functions for computational modelling of hyperbolic-type wave equations. These types of equations describe a variety of physical models in the vibrations of structures, nonlinear optics, quantum field theory and solid-state physics, etc. Design/methodology/approach Dirichlet boundary conditions cannot be handled easily by cubic trigonometric B-spline functions. Then, a modification is made in cubic trigonometric B-spline functions to handle the Dirichlet boundary conditions and a numerical algorithm is developed. The proposed algorithm reduced the hyperbolic-type wave equations into a system of first-order ordinary differential equations (ODEs) in time variable. Then, stability-preserving SSP-RK54 scheme and the Thomas algorithm are used to solve the obtained system. The stability of the algorithm is also discussed. Findings A different technique based on modified cubic trigonometric B-spline functions is proposed which is quite different from the schemes developed (Abbas et al., 2014; Nazir et al., 2016) and depicts the computational modelling of hyperbolic-type wave equations. Originality/value To the best of the authors’ knowledge, this technique is novel for solving hyperbolic-type wave equations and the developed algorithm is free from quasi-linearization process and finite difference operators for time derivatives. This algorithm gives better results than the results discussed in literature (Dehghan and Shokri, 2008; Batiha et al., 2007; Mittal and Bhatia, 2013; Jiwari, 2015).

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16

Viswanathan, K. K., Kyung Su Kim, Kyung Ho Lee, and Jang Hyun Lee. "Free Vibration of Layered Circular Cylindrical Shells of Variable Thickness Using Spline Function Approximation." Mathematical Problems in Engineering 2010 (2010): 1–14. http://dx.doi.org/10.1155/2010/547956.

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Free vibration of layered circular cylindrical shells of variable thickness is studied using spline function approximation by applying a point collocation method. The shell is made up of uniform layers of isotropic or specially orthotropic materials. The equations of motions in longitudinal, circumferential and transverse displacement components, are derived using extension of Love's first approximation theory. The coupled differential equations are solved using Bickley-type splines of suitable order, which are cubic and quintic, by applying the point collocation method. This results in the generalized eigenvalue problem by combining the suitable boundary conditions. The effect of frequency parameters and the corresponding mode shapes of vibration are studied with different thickness variation coefficients, and other parameters. The thickness variations are assumed to be linear, exponential, and sinusoidal along the axial direction. The results are given graphically and comparisons are made with those results obtained using finite element method.

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17

Morrison, John C., Kyle Steffen, Blake Pantoja, Asha Nagaiya, Jacek Kobus, and Thomas Ericsson. "Numerical Methods for Solving the Hartree-Fock Equations of Diatomic Molecules II." Communications in Computational Physics 19, no. 3 (March 2016): 632–47. http://dx.doi.org/10.4208/cicp.101114.170615a.

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AbstractIn order to solve the partial differential equations that arise in the Hartree- Fock theory for diatomicmolecules and inmolecular theories that include electron correlation, one needs efficient methods for solving partial differential equations. In this article, we present numerical results for a two-variablemodel problem of the kind that arises when one solves the Hartree-Fock equations for a diatomic molecule. We compare results obtained using the spline collocation and domain decomposition methods with third-order Hermite splines to results obtained using the more-established finite difference approximation and the successive over-relaxation method. The theory of domain decomposition presented earlier is extended to treat regions that are divided into an arbitrary number of subregions by families of lines parallel to the two coordinate axes. While the domain decomposition method and the finite difference approach both yield results at the micro-Hartree level, the finite difference approach with a 9- point difference formula produces the same level of accuracy with fewer points. The domain decompositionmethod has the strength that it can be applied to problemswith a large number of grid points. The time required to solve a partial differential equation for a fine grid with a large number of points goes down as the number of partitions increases. The reason for this is that the length of time necessary for solving a set of linear equations in each subregion is very much dependent upon the number of equations. Even though a finer partition of the region has more subregions, the time for solving the set of linear equations in each subregion is very much smaller. This feature of the theory may well prove to be a decisive factor for solving the two-electron pair equation, which – for a diatomic molecule – involves solving partial differential equations with five independent variables. The domain decomposition theory also makes it possible to study complex molecules by dividing them into smaller fragments that are calculated independently. Since the domain decomposition approachmakes it possible to decompose the variable space into separate regions in which the equations are solved independently, this approach is well-suited to parallel computing.

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18

Ramadan, Mohamed A., Abd-El-Aziz El-Sherbeiny, and Mahmoud N. Sherif. "Numerical solution of system of first-order delay differential equations using polynomial spline functions." International Journal of Computer Mathematics 83, no. 12 (December 2006): 925–37. http://dx.doi.org/10.1080/00207160601138889.

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19

Liu, Chengzhi, Xuli Han, and Juncheng Li. "A Class of Spline Functions for Solving 2-Order Linear Differential Equations with Boundary Conditions." Algorithms 13, no. 9 (September 15, 2020): 231. http://dx.doi.org/10.3390/a13090231.

