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Relevant bibliographies by topics / Spline theory. Differential equations

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Academic literature on the topic 'Spline theory. Differential equations'

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Published: 4 June 2021

Last updated: 1 February 2022

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

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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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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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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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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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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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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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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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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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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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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Dissertations / Theses on the topic "Spline theory. Differential equations"

1

Tarang, Mare. "Stability of the spline collocation method for Volterra integro-differential equations." Online version, 2004. http://dspace.utlib.ee/dspace/bitstream/10062/793/5/Tarang.pdf.

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Kadhum, Nashat Ibrahim. "The spline approach to the numerical solution of parabolic partial differential equations." Thesis, Loughborough University, 1988. https://dspace.lboro.ac.uk/2134/6725.

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This thesis is concerned with the Numerical Solution of Partial Differential Equations. Initially some definitions and mathematical background are given, accompanied by the basic theories of solving linear systems and other related topics. Also, an introduction to splines, particularly cubic splines and their identities are presented. The methods used to solve parabolic partial differential equations are surveyed and classified into explicit or implicit (direct and iterative) methods. We concentrate on the Alternating Direction Implicit (ADI), the Group Explicit (GE) and the Crank-Nicolson (C-N) methods. A new method, the Splines Group Explicit Iterative Method is derived, and a theoretical analysis is given. An optimum single parameter is found for a special case. Two criteria for the acceleration parameters are considered; they are the Peaceman-Rachford and the Wachspress criteria. The method is tested for different numbers of both parameters. The method is also tested using single parameters, i. e. when used as a direct method. The numerical results and the computational complexity analysis are compared with other methods, and are shown to be competitive. The method is shown to have good stability property and achieves high accuracy in the numerical results. Another direct explicit method is developed from cubic splines; the splines Group Explicit Method which includes a parameter that can be chosen to give optimum results. Some analysis and the computational complexity of the method is given, with some numerical results shown to confirm the efficiency and compatibility of the method. Extensions to two dimensional parabolic problems are given in a further chapter. In this thesis the Dirichlet, the Neumann and the periodic boundary conditions for linear parabolic partial differential equations are considered. The thesis concludes with some conclusions and suggestions for further work.

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Roeser, Markus Karl. "The ASD equations in split signature and hypersymplectic geometry." Thesis, University of Oxford, 2012. http://ora.ox.ac.uk/objects/uuid:7d46ffc8-6d12-4fec-9450-13d2c726885c.

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This thesis is mainly concerned with the study of hypersymplectic structures in gauge theory. These structures arise via applications of the hypersymplectic quotient construction to the action of the gauge group on certain spaces of connections and Higgs fields. Motivated by Kobayashi-Hitchin correspondences in the case of hyperkähler moduli spaces, we first study the relationship between hypersymplectic, complex and paracomplex quotients in the spirit of Kirwan's work relating Kähler quotients to GIT quotients. We then study dimensional reductions of the ASD equations on $mathbb R^{2,2}$. We discuss a version of twistor theory for hypersymplectic manifolds, which we use to put the ASD equations into Lax form. Next, we study Schmid's equations from the viewpoint of hypersymplectic quotients and examine the local product structure of the moduli space. Then we turn towards the integrability aspects of this system. We deduce various properties of the spectral curve associated to a solution and provide explicit solutions with cyclic symmetry. Hitchin's harmonic map equations are the split signature analogue of the self-duality equations on a Riemann surface, in which case it is known that there is a smooth hyperkähler moduli space. In the case at hand, we cannot expect to obtain a globally well-behaved moduli space. However, we are able to construct a smooth open set of solutions with small Higgs field, on which we then analyse the hypersymplectic geometry. In particular, we exhibit the local product structures and the family of complex structures. This is done by interpreting the equations as describing certain geodesics on the moduli space of unitary connections. Using this picture we relate the degeneracy locus to geodesics with conjugate endpoints. Finally, we present a split signature version of the ADHM construction for so-called split signature instantons on $S^2 imes S^2$, which can be given an interpretation as a hypersymplectic quotient.

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Castro, Douglas Azevedo 1982. "Esquemas de aproximação em multinível e aplicações." [s.n.], 2011. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306587.

