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Articoli di riviste sul tema "Partial Numerical solutions":
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Zhang, Zhao. "Numerical Analysis and Comparison of Gridless Partial Differential Equations". International Journal of Circuits, Systems and Signal Processing 15 (31 agosto 2021): 1223–31. http://dx.doi.org/10.46300/9106.2021.15.133.
In the field of science and engineering, partial differential equations play an important role in the process of transforming physical phenomena into mathematical models. Therefore, it is very important to get a numerical solution with high accuracy. In solving linear partial differential equations, meshless solution is a very important method. Based on this, we propose the numerical solution analysis and comparison of meshless partial differential equations (PDEs). It is found that the interaction between the numerical solutions of gridless PDEs is better, and the absolute error and relative error are lower, which proves the superiority of the numerical solutions of gridless PDEs
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Wu, G., Eric Wai Ming Lee e Gao Li. "Numerical solutions of the reaction-diffusion equation". International Journal of Numerical Methods for Heat & Fluid Flow 25, n. 2 (2 marzo 2015): 265–71. http://dx.doi.org/10.1108/hff-04-2014-0113.
Purpose – The purpose of this paper is to introduce variational iteration method (VIM) to construct equivalent integral equations for initial-boundary value problems of nonlinear partial differential equations. The Lagrange multipliers become the integral kernels. Design/methodology/approach – Using the discrete numerical integral formula, the general way is given to solve the famous reaction-diffusion equation numerically. Findings – With the given explicit solution, the results show the conveniences of the general numerical schemes and numerical simulation of the reaction-diffusion is finally presented in the cases of various coefficients. Originality/value – The method avoids the treatment of the time derivative as that in the classical finite difference method and the VIM is introduced to construct equivalent integral equations for initial-boundary value problems of nonlinear partial differential equations.
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Wang, Zhigang, Xiaoting Liu, Lijun Su e Baoyan Fang. "Numerical Solutions of Convective Diffusion Equations using Wavelet Collocation Method". Advances in Engineering Technology Research 1, n. 1 (17 maggio 2022): 192. http://dx.doi.org/10.56028/aetr.1.1.192.
Some partial differential equations appear in many application fields. Therefore, the discussion of numerical solutions of those partial differential equations using numerical methods becomes a valuable and important issue in numerical simulation. In numerical methods, the wavelet-collocation method has been frequently developed for solving PDEs, and the algorithm has yielded substantial results. However, theoretical research of the numerical solution has been rarely discussed yet. In this paper, the numerical solution of convective diffusion equations using the wavelet-collocation method is established, and its existence and uniqueness are derived.
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NAKAO, Mitsuhiro. "Numerical Verification of Solutions for Partial Differential Equations". IEICE ESS FUNDAMENTALS REVIEW 2, n. 3 (2009): 19–28. http://dx.doi.org/10.1587/essfr.2.3_19.
Nakao, Mitsuhiro T. "Numerical verification for solutions to partial differential equations". Sugaku Expositions 30, n. 1 (17 marzo 2017): 89–109. http://dx.doi.org/10.1090/suga/419.
Computational simulations of ocean circulation rely on the numerical solution of partial differential equations of fluid dynamics, as applied to a relatively thin layer of stratified fluid on a rotating globe. This paper describes some of the physical and mathematical properties of the solutions being sought, some of the issues that are encountered when the governing equations are solved numerically, and some of the numerical methods that are being used in this area.
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Seth, G. S., S. Sarkar e R. Sharma. "Effects of Hall current on unsteady hydromagnetic free convection flow past an impulsively moving vertical plate with Newtonian heating". International Journal of Applied Mechanics and Engineering 21, n. 1 (1 febbraio 2016): 187–203. http://dx.doi.org/10.1515/ijame-2016-0012.
Abstract An investigation of unsteady hydromagnetic free convection flow of a viscous, incompressible and electrically conducting fluid past an impulsively moving vertical plate with Newtonian surface heating embedded in a porous medium taking into account the effects of Hall current is carried out. The governing partial differential equations are first subjected to the Laplace transformation and then inverted numerically using INVLAP routine of Matlab. The governing partial differential equations are also solved numerically by the Crank-Nicolson implicit finite difference scheme and a comparison has been provided between the two solutions. The numerical solutions for velocity and temperature are plotted graphically whereas the numerical results of skin friction and the Nusselt number are presented in tabular form for various parameters of interest. The present solution in special case is compared with a previously obtained solution and is found to be in excellent agreement.