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In this paper, we exploit an numerical method for solving second order differential equations with boundary conditions. Based on the theory of the analytic solution, a series of spline functions are presented to find approximate solutions, and one of them is selected to approximate the solution automatically. Compared with the other methods, we only need to solve a tri-diagonal system, which is much easier to implement. This method has the advantages of high precision and less computational cost. The analysis of local truncation error is also discussed in this paper. At the end, some numerical examples are given to illustrate the effectiveness of the proposed method.

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20

Assari, Pouria. "The thin plate spline collocation method for solving integro-differential equations arisen from the charged particle motion in oscillating magnetic fields." Engineering Computations 35, no. 4 (June 11, 2018): 1706–26. http://dx.doi.org/10.1108/ec-08-2017-0330.

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Purpose The purpose of this study is to obtain a scheme for the numerical solution of Volterra integro-differential equations with time periodic coefficients deduced from the charged particle motion for certain configurations of oscillating magnetic fields. Design/methodology/approach The method reduces the solution of these types of integro-differential equations to the solution of two-dimensional Volterra integral equations of the second kind. The new method uses the discrete collocation method together with thin plate splines constructed on a set of scattered points as a basis. Findings The scheme can be easily implemented on a computer and has a computationally attractive algorithm. Numerical examples are included to show the validity and efficiency of the new technique. Originality/value The author uses thin plate splines as a type of free-shape parameter radial basis functions which establish an effective and stable method to solve electromagnetic integro-differential equations. As the scheme does not need any background meshes, it can be identified as a meshless method.

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21

El-Hawary, H. M., and S. M. Mahmoud*. "The numerical solution of higher index differential–algebraic equations by 4-point spline collocation methods." International Journal of Computer Mathematics 80, no. 10 (October 2003): 1299–312. http://dx.doi.org/10.1080/0020716031000070580.

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22

Su, Hong, Shui-Ping Yang, and Li-Ping Wen. "Stability and convergence of the two parameter cubic spline collocation method for delay differential equations." Computers & Mathematics with Applications 62, no. 6 (September 2011): 2580–90. http://dx.doi.org/10.1016/j.camwa.2011.07.057.

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23

Annunziato, Mario, and Hanno Gottschalk. "CALIBRATION OF LÉVY PROCESSES USING OPTIMAL CONTROL OF KOLMOGOROV EQUATIONS WITH PERIODIC BOUNDARY CONDITIONS." Mathematical Modelling and Analysis 23, no. 3 (June 14, 2018): 390–413. http://dx.doi.org/10.3846/mma.2018.024.

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We present an optimal control approach to the problem of model calibration for Lévy processes based on an non-parametric estimation procedure of the measure. The optimization problem is related to the maximum likelihood theory of sieves [25] and is formulated with the Fokker-Planck-Kolmogorov approach [3, 4]. We use a generic spline discretization of the Lévy jump measure and select an adequate size of the spline basis using the Akaike Information Criterion (AIC) [12]. The first order necessary optimality conditions are derived based on the Lagrange multiplier technique in a functional space. The resulting Partial Integral-Differential Equations (PIDE) are discretized, numerically solved using a scheme composed of Chang-Cooper, BDF2 and direct quadrature methods, jointly to a non-linear conjugate gradient method. For the numerical solver of the Kolmogorov's forward equation we prove conditions for non-negativity and stability in the L1 norm of the discrete solution.

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24

Boykov, Ilya V., Pavel V. Aykashev, and Alla I. Boykova. "Approximate solution of hypersingular integral equations on the number axis." Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva 22, no. 4 (December 31, 2020): 405–23. http://dx.doi.org/10.15507/2079-6900.22.202004.405-423.

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In the paper we investigate approximate methods for solving linear and nonlinear hypersingular integral equations defined on the number axis. We study equations with the second-order singularities because such equations are widely used in problems of natural science and technology. Three computational schemes are proposed for solving linear hypersingular integral equations. The first one is based on the mechanical quadrature method. We used rational functions as the basic ones. The second computational scheme is based on the spline-collocation method with the first-order splines. The third computational scheme uses the zero-order splines. Continuous method for solving operator equations has been used for justification and implementation of the proposed schemes. The application of the method allows to weaken the requirements imposed on the original equation. It is sufficient to require solvability for a given right-hand side. The continuous operator method is based on Lyapunov's stability for solutions of systems of ordinary differential equations. Thus it is stable for perturbations of coefficients and of right-hand sides. Approximate methods for solving nonlinear hypersingular integral equations are presented by the example of the Peierls - Naborro equation of dislocation theory. By analogy with linear hypersingular integral equations, three computational schemes have been constructed to solve this equation. The justification and implementation are based on continuous method for solving operator equations. The effectiveness of the proposed schemes is shown on solving the Peierls - Naborro equation.

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25

Hafizah, A. K. Nor, J. H. Lee, Z. A. Aziz, and K. K. Viswanathan. "Vibration of Antisymmetric Angle-Ply Laminated Plates of Higher-Order Theory with Variable Thickness." Mathematical Problems in Engineering 2018 (2018): 1–14. http://dx.doi.org/10.1155/2018/7323628.