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Orientador: Sônia Maria Gomes, Jorge Stolfi
Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica
Made available in DSpace on 2018-08-19T12:39:30Z (GMT). No. of bitstreams: 1 Castro_DouglasAzevedo_D.pdf: 8872633 bytes, checksum: a17b2761789c6a831631ac143fdf5ca7 (MD5) Previous issue date: 2011
Resumo: O objetivo desta tese é desenvolver algoritmos baseados em malhas e bases funcionais inovadoras usando técnicas de multiescala para aproximação de funções e resolução de problemas de equações diferenciais. Para certas classes de problemas, é possível incrementar a eficiência dos algoritmos de multiescala usando bases adaptativas, associadas a malhas construídas de forma a se ajustarem com o fenômeno a ser modelado. Nesta abordagem, em cada nível da hierarquia, os detalhes entre a aproximação desse nível e a aproximação definida no próximo nível menos refinado pode ser usada como indicador de regiões que necessitam de mais ou menos refinamento. Desta forma, em regiões onde a solução é suave, basta utilizar os elementos dos níveis menos refinados da hierarquia, enquanto que o maior refinamento é feito apenas onde a solução tiver variações bruscas. Consideramos dois tipos de formulações para representações multiescala, dependendo das bases adotadas: splines diádicos e wavelets. A primeira abordagem considera espaços aproximantes por funções splines sobre uma hierarquia de malhas cuja resolução depende do nível. A outra abordagem considera ferramentas da analise wavelet para representações em multirresolução de médias celulares. O enfoque está no desenvolvimento de algoritmos baseados em dados amostrais d-dimensionais em malhas diádicas que são armazenados em uma estrutura de árvore binária. A adaptatividade ocorre quando o refinamento é interrompido em algumas regiões do domínio, onde os detalhes entre dois níveis consecutivos são suficientemente pequenos. Um importante aspecto deste tipo de representação é que a mesma estrutura de dados é usada em qualquer dimensão, além de facilitar o acesso aos dados nela armezenados. Utilizamos as técnicas desenvolvidas na construção de um método adaptativo de volumes finitos em malhas diádicas para a solução de problemas diferenciais. Analisamos o desempenho do método adaptativo em termos da compressão de memória e tempo de CPU em comparação com os resultados do esquema de referência em malha uniforme no nível mais refinado. Neste sentido, comprovamos a eficiência do método adaptativo, que foi avaliada levando-se em consideração os efeitos da escolha de diferentes tipos de fluxo numérico e dos parâmetros de truncamento
Abstract: The goal of this thesis is to develop algorithms based on innovative meshes and functional bases using multiscale techniques for function approximation and solution of differential equation problems. For certain classes of problems, one can increase the efficiency of multiscale algorithms using hierarchical adaptive bases, associated to meshes whose resolution varies according to the local features of the phenomenon to be modeled. In this approach, at each level of the hierarchy the details-differences between the approximation for that level and that of the next coarser level-can be used as indicators of regions that need more or less refinement. In this way, in regions where the solution is smooth, it suffices to use elements of the less refined levels of the hierarchy, while the maximum refinement is used only where the solution has sharp variations. We consider two classes of formulations for multiscale representations, depending on the bases used: dyadic splines and wavelets. The first approach uses approximation spaces consisting of spline functions defined over a mesh hierarchy whose resolution depends on the level. The other approach uses tools from wavelet analysis for multiresolu-tion representations of cell averages. The focus is on the development of algorithms based on sampled d-dimensional data on dyadic meshes which are stored in a binary tree structures. The adaptivity happens when the refinement is interrupted in certain regions of the domain, where the details between two consecutive levels are sufficiently small. This representation greatly simplifies the access to the data and it can be used in any dimension. We use these techniques to build an adaptive finite volume method on dyadic grids for the solution of differential problems. We analyze the performance of the method in terms of memory compression and CPU time, comparing it with the reference scheme (which uses a uniform mesh at the maximum refinement level). In these tests, we confirmed the efficiency of the adaptive method for various numeric flow formulas and various choices of the thresholding parameters
Doutorado
Matematica Aplicada
Doutor em Matemática Aplicada

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Tung, Michael Ming-Sha. "Spline approximations for systems of ordinary differential equations." Doctoral thesis, Universitat Politècnica de València, 2013. http://hdl.handle.net/10251/31658.