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Iqbal, Mazhar, M. T. Mustafa e Azad A. Siddiqui. "A Method for Generating Approximate Similarity Solutions of Nonlinear Partial Differential Equations". Abstract and Applied Analysis 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/105414.
Standard application of similarity method to find solutions of PDEs mostly results in reduction to ODEs which are not easily integrable in terms of elementary or tabulated functions. Such situations usually demand solving reduced ODEs numerically. However, there are no systematic procedures available to utilize these numerical solutions of reduced ODE to obtain the solution of original PDE. A practical and tractable approach is proposed to deal with such situations and is applied to obtain approximate similarity solutions to different cases of an initial-boundary value problem of unsteady gas flow through a semi-infinite porous medium.
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Zou, Guang-an. "Numerical solutions to time-fractional stochastic partial differential equations". Numerical Algorithms 82, n. 2 (5 novembre 2018): 553–71. http://dx.doi.org/10.1007/s11075-018-0613-0.
ARLUKOWICZ, P., e W. CZERNOUS. "A numerical method of bicharacteristics For quasi-linear partial functional Differential equations". Computational Methods in Applied Mathematics 8, n. 1 (2008): 21–38. http://dx.doi.org/10.2478/cmam-2008-0002.
Abstract Classical solutions of mixed problems for first order partial functional differential equations in several independent variables are approximated by solutions of an Euler-type difference problem. The mesh for the approximate solutions is obtained by the numerical solution of equations of bicharacteristics. The convergence of explicit difference schemes is proved by means of consistency and stability arguments. It is assumed that the given functions satisfy the nonlinear estimates of the Perron type. Differential systems with deviated variables and differential integral systems can be obtained from the general model by specializing the given operators.
Bratsos, A. G. "Numerical solutions of nonlinear partial differential equations". Thesis, Brunel University, 1993. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.332806.
Sundqvist, Per. "Numerical Computations with Fundamental Solutions". Doctoral thesis, Uppsala : Acta Universitatis Upsaliensis : Univ.-bibl. [distributör], 2005. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-5757.
Cheung, Ka Chun. "Meshless algorithm for partial differential equations on open and singular surfaces". HKBU Institutional Repository, 2016. https://repository.hkbu.edu.hk/etd_oa/278.
Radial Basis function (RBF) method for solving partial differential equation (PDE) has a lot of applications in many areas. One of the advantages of RBF method is meshless. The cost of mesh generation can be reduced by playing with scattered data. It can also allow adaptivity to solve some problems with special feature. In this thesis, RBF method will be considered to solve several problems. Firstly, we solve the PDEs on surface with singularity (folded surface) by a localized method. The localized method is a generalization of finite difference method. A priori error estimate for the discreitzation of Laplace operator is given for points selection. A stable solver (RBF-QR) is used to avoid ill-conditioning for the numerical simulation. Secondly, a {dollar}H^2{dollar} convergence study for the least-squares kernel collocation method, a.k.a. least-square Kansa's method will be discussed. This chapter can be separated into two main parts: constraint least-square method and weighted least-square method. For both methods, stability and consistency analysis are considered. Error estimate for both methods are also provided. For the case of weighted least-square Kansa's method, we figured out a suitable weighting for optimal error estimation. In Chapter two, we solve partial differential equation on smooth surface by an embedding method in the embedding space {dollar}\R^d{dollar}. Therefore, one can apply any numerical method in {dollar}\R^d{dollar} to solve the embedding problem. Thus, as an application of previous result, we solve embedding problem by least-squares kernel collocation. Moreover, we propose a new embedding condition in this chapter which has high order of convergence. As a result, we solve partial differential equation on smooth surface with a high order kernel collocation method. Similar to chapter two, we also provide error estimate for the numerical solution. Some applications such as pattern formation in the Brusselator system and excitable media in FitzHughNagumo model are also studied.
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Al-Muslimawi, Alaa Hasan A. "Numerical analysis of partial differential equations for viscoelastic and free surface flows". Thesis, Swansea University, 2013. https://cronfa.swan.ac.uk/Record/cronfa42876.