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Free vibration of antisymmetric angle-ply laminated plates with variable thickness is studied. Higher-order shear deformation plate theory (HSDT) is introduced in the present method to remove the shear correction factors and improve the accuracy of transverse shear stresses. The thickness variations are assumed to be linear, exponential, and sinusoidal. The coupled differential equations are obtained in terms of displacement and rotational functions and approximated using cubic and quantic spline. A generalized eigenvalue problem is obtained and solved numerically by employing the eigensolution techniques with eigenvectors as spline coefficients to obtain the required frequencies. The results of numerical calculations are presented for laminated plates with simply supported boundary conditions. Comparisons of the current solutions and those reported in literature are provided to verify the accuracy of the proposed method. The effects of aspect ratio, number of layers, ply-angles, side-to-thickness ratio, and materials on the free vibration of cylindrical plates are discussed in detail.

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26

Liu, Jun, Hui Guo, Yao-Lin Jiang, and Yan Wang. "High order numerical algorithms based on biquadratic spline collocation for two-dimensional parabolic partial differential equations." International Journal of Computer Mathematics 96, no. 3 (February 16, 2018): 500–536. http://dx.doi.org/10.1080/00207160.2018.1437260.

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27

ACKLEH, AZMY S., BEN G. FITZPATRICK, and THOMAS G. HALLAM. "APPROXIMATION AND PARAMETER ESTIMATION PROBLEMS FOR ALGAL AGGREGATION MODELS." Mathematical Models and Methods in Applied Sciences 04, no. 03 (June 1994): 291–311. http://dx.doi.org/10.1142/s0218202594000182.

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Aggregation processes are intrinsic to many biological phenomena including sedimentation and coagulation of algae during bloom periods. A fundamental but unresolved problem associated with aggregate processes is the determination of the “stickiness function,” a measure of the ability of particles to adhere to other particles. This leads to an inverse problem associated with a class of nonlinear integro-differential equations. The purpose of this article is to develop convergence theory for this algal coagulation model utilizing a spline-based collocation scheme within the context of the parameter identification problem.

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28

Chang-Jiang, Liu, Zheng Zhou-Lian, Huang Cong-Bing, He Xiao-Ting, Sun Jun-Yi, and Chen Shan-Lin. "The Nonlinear Instability Modes of Dished Shallow Shells under Circular Line Loads." Mathematical Problems in Engineering 2011 (2011): 1–21. http://dx.doi.org/10.1155/2011/793798.

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This paper investigated the nonlinear stability problem of dished shallow shells under circular line loads. We derived the dimensionless governing differential equations of dished shallow shell under circular line loads according to the nonlinear theory of plates and shells and solved the governing differential equations by combing the free-parameter perturbation method (FPPM) with spline function method (SFM) to analyze the nonlinear instability modes of dished shallow shell under circular line loads. By analyzing the nonlinear instability modes and combining with concrete computational examples, we obtained the variation rules of the maximum deflection area of initial instability with different geometric parameters and loading action positions and discussed the relationship between the initial instability area and the maximum deflection area of initial instability. The results obtained from this paper provide some theoretical basis for engineering design and instability prediction and control of shallow-shell structures.

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Blaga, P., G. Micula, and H. Akça. "On the use of spline functions of even degree for the numerical solution of the delay differential equations." Calcolo 32, no. 1-2 (March 1995): 83–101. http://dx.doi.org/10.1007/bf02576544.

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30

Huang, X., and X. Lu. "The Use of Fractional B-Splines Wavelets in Multiterms Fractional Ordinary Differential Equations." International Journal of Differential Equations 2010 (2010): 1–13. http://dx.doi.org/10.1155/2010/968186.

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We discuss the existence and uniqueness of the solutions of the nonhomogeneous linear differential equations of arbitrary positive real order by using the fractional B-Splines wavelets and the Mittag-Leffler function. The differential operators are taken in the Riemann-Liouville sense and the initial values are zeros. The scheme of solving the fractional differential equations and the explicit expression of the solution is given in this paper. At last, we show the asymptotic solution of the differential equations of fractional order and corresponding truncated error in theory.

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31

Ramadan, Mohamed A., Abd El Aziz El-Sherbeiny, and Mahmoud N. Sherif. "The use of polynomial spline functions for the solution of system of second order delay differential equations." International Journal of Computer Mathematics 86, no. 7 (July 2009): 1167–81. http://dx.doi.org/10.1080/00207160701769617.

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Blaga, P., G. Micula, and M. Micula. "The numerical solution of differential equations with retarded argument by means of natural spline functions of even degree." International Journal of Computer Mathematics 64, no. 3-4 (January 1997): 245–62. http://dx.doi.org/10.1080/00207169708804588.

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33

Bodduluri, R. M. C., and B. Ravani. "Geometric Design and Fabrication of Developable Bezier and B-spline Surfaces." Journal of Mechanical Design 116, no. 4 (December 1, 1994): 1042–48. http://dx.doi.org/10.1115/1.2919485.