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El objetivo de esta tesis doctoral es desarrollar nuevos métodos basados en splines para la resolución de sistemas de ecuaciones diferenciales del tipo Y'(x)=f(x,Y(x)) , aTung, MM. (2013). Spline approximations for systems of ordinary differential equations [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/31658
TESIS
Premiado

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Kirby, P. J. "The theory of exponential differential equations." Thesis, University of Oxford, 2006. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.433471.

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This thesis is a model-theoretic study of exponential differential equations in the context of differential algebra. I define the theory of a set of differential equations and give an axiomatization for the theory of the exponential differential equations of split semiabelian varieties. In particular, this includes the theory of the equations satisfied by the usual complex exponential function and the Weierstrass p-functions. The theory consists of a description of the algebraic structure on the solution sets together with necessary and sufficient conditions for a system of equations to have solutions. These conditions are stated in terms of a dimension theory; their necessity generalizes Ax’s differential field version of Schanuel’s conjecture and their sufficiency generalizes recent work of Crampin. They are shown to apply to the solving of systems of equations in holomorphic functions away from singularities, as well as in the abstract setting. The theory can also be obtained by means of a Hrushovski-style amalgamation construction, and I give a category-theoretic account of the method. Restricting to the usual exponential differential equation, I show that a “blurring” of Zilber’s pseudo-exponentiation satisfies the same theory. I conjecture that this theory also holds for a suitable blurring of the complex exponential maps and partially resolve the question, proving the necessity but not the sufficiency of the aforementioned conditions. As an algebraic application, I prove a weak form of Zilber’s conjecture on intersections with subgroups (known as CIT) for semiabelian varieties. This in turn is used to show that the necessary and sufficient conditions are expressible in the appropriate first order language.

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Zhu, Wei. "Fractional differential equations in risk theory." Thesis, University of Liverpool, 2018. http://livrepository.liverpool.ac.uk/3018514/.

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This thesis considers one of the central topics in the actuarial mathematics literature, deriving the probability of ruin in the collective risk model. The classical risk model and renewal risk models are focused in this project, where the claim number processes are assumed to be Poisson counting processes and any general renewal counting processes, respectively. The first part of this project is about the classical risk model. We look at the case when claim sizes follow a gamma distribution. Explicit expressions for ruin probabilities are derived via Laplace transform and inverse Laplace transform approach. The second half is about the renewal risk model. Very general assumptions on inter-arrival times are possible for the renewal risk model, which includes the classical risk model, Erlang risk model and fractional Poisson risk model. A new family of differential operators are de ned in order to construct the fractional integro-differential equations for ruin probabilities in such renewal risk models. Through the characteristic equation approach, specific fractional differential equations for the ruin probabilities can be solved explicitly, allowing for the analysis of the ruin probabilities.

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Nagloo, Joel Chris Ronnie. "Model theory, algebra and differential equations." Thesis, University of Leeds, 2014. http://etheses.whiterose.ac.uk/6813/.

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In this thesis, we applied ideas and techniques from model theory, to study the structure of the sets of solutions XII - XV I , in a differentially closed field, of the Painlevé equations. First we show that the generic XII - XV I , that is those with parameters in general positions, are strongly minimal and geometrically trivial. Then, we prove that the generic XII , XIV and XV are strictly disintegrated and that the generic XIII and XV I are ω-categorical. These results, already known for XI , are the culmination of the work started by P. Painlevé (over 100 years ago), the Japanese school and many others on transcendence and the Painlevé equations. We also look at the non generic second Painlevé equations and show that all the strongly minimal ones are geometrically trivial.

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Whitehead, Andrew John. "Differential equations and differential polynomials in the complex plane." Thesis, University of Nottingham, 2002. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.273112.