ROEHL, NITZI MESQUITA. "NUMERICAL SOLUTIONS FOR SHAPE OPTIMIZATION PROBLEMS ASSOCIATED WITH ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS". PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 1991. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=9277@1.
COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR Essa dissertação visa à obtenção de soluções numéricas
para problemas de otimização de formas geométricas
associados a equações diferenciais parciais elípticas. A
principal motivação é um problema termal, onde deseja-se
determinar a fronteira ótima, para um volume de material
isolante fixo, tal que a perda de calor de um corpo seja
minimizada. Realiza-se a análise e implementação numérica
de uma abordagem via método das penalidades dos problemas
de minimização. O método de elementos finitos é utilizado
para discretizar o domínio em questão. A formulação
empregada possui a característica atrativa da minimização
ser conduzida sobre um espaço de funções lineares. Uma
série de resultados numéricos são obtidos. Propõe-se,
ainda, um algoritmo para a solução de problemas termais
que envolvem material isolante composto. This work is directed at the problem of determining
numerical solutions for shape optimization problems
associated with elliptic partial differential equations.
Our primarily motivation is the problem of determining
optimal shapes in order to minimize the heat lost of a
body, given a fixed volume of insulation and a fixed
internal (or external) geometry. The analysis and
implementation of a penaly approach of the heat loss
minimization problem are achieved. The formulation
employed has the attractive feature that minimization is
conducted over a linear function space. The algrithm
adopted is based on the finite element method. Many
numerical results are presented. We also propose an
algorithm for the numerical solution of termal problems
wich are concerned with multiple insulation layers.
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Zeng, Suxing. "Numerical solutions of boundary inverse problems for some elliptic partial differential equations". Morgantown, W. Va. : [West Virginia University Libraries], 2009. http://hdl.handle.net/10450/10345.
Thesis (Ph. D.)--West Virginia University, 2009. Title from document title page. Document formatted into pages; contains v, 58 p. : ill. (some col.). Includes abstract. Includes bibliographical references (p. 56-58).
Capitoli di libri sul tema "Partial Numerical solutions":
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Logan, J. David. "Numerical Computation of Solutions". In Applied Partial Differential Equations, 257–77. Cham: Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-12493-3_6.
Bleecker, David, e George Csordas. "Numerical Solutions of PDEs — An Introduction". In Basic Partial Differential Equations, 503–58. Boston, MA: Springer US, 1992. http://dx.doi.org/10.1007/978-1-4684-1434-9_8.
Saha Ray, Santanu. "Numerical Solutions of Partial Differential Equations". In Numerical Analysis with Algorithms and Programming, 591–640. Boca Raton : Taylor & Francis, 2016. | “A CRC title.”: Chapman and Hall/CRC, 2018. http://dx.doi.org/10.1201/9781315369174-10.
Stroud, K. A., e Dexter Booth. "Numerical solutions of partial differential equations". In Advanced Engineering Mathematics, 593–641. London: Macmillan Education UK, 2011. http://dx.doi.org/10.1057/978-0-230-34474-7_18.
Leung, Anthony W. "Systems of Finite Difference Equations, Numerical Solutions". In Systems of Nonlinear Partial Differential Equations, 271–323. Dordrecht: Springer Netherlands, 1989. http://dx.doi.org/10.1007/978-94-015-3937-1_6.
Dong, Gang Nathan. "Numerical Solutions of Financial Partial Differential Equations". In Handbook of Quantitative Finance and Risk Management, 1209–21. Boston, MA: Springer US, 2010. http://dx.doi.org/10.1007/978-0-387-77117-5_79.
Rathish Kumar, B. V., e Gopal Priyadarshi. "Wavelet Galerkin Methods for Higher Order Partial Differential Equations". In Mathematical Modelling, Optimization, Analytic and Numerical Solutions, 231–53. Singapore: Springer Singapore, 2020. http://dx.doi.org/10.1007/978-981-15-0928-5_11.
Akilandeeswari, A., K. Balachandran e N. Annapoorani. "On Fractional Partial Differential Equations of Diffusion Type with Integral Kernel". In Mathematical Modelling, Optimization, Analytic and Numerical Solutions, 333–49. Singapore: Springer Singapore, 2020. http://dx.doi.org/10.1007/978-981-15-0928-5_16.