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In this paper we study Computer Aided Geometric Design (CAGD) and Manufacturing (CAM) of developable surfaces. We develop a direct representation of developable surfaces in terms of plane geometry. It uses control planes to determine a surface which is a Bezier or a B-spline interpolation of the control planes. In the Bezier case, a de Casteljau type construction method is presented for geometric design of developable Bezier surfaces. In the B-spline case, de Boor type construction for the geometric design of the developable surface and Boehm type knot insertion algorithm are presented. In the area of manufacturing or fabrication of developable surfaces, we present simple methods for both development of a surface into a plane and bending of a flat plane into a desired developable surface. The approach presented uses plane and line geometries and eliminates the need for solving differential equations of Riccatti type used in previous methods. The results are illustrated using an example generated by a CAD/CAM system implemented based on the theory presented.

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34

Jiang, Wei, and Beibei Guo. "A new numerical method for solving two-dimensional variable-order anomalous sub-diffusion equation." Thermal Science 20, suppl. 3 (2016): 701–10. http://dx.doi.org/10.2298/tsci16s3701j.

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The novelty and innovativeness of this paper are the combination of reproducing kernel theory and spline, this leads to a new simple but effective numerical method for solving variable-order anomalous sub-diffusion equation successfully. This combination overcomes the weaknesses of piecewise polynomials that can not be used to solve differential equations directly because of lack of the smoothness. Moreover, new bases of reproducing kernel spaces are constructed. On the other hand, the existence of any ?-approximate solution is proved and an effective method for obtaining the ?-approximate solution is established. A numerical example is given to show the accuracy and effectiveness of theoretical results.

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Mittal, Ramesh Chand, and Sumita Dahiya. "A comparative study of modified cubic B-spline differential quadrature methods for a class of nonlinear viscous wave equations." Engineering Computations 35, no. 1 (March 5, 2018): 315–33. http://dx.doi.org/10.1108/ec-06-2016-0188.

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Purpose In this study, a second-order standard wave equation extended to a two-dimensional viscous wave equation with timely differentiated advection-diffusion terms has been solved by differential quadrature methods (DQM) using a modification of cubic B-spline functions. Two numerical schemes are proposed and compared to achieve numerical approximations for the solutions of nonlinear viscous wave equations. Design/methodology/approach Two schemes are adopted to reduce the given system into two systems of nonlinear first-order partial differential equations (PDE). For each scheme, modified cubic B-spline (MCB)-DQM is used for calculating the spatial variables and their derivatives that reduces the system of PDEs into a system of nonlinear ODEs. The solutions of these systems of ODEs are determined by SSP-RK43 scheme. The CPU time is also calculated and compared. Matrix stability analysis has been performed for each scheme and both are found to be unconditionally stable. The results of both schemes have been extensively discussed and compared. The accuracy and reliability of the methods have been successfully tested on several examples. Findings A comparative study has been carried out for two different schemes. Results from both schemes are also compared with analytical solutions and the results available in literature. Experiments show that MCB-DQM with Scheme II yield more accurate and reliable results in solving viscous wave equations. But Scheme I is comparatively less expensive in terms of CPU time. For MCB-DQM, less depository requirements lead to less aggregation of approximation errors which in turn enhances the correctness and readiness of the numerical techniques. Approximate solutions to the two-dimensional nonlinear viscous wave equation have been found without linearizing the equation. Ease of implementation and low computation cost are the strengths of the method. Originality/value For the first time, a comparative study has been carried out for the solution of nonlinear viscous wave equation. Comparisons are done in terms of accuracy and CPU time. It is concluded that Scheme II is more suitable.

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Defez, Emilio, Michael M. Tung, Francisco J. Solis, and Javier Ibáñez. "Numerical approximations of second-order matrix differential equations using higher degree splines." Linear and Multilinear Algebra 63, no. 3 (February 14, 2014): 472–89. http://dx.doi.org/10.1080/03081087.2013.873427.

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Moghaddam, B. P., and J. A. T. Machado. "A stable three-level explicit spline finite difference scheme for a class of nonlinear time variable order fractional partial differential equations." Computers & Mathematics with Applications 73, no. 6 (March 2017): 1262–69. http://dx.doi.org/10.1016/j.camwa.2016.07.010.

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EVANS, JOHN A., and THOMAS J. R. HUGHES. "ISOGEOMETRIC DIVERGENCE-CONFORMING B-SPLINES FOR THE DARCY–STOKES–BRINKMAN EQUATIONS." Mathematical Models and Methods in Applied Sciences 23, no. 04 (January 31, 2013): 671–741. http://dx.doi.org/10.1142/s0218202512500583.

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We develop divergence-conforming B-spline discretizations for the numerical solution of the Darcy–Stokes–Brinkman equations. These discretizations are motivated by the recent theory of isogeometric discrete differential forms and may be interpreted as smooth generalizations of Raviart–Thomas elements. The new discretizations are (at least) patchwise C0 and can be directly utilized in the Galerkin solution of Darcy–Stokes–Brinkman flow for single-patch configurations. When applied to incompressible flows, these discretizations produce pointwise divergence-free velocity fields and hence exactly satisfy mass conservation. In the presence of no-slip boundary conditions and multi-patch geometries, the discontinuous Galerkin framework is invoked to enforce tangential continuity without upsetting the conservation or stability properties of the method across patch boundaries. Furthermore, as no-slip boundary conditions are enforced weakly, the method automatically defaults to a compatible discretization of Darcy flow in the limit of vanishing viscosity. The proposed discretizations are extended to general mapped geometries using divergence-preserving transformations. For sufficiently regular single-patch solutions, we prove a priori error estimates which are optimal for the discrete velocity field and suboptimal, by one order, for the discrete pressure field. Our estimates are in addition robust with respect to the parameters of the Darcy–Stokes–Brinkman problem. We present a comprehensive suite of numerical experiments which indicate optimal convergence rates for both the discrete velocity and pressure fields for general configurations, suggesting that our a priori estimates may be conservative. The focus of this paper is strictly on incompressible flows, but our theoretical results naturally extend to flows characterized by mass sources and sinks.