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Ozbekler, Abdullah. "Sturm Comparison Theory For Impulsive Differential Equations." Phd thesis, METU, 2005. http://etd.lib.metu.edu.tr/upload/3/12606894/index.pdf.

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In this thesis, we investigate Sturmian comparison theory and oscillation for second order impulsive differential equations with fixed moments of impulse actions. It is shown that impulse actions may greatly alter the oscillation behavior of solutions. In chapter two, besides Sturmian type comparison results, we give Leightonian type comparison theorems and obtain Wirtinger type inequalities for linear, half-linear and non-selfadjoint equations. We present analogous results for forced super linear and super half-linear equations with damping. In chapter three, we derive sufficient conditions for oscillation of nonlinear equations. Integral averaging, function averaging techniques as well as interval criteria for oscillation are discussed. Oscillation criteria for solutions of impulsive Hill&
#8217
s equation with damping and forced linear equations with damping are established.

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Books on the topic "Spline theory. Differential equations"

1

Frank, Schneider. Inverse problems in satellite geodesy and their approximate solution by splines and wavelets. Aachen: Shaker, 1997.

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2

Schiesser, William E. Spline Collocation Methods for Partial Differential Equations. Hoboken, NJ, USA: John Wiley & Sons, Inc., 2017. http://dx.doi.org/10.1002/9781119301066.

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Multivalued differential equations. Berlin: W. de Gruyter, 1992.

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1889-1950, Stepanov V. V., ed. Qualitative theory of differential equations. New York: Dover Publications, 1989.

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Ordinary differential equations: Qualitative theory. Providence, R.I: American Mathematical Society, 2010.

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Dan, Port, ed. Differential equations: Theory and applications. Boston: Jones and Bartlett Publishers, 1991.

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Bellman, Richard Ernest. Stability theory of differential equations. Mineola, N.Y: Dover Publications, 2008.

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Differential equations: Theory and applications. 2nd ed. New York: Springer, 2010.

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Taylor, Michael Eugene. Partial differential equations: Basic theory. New York: Springer, 1996.

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Lakshmikantham, V. Theory of integro-differential equations. Lausanne, Switzerland: Gordon and Breach Science Publishers, 1995.

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Book chapters on the topic "Spline theory. Differential equations"

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Pedas, Arvet, Enn Tamme, and Mikk Vikerpuur. "Spline Collocation for Fractional Integro-Differential Equations." In Finite Difference Methods,Theory and Applications, 315–22. Cham: Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-20239-6_34.

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Bica, Alexandru Mihai, Mircea Curila, and Sorin Curila. "Spline Iterative Method for Pantograph Type Functional Differential Equations." In Finite Difference Methods. Theory and Applications, 159–66. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-11539-5_16.

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Didenko, Victor D., and Anh My Vu. "Spline Galerkin Methods for the Double Layer Potential Equations on Contours with Corners." In Recent Trends in Operator Theory and Partial Differential Equations, 129–44. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-47079-5_7.

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Yuan, Haiyan, Jihong Shen, and Cheng Song. "Mean Square Stability and Dissipativity of Split-Step Theta Method for Stochastic Delay Differential Equations with Poisson White Noise Excitations." In Automation Control Theory Perspectives in Intelligent Systems, 87–97. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-33389-2_9.

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Tikhonov, Andrei N., Adelaida B. Vasil’eva, and Alexei G. Sveshnikov. "General Theory." In Differential Equations, 18–65. Berlin, Heidelberg: Springer Berlin Heidelberg, 1985. http://dx.doi.org/10.1007/978-3-642-82175-2_2.

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Tikhonov, Andrei N., Adelaida B. Vasil’eva, and Alexei G. Sveshnikov. "Stability Theory." In Differential Equations, 130–53. Berlin, Heidelberg: Springer Berlin Heidelberg, 1985. http://dx.doi.org/10.1007/978-3-642-82175-2_5.

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Schechter, Martin. "Differential Equations." In Critical Point Theory, 83–93. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-45603-0_6.

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Ross, Clay C. "Differential Systems: Theory." In Differential Equations, 297–368. New York, NY: Springer New York, 2004. http://dx.doi.org/10.1007/978-1-4757-3949-7_9.