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EVANS, JOHN A., and THOMAS J. R. HUGHES. "ISOGEOMETRIC DIVERGENCE-CONFORMING B-SPLINES FOR THE STEADY NAVIER–STOKES EQUATIONS." Mathematical Models and Methods in Applied Sciences 23, no. 08 (April 22, 2013): 1421–78. http://dx.doi.org/10.1142/s0218202513500139.

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We develop divergence-conforming B-spline discretizations for the numerical solution of the steady Navier–Stokes equations. These discretizations are motivated by the recent theory of isogeometric discrete differential forms and may be interpreted as smooth generalizations of Raviart–Thomas elements. They are (at least) patchwise C0 and can be directly utilized in the Galerkin solution of steady Navier–Stokes flow for single-patch configurations. When applied to incompressible flows, these discretizations produce pointwise divergence-free velocity fields and hence exactly satisfy mass conservation. Consequently, discrete variational formulations employing the new discretization scheme are automatically momentum-conservative and energy-stable. In the presence of no-slip boundary conditions and multi-patch geometries, the discontinuous Galerkin framework is invoked to enforce tangential continuity without upsetting the conservation or stability properties of the method across patch boundaries. Furthermore, as no-slip boundary conditions are enforced weakly, the method automatically defaults to a compatible discretization of Euler flow in the limit of vanishing viscosity. The proposed discretizations are extended to general mapped geometries using divergence-preserving transformations. For sufficiently regular single-patch solutions subject to a smallness condition, we prove a priori error estimates which are optimal for the discrete velocity field and suboptimal, by one order, for the discrete pressure field. We present a comprehensive suite of numerical experiments which indicate optimal convergence rates for both the discrete velocity and pressure fields for general configurations, suggesting that our a priori estimates may be conservative. These numerical experiments also suggest our discretization methodology is robust with respect to Reynolds number and more accurate than classical numerical methods for the steady Navier–Stokes equations.

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Wang, Meiqing, Chensi Huang, Chao Zeng, and Choi-Hong Lai. "Two-Phase Image Inpainting: Combine Edge-Fitting with PDE Inpainting." Advances in Applied Mathematics and Mechanics 4, no. 06 (December 2012): 769–79. http://dx.doi.org/10.4208/aamm.12-12s08.

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AbstractThe digital image inpainting technology based on partial differential equations (PDEs) has become an intensive research topic over the last few years due to the mature theory and prolific numerical algorithms of PDEs. However, PDE based models are not effective when used to inpaint large missing areas of images, such as that produced by object removal. To overcome this problem, in this paper, a two-phase image inpainting method is proposed. First, some edges which cross the damaged regions are located and the missing parts of these edges are fitted by using the cubic spline interpolation. These fitted edges partition the damaged regions into some smaller damaged regions. Then these smaller regions may be inpainted by using classical PDE models. Experiment results show that the inpainting results by using the proposed method are better than those of BSCB model and TV model.

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Firsanov, V. V., and V. T. Pham. "STRESS-STRAIN STATE OF THE SPHERICAL SHELL EXPOSED TO AN ARBITRARY LOAD BASED ON A NON-CLASSICAL THEORY." Problems of strenght and plasticity 81, no. 3 (2019): 359–68. http://dx.doi.org/10.32326/1814-9146-2019-81-3-359-368.

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Considered is the stress state of an isotropic spherical shell exposed to an arbitrary load based on a non-classical theory. When building a mathematical model of the shell, three-dimensional equations of the theory of elasticity are applied. Displacements are represented in the form of polynomials along the coordinate normal to the middle surface two degrees higher relative to the classical theory of the Kirchhoff-Love type. As a result of minimization of the refined value of the Lagrange energy functional, a system of differential equilibrium equations in displacements and natural boundary conditions are obtained. The task of reducing two-dimensional equations to ordinary differential equations is carried out by decomposing the components of displacements and external loads into trigonometric series in the circumferential coordinate. Displacements are represented in form of polynomials along the coordinate normal to the middle surface by two degrees higher relative to the classical theory of the Kirchhoff-Love type. Resulting from minimization of the refined value of the Lagrange energy functional, a system of differential equilibrium equations in displacements and natural boundary conditions are received. The task of reducing two-dimensional equations to ordinary differential equations is carried out by decomposing the components of the displacements and the external loads into trigonometric series as per the circumferential coordinate. The formulated boundary problem is solved by the methods of finite differences and matrix sweep. As a result, displacements are obtained in the grid nodes, for approximation of which splines are used. The shell deformations are found using geometric relationship; tangential stresses are received from the correlations of Hooke's law. One of the features of this paper lies in the fact that the transverse stresses are determined by the direct integration of the equilibrium equations of the three-dimensional theory of elasticity. An example of the calculation of a hemispherical shell rigidly restrained along the lower base contour is brought. The shell is exposed to the wind load. Comparison of the results received by the refined theory with the data of the classical theory has shown that in the zone of distortion of the stressed state, the normal tangential stresses are substantially revised and the transverse normal stresses, which are neglected in the classical theory, are of the same magnitude with the maximum values of the main bending stress. Considered is the influence of the relative thickness on the stress state of the shell. It was discovered that the shell thickness significantly increases the error of the classical theory, while determining the stresses and assessing the strength of the elements of the aircraft structures.

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Koubaiti, Ouadie, Said EL Fakkoussi, Jaouad El-Mekkaoui, Hassan Moustachir, Ahmed Elkhalfi, and Catalin I. Pruncu. "The treatment of constraints due to standard boundary conditions in the context of the mixed Web-spline finite element method." Engineering Computations 38, no. 7 (February 8, 2021): 2937–68. http://dx.doi.org/10.1108/ec-02-2020-0078.

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Purpose This paper aims to propose a new boundary condition and a web-spline basis of finite element space approximation to remedy the problems of constraints due to homogeneous and non-homogeneous; Dirichlet boundary conditions. This paper considered the two-dimensional linear elasticity equation of Navier–Lamé with the condition CAB. The latter allows to have a total insertion of the essential boundary condition in the linear system obtained; without using a numerical method as Lagrange multiplier. This study have developed mixed finite element; method using the B-splines Web-spline space. These provide an exact implementation of the homogeneous; Dirichlet boundary conditions, which removes the constraints caused by the standard; conditions. This paper showed the existence and the uniqueness of the weak solution, as well as the convergence of the numerical solution for the quadratic case are proved. The weighted extended B-spline; approach have become a much more workmanlike solution. Design/methodology/approach In this paper, this study used the implementation of weighted finite element methods to solve the Navier–Lamé system with a new boundary condition CA, B (Koubaiti et al., 2020), that generalises the well-known basis, especially the Dirichlet and the Neumann conditions. The novel proposed boundary condition permits to use a single Matlab code, which summarises all kind of boundary conditions encountered in the system. By using this model is possible to save time and programming recourses while reap several programs in a single directory. Findings The results have shown that the Web-spline-based quadratic-linear finite elements satisfy the inf–sup condition, which is necessary for existence and uniqueness of the solution. It was demonstrated by the existence of the discrete solution. A full convergence was established using the numerical solution for the quadratic case. Due to limited regularity of the Navier–Lamé problem, it will not change by increasing the degree of the Web-spline. The computed relative errors and their rates indicate that they are of order 1/H. Thus, it was provided their theoretical validity for the numerical solution stability. The advantage of this problem that uses the CA, B boundary condition is associated to reduce Matlab programming complexity. Originality/value The mixed finite element method is a robust technique to solve difficult challenges from engineering and physical sciences using the partial differential equations. Some of the important applications include structural mechanics, fluid flow, thermodynamics and electromagnetic fields (Zienkiewicz and Taylor, 2000) that are mainly based on the approximation of Lagrange. However, this type of approximation has experienced a great restriction in the level of domain modelling, especially in the case of complicated boundaries such as that in the form of curvilinear graphs. Recently, the research community tried to develop a new way of approximation based on the so-called B-spline that seems to have superior results in solving the engineering problems.

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Su, Liyun, Tianshun Yan, Yanyong Zhao, and Fenglan Li. "Local Polynomial Regression Solution for Partial Differential Equations with Initial and Boundary Values." Discrete Dynamics in Nature and Society 2012 (2012): 1–11. http://dx.doi.org/10.1155/2012/201678.

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Local polynomial regression (LPR) is applied to solve the partial differential equations (PDEs). Usually, the solutions of the problems are separation of variables and eigenfunction expansion methods, so we are rarely able to find analytical solutions. Consequently, we must try to find numerical solutions. In this paper, two test problems are considered for the numerical illustration of the method. Comparisons are made between the exact solutions and the results of the LPR. The results of applying this theory to the PDEs reveal that LPR method possesses very high accuracy, adaptability, and efficiency; more importantly, numerical illustrations indicate that the new method is much more efficient than B-splines and AGE methods derived for the same purpose.

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Mohanty, Ranjan Kumar, and Gunjan Khurana. "A new high accuracy cubic spline method based on half-step discretization for the system of 1D non-linear wave equations." Engineering Computations 36, no. 3 (April 8, 2019): 930–57. http://dx.doi.org/10.1108/ec-04-2018-0194.