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Klenke, Achim. "Stochastic Differential Equations." In Probability Theory, 589–611. London: Springer London, 2014. http://dx.doi.org/10.1007/978-1-4471-5361-0_26.

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Klenke, Achim. "Stochastic Differential Equations." In Probability Theory, 665–90. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-56402-5_26.

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Conference papers on the topic "Spline theory. Differential equations"

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Huo, Xiaoming. "Identification of Underlying Partial Differential Equations from Noisy Data with Splines." In 3nd International Conference on Statistics: Theory and Applications (ICSTA'21). Avestia Publishing, 2021. http://dx.doi.org/10.11159/icsta21.005.

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Mashrouteh, Shamim, Ahmad Barari, and Ebrahim Esmailzadeh. "Flow-Induced Nonlinear Vibration of Non-Uniform Nanotubes." In ASME 2017 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2017. http://dx.doi.org/10.1115/detc2017-68208.

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This paper focuses on nonlinear forced vibration analysis of a free-form conveying fluid nanotube. Non-Uniform Rational B-Splines (NURBS) is used to model the free-form curvature of the nanotube, analytically. In order to develop the ordinary differential equations of motion, the Euler-Bernoulli beam theory and Galerkin method are implemented and the frequency response and the primary resonance of the nanotube under a harmonic excitation are studied. The effects of the free-form curvature of the nanotube and its physical characteristic on the nonlinear vibration behavior of the system are discussed as a parametric study.

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Żołądek, Henryk. "Polynomial Riccati equations with algebraic solutions." In Differential Galois Theory. Warsaw: Institute of Mathematics Polish Academy of Sciences, 2002. http://dx.doi.org/10.4064/bc58-0-17.

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Mutou, A., S. Mizuki, Y. Komatsubara, and H. Tsujita. "Behavior of Attractors During Surge in Centrifugal Compression System." In ASME 1998 International Gas Turbine and Aeroengine Congress and Exhibition. American Society of Mechanical Engineers, 1998. http://dx.doi.org/10.1115/98-gt-301.

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A dynamical system analysis method is presented, that permits the characterization of unsteady phenomena in a centrifugal compression system. The method maps one experimental time series of data into a state space in which behaviors of the compression system should be represented, and reconstructs an attractor that geometrically characterizes a state of the compression system. The time series of data were obtained by using a high response pressure transducer and an analog to digital converter at surge condition. For the reconstruction of attractors, a noise free differentiation method in time was employed. The differentiation was made by high order finite difference methods. To remove the influence of noise, the data were passed through a filter using a third order spline interpolation. In this study, the dimension of the state space was restricted to three. The measured data itself and their first and second derivatives in time are used to represent an attractor in the state space. The modeling of the system behavior from the time series of data by second order ordinary differential equations was attempted. It is assumed that the data and their derivatives satisfy the equations at each time. Then, appropriate coefficients are determined by a least square method. The reconstructed attractor showed complex cyclic trajectories at a first glance. However, by applying a band pass filter to the original signal, it was found that the attractor consisted of three independent wave forms and formed an attractor with torus-like behavior. In contrast, the solution by the modeled equations showed a type of limit cycle.

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5

Askari, Hassan, Zia Saadatnia, and Ebrahim Esmailzadeh. "Nonlinear Vibration of Nanobeam With Quadratic Rational Bezier Arc Curvature." In ASME 2014 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2014. http://dx.doi.org/10.1115/imece2014-37986.

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Nonlinear vibration of nanobeam with the quadratic rational Bezier arc curvature is investigated. The governing equation of motion of the nanobeam based on the Euler-Bernoulli beam theory is developed. Accordingly, the non-uniform rational B-spline (NURBS) is implemented in order to write the implicit form of the governing equation of the structure. The simply-supported boundary conditions are assumed and the Galerkin procedure is utilized to find the nonlinear ordinary differential equation of the system. The nonlinear natural frequency of the system is found and the effects of different parameters, namely, the waviness amplitude, oscillation amplitude, aspect ratio, curvature shape and the Pasternak foundation coefficient are fully investigated. The hardening and softening responses of the natural frequency of structure are detected for variations of the shape and amplitude of the curvature waviness. It is revealed that the ratio of nonlinear to linear frequency increases with an increase in the oscillation amplitudes. It is found that by increasing the Pasternak foundation coefficient, the ratio of nonlinear to linear frequency decreases.