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PurposeThis paper aims to develop a new 3-level implicit numerical method of order 2 in time and 4 in space based on half-step cubic polynomial approximations for the solution of 1D quasi-linear hyperbolic partial differential equations. The method is derived directly from the consistency condition of spline function which is fourth-order accurate. The method is directly applied to hyperbolic equations, irrespective of coordinate system, and fourth-order nonlinear hyperbolic equation, which is main advantage of the work.Design/methodology/approachIn this method, three grid points for the unknown function w(x,t) and two half-step points for the known variablexin spatial direction are used. The methodology followed in this paper is construction of a cubic spline polynomial and using its continuity properties to obtain fourth-order consistency condition. The proposed method, when applied to a linear equation is shown to be unconditionally stable. The technique is extended to solve system of quasi-linear hyperbolic equations. To assess the validity and accuracy, the method is applied to solve several benchmark problems, and numerical results are provided to demonstrate the usefulness of the method.FindingsThe paper provides a fourth-order numerical scheme obtained directly from fourth-order consistency condition. In earlier methods, consistency conditions were only second-order accurate. This brings an edge over other past methods. In addition, the method is directly applicable to physical problems involving singular coefficients. Therefore, no modification in the method is required at singular points. This saves CPU time, as well as computational costs.Research limitations/implicationsThere are no limitations. Obtaining a fourth-order method directly from consistency condition is a new work. In addition, being an implicit method, this method is unconditionally stable for a linear test equation.Practical implicationsPhysical problems with singular and nonsingular coefficients are directly solved by this method.Originality/valueThe paper develops a new fourth-order implicit method which is original and has substantial value because many benchmark problems of physical significance are solved in this method.

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Akimov, Pavel A., Marina L. Mozgaleva, and Taymuraz B. Kaytukov. "About the B-spline wavelet discrete-continual finite element method of the local plate analysis." Vestnik MGSU, no. 6 (June 2021): 666–75. http://dx.doi.org/10.22227/1997-0935.2021.6.666-675.

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Introduction. This distinctive paper addresses the local semi-analytical solution to the problem of plate analysis. Isotropic plates featuring the regularity (constancy) of physical and geometric parameters (modulus of elasticity of the plate material, Poisson’s ratio of the plate material, dimensions of the cross section of the plate) along one direction (dimension) are under consideration. This direction is conventionally called the basic direction. Materials and methods. The B-spline wavelet discrete-continual finite element method (DCFEM) is used. The initial operational formulation of the problem was constructed using the theory of distribution and the so-called method of extended domain, proposed by Prof. Alexander B. Zolotov. Results. Some relevant issues of construction of normalized basis functions of the B-spline are considered; the technique of approximation of corresponding vector functions and operators within DCFEM is described. The problem remains continual if analyzed along the basic direction, and its exact analytical solution can be obtained, whereas the finite element approximation is used in combination with a wavelet analysis apparatus in respect of the non-basic direction. As a result, we can obtain a discrete-continual formulation of the problem. Thus, we have a multi-point (in particular, two-point) boundary problem for the first-order system of ordinary differential equations with constant coefficients. A special correct analytical method of solving such problems was developed, described and verified in the numerous papers of the co-authors. In particular, we consider the simplest sample analysis of a plate (rectangular in plan) fixed along the side faces exposed to the influence of the load concentrated in the center of the plate. Conclusions. The solution to the verification problem obtained using the proposed version of wavelet-based DCFEM was in good agreement with the solution obtained using the conventional finite element method (the corresponding solutions were constructed with and without localization; these solutions almost completely coincided, while the advantages of the numerical-analytical approach were quite obvious). It is shown that the use of B-splines of various degrees within wavelet-based DCFEM leads to a significant reduction in the number of unknowns.

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Kipiani, Gela, and Nika Botchorishvili. "Analysis of Lamellar Structures with Application of Generalized Functions." Transactions of the VŠB – Technical University of Ostrava, Civil Engineering Series. 16, no. 2 (December 1, 2016): 49–60. http://dx.doi.org/10.1515/tvsb-2016-0014.

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Abstract Theory of differential equations in respect of the functional area is based on the basic concepts on generalized functions and splines. There are some basic concepts related to the theory of generalized functions and their properties are considered in relation to the rod systems and lamellar structures. The application of generalized functions gives the possibility to effectively calculate step-variable stiffness lamellar structures. There are also widely applied structures, in that several in which a number of parallel load bearing layers are interconnected by discrete-elastic links. For analysis of system under study, such as design diagrams, there are applied discrete and discrete-continual models.

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Rehme, Michael, Stephen Roberts, and Dirk Pflüger. "Uncertainty quantification for the Hokkaido Nansei-Oki tsunami using B-splines on adaptive sparse grids." ANZIAM Journal 62 (June 29, 2021): C30—C44. http://dx.doi.org/10.21914/anziamj.v62.16121.

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Modeling uncertainties in the input parameters of computer simulations is an established way to account for inevitably limited knowledge. To overcome long run-times and high demand for computational resources, a surrogate model can replace the original simulation. We use spatially adaptive sparse grids for the creation of this surrogate model. Sparse grids are a discretization scheme designed to mitigate the curse of dimensionality, and spatial adaptivity further decreases the necessary number of expensive simulations. We combine this with B-spline basis functions which provide gradients and are exactly integrable. We demonstrate the capability of this uncertainty quantification approach for a simulation of the Hokkaido Nansei–Oki Tsunami with anuga. We develop a better understanding of the tsunami behavior by calculating key quantities such as mean, percentiles and maximum run-up. We compare our approach to the popular Dakota toolbox and reach slightly better results for all quantities of interest. References B. M. Adams, M. S. Ebeida, et al. Dakota. Sandia Technical Report, SAND2014-4633, Version 6.11 User’s Manual, July 2014. 2019. https://dakota.sandia.gov/content/manuals. J. H. S. de Baar and S. G. Roberts. Multifidelity sparse-grid-based uncertainty quantification for the Hokkaido Nansei–Oki tsunami. Pure Appl. Geophys. 174 (2017), pp. 3107–3121. doi: 10.1007/s00024-017-1606-y. H.-J. Bungartz and M. Griebel. Sparse grids. Acta Numer. 13 (2004), pp. 147–269. doi: 10.1017/S0962492904000182. M. Eldred and J. Burkardt. Comparison of non-intrusive polynomial chaos and stochastic collocation methods for uncertainty quantification. 47th AIAA. 2009. doi: 10.2514/6.2009-976. K. Höllig and J. Hörner. Approximation and modeling with B-splines. Philadelphia: SIAM, 2013. doi: 10.1137/1.9781611972955. M. Matsuyama and H. Tanaka. An experimental study of the highest run-up height in the 1993 Hokkaido Nansei–Oki earthquake tsunami. National Tsunami Hazard Mitigation Program Review and International Tsunami Symposium (ITS). 2001. O. Nielsen, S. Roberts, D. Gray, A. McPherson, and A. Hitchman. Hydrodymamic modelling of coastal inundation. MODSIM 2005. 2005, pp. 518–523. https://www.mssanz.org.au/modsim05/papers/nielsen.pdf. J. Nocedal and S. J. Wright. Numerical optimization. Springer, 2006. doi: 10.1007/978-0-387-40065-5. D. Pflüger. Spatially Adaptive Sparse Grids for High-Dimensional Problems. Dr. rer. nat., Technische Universität München, Aug. 2010. https://www5.in.tum.de/pub/pflueger10spatially.pdf. M. F. Rehme, F. Franzelin, and D. Pflüger. B-splines on sparse grids for surrogates in uncertainty quantification. Reliab. Eng. Sys. Saf. 209 (2021), p. 107430. doi: 10.1016/j.ress.2021.107430. M. F. Rehme and D. Pflüger. Stochastic collocation with hierarchical extended B-splines on Sparse Grids. Approximation Theory XVI, AT 2019. Springer Proc. Math. Stats. Vol. 336. Springer, 2020. doi: 10.1007/978-3-030-57464-2_12. S Roberts, O. Nielsen, D. Gray, J. Sexton, and G. Davies. ANUGA. Geoscience Australia. 2015. doi: 10.13140/RG.2.2.12401.99686. I. J. Schoenberg and A. Whitney. On Pólya frequence functions. III. The positivity of translation determinants with an application to the interpolation problem by spline curves. Trans. Am. Math. Soc. 74.2 (1953), pp. 246–259. doi: 10.2307/1990881. W. Sickel and T. Ullrich. Spline interpolation on sparse grids. Appl. Anal. 90.3–4 (2011), pp. 337–383. doi: 10.1080/00036811.2010.495336. C. E. Synolakis, E. N. Bernard, V. V. Titov, U. Kânoğlu, and F. I. González. Standards, criteria, and procedures for NOAA evaluation of tsunami numerical models. NOAA/Pacific Marine Environmental Laboratory. 2007. https://nctr.pmel.noaa.gov/benchmark/. J. Valentin and D. Pflüger. Hierarchical gradient-based optimization with B-splines on sparse grids. Sparse Grids and Applications—Stuttgart 2014. Lecture Notes in Computational Science and Engineering. Vol. 109. Springer, 2016, pp. 315–336. doi: 10.1007/978-3-319-28262-6_13. D. Xiu and G. E. Karniadakis. The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24.2 (2002), pp. 619–644. doi: 10.1137/S1064827501387826.

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El-Hawary, H. M., and S. M. Mahmoud. "Spline collocation methods for solving delay-differential equations." Applied Mathematics and Computation 146, no. 2-3 (December 2003): 359–72. http://dx.doi.org/10.1016/s0096-3003(02)00586-6.

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Bialecki, B., and G. Fairweather. "Orthogonal spline collocation methods for partial differential equations." Journal of Computational and Applied Mathematics 128, no. 1-2 (March 2001): 55–82. http://dx.doi.org/10.1016/s0377-0427(00)00509-4.

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Yang, Shui-Ping, and Ai-Guo Xiao. "Cubic Spline Collocation Method for Fractional Differential Equations." Journal of Applied Mathematics 2013 (2013): 1–20. http://dx.doi.org/10.1155/2013/864025.

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We discuss the cubic spline collocation method with two parameters for solving the initial value problems (IVPs) of fractional differential equations (FDEs). Some results of the local truncation error, the convergence, and the stability of this method for IVPs of FDEs are obtained. Some numerical examples verify our theoretical results.

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