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Grill, Thomas, Manfred Knebusch, and Marcus Tressl. "An existence theorem for systems of implicit differential equations." In Differential Galois Theory. Warsaw: Institute of Mathematics Polish Academy of Sciences, 2002. http://dx.doi.org/10.4064/bc58-0-6.

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7

Hall, Andrew, Chad Schmitke, and John McPhee. "Symbolic Formulation of a Path-Following Joint for Multibody Dynamics." In ASME 2014 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2014. http://dx.doi.org/10.1115/detc2014-35082.

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We present a specialized multibody joint that constrains motion to a spatial path. The joint is used in the reduction of 1 degree-of-freedom systems with complex kinematics. Example applications of the joint are: the reduction of vehicle suspension systems, or the representation of biological joints. The new joint is implemented in the graph-theoretic symbolic multibody modeling environment of MapleSim and is formulated in such a way that a single ordinary differential equation is used to describe the resulting kinematic pair. A particle moving along a planar semi-circular path was chosen as the first example for successful validation of the new joint since a simple closed-form solution in terms of the path length exists. To represent arbitrary curves, the path must first be parameterized in terms of its path length. Next, a differentiable mathematical definition of the curve must be generated. B-splines are generated to define the path. For best performance we minimize the number of knots in the splines and find their optimal locations. Using the spline fitting approach, a planar parabolic path is generated and used to further analyze the performance of our implementation.

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Christara, C. C., E. N. Houstis, and J. R. Rice. "A parallel spline collocation-capacitance method for elliptic partial differential equations." In the 2nd international conference. New York, New York, USA: ACM Press, 1988. http://dx.doi.org/10.1145/55364.55401.

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Halmschlager, Andrea, László Szenthe, and János Tóth. "Invariants of kinetic differential equations." In The 7'th Colloquium on the Qualitative Theory of Differential Equations. Szeged: Bolyai Institute, SZTE, 2003. http://dx.doi.org/10.14232/ejqtde.2003.6.14.

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Malinowski, Marek T. "On Bipartite Fuzzy Stochastic Differential Equations." In 8th International Conference on Fuzzy Computation Theory and Applications. SCITEPRESS - Science and Technology Publications, 2016. http://dx.doi.org/10.5220/0006079501090114.

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Reports on the topic "Spline theory. Differential equations"

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Ostashev, Vladimir, Michael Muhlestein, and D. Wilson. Extra-wide-angle parabolic equations in motionless and moving media. Engineer Research and Development Center (U.S.), September 2021. http://dx.doi.org/10.21079/11681/42043.

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Wide-angle parabolic equations (WAPEs) play an important role in physics. They are derived by an expansion of a square-root pseudo-differential operator in one-way wave equations, and then solved by finite-difference techniques. In the present paper, a different approach is suggested. The starting point is an extra-wide-angle parabolic equation (EWAPE) valid for small variations of the refractive index of a medium. This equation is written in an integral form, solved by a perturbation technique, and transformed to the spectral domain. The resulting split-step spectral algorithm for the EWAPE accounts for the propagation angles up to 90° with respect to the nominal direction. This EWAPE is also generalized to large variations in the refractive index. It is shown that WAPEs known in the literature are particular cases of the two EWAPEs. This provides an alternative derivation of the WAPEs, enables a better understanding of the underlying physics and ranges of their applicability, and opens an opportunity for innovative algorithms. Sound propagation in both motionless and moving media is considered. The split-step spectral algorithm is particularly useful in the latter case since complicated partial derivatives of the sound pressure and medium velocity reduce to wave vectors (essentially, propagation angles) in the spectral domain.

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Pieper, G. W. Proceedings of the focused research program on spectral theory and boundary value problems: Volume 3, Linear differential equations and systems. Office of Scientific and Technical Information (OSTI), April 1989. http://dx.doi.org/10.2172/6023178.

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