Hou, Thomas Y. "Numerical Approximations to Multiscale Solutions in Partial Differential Equations". In Universitext, 241–301. Berlin, Heidelberg: Springer Berlin Heidelberg, 2003. http://dx.doi.org/10.1007/978-3-642-55692-0_6.
Atti di convegni sul tema "Partial Numerical solutions":
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Siddique, Mohammad, Theodore E. Simos, George Psihoyios e Ch Tsitouras. "Symposium: Advances in the Numerical Solutions of Partial Differential Equations". In ICNAAM 2010: International Conference of Numerical Analysis and Applied Mathematics 2010. AIP, 2010. http://dx.doi.org/10.1063/1.3498011.
Kudryashov, N. A., e A. K. Volkov. "Concatenons as the solutions for non-linear partial differential equations". In INTERNATIONAL CONFERENCE OF NUMERICAL ANALYSIS AND APPLIED MATHEMATICS (ICNAAM 2016). Author(s), 2017. http://dx.doi.org/10.1063/1.4992559.
Aleixo, Rafael, e Daniela Amazonas. "Noise Reduction on Numerical Solutions of Partial Differential Equations using Fuzzy Transform". In CNMAC 2017 - XXXVII Congresso Nacional de Matemática Aplicada e Computacional. SBMAC, 2018. http://dx.doi.org/10.5540/03.2018.006.01.0402.
Wang, Zhanjiang, Xiaoqing Jin, Leon M. Keer e Qian Wang. "Numerical Modeling of Partial Slip Contact Involving Inhomogeneous Materials". In ASME/STLE 2012 International Joint Tribology Conference. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/ijtc2012-61108.
When solving the problems involving inhomogeneous materials, the influence of the inhomogeneity upon contact behavior should be properly considered. This research proposes a fast and novel method, based on the equivalent inclusion method where inhomogeneity is replaced by an inclusion with properly chosen eigenstrains, to simulate contact partial slip of the interface involving inhomogeneous materials. The total stress and displacement fields represent the superposition of homogeneous solutions and perturbed solutions due to the chosen eigenstrains. In the present numerical simulation, the half space is meshed into a number of cuboids of the same size, where each cuboid is has a uniform eigenstrain. The stress and displacement fields due to eigenstrains are formulated by employing the recent half-space inclusion solutions derived by the authors and solved using a three-dimensional fast Fourier transform algorithm. The partial slip contact between an elastic ball and an elastic half space containing a cuboidal inhomogeneity was investigated.
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Ashyralyev, Allaberen, Evren Hincal e Bilgen Kaymakamzade. "Numerical solutions of the system of partial differential equations for observing epidemic models". In INTERNATIONAL CONFERENCE ON ANALYSIS AND APPLIED MATHEMATICS (ICAAM 2018). Author(s), 2018. http://dx.doi.org/10.1063/1.5049044.
McMasters, Robert L., Filippo de Monte, James V. Beck e Donald E. Amos. "Transient Two-Dimensional Heat Conduction Problem With Partial Heating Near Corners". In ASME 2016 Heat Transfer Summer Conference collocated with the ASME 2016 Fluids Engineering Division Summer Meeting and the ASME 2016 14th International Conference on Nanochannels, Microchannels, and Minichannels. American Society of Mechanical Engineers, 2016. http://dx.doi.org/10.1115/ht2016-7103.
This paper provides a solution for two-dimensional heating over a rectangular region on a homogeneous plate. It has application to verification of numerical conduction codes as well as direct application for heating and cooling of electronic equipment. Additionally, it can be applied as a direct solution for the inverse heat conduction problem, most notably used in thermal protection systems for re-entry vehicles. The solutions used in this work are generated using Green’s functions. Two approaches are used which provide solutions for either semi-infinite plates or finite plates with isothermal conditions which are located a long distance from the heating. The methods are both efficient numerically and have extreme accuracy, which can be used to provide additional solution verification. The solutions have components that are shown to have physical significance. The extremely precise nature of analytical solutions allows them to be used as prime standards for their respective transient conduction cases. This extreme precision also allows an accurate calculation of heat flux by finite differences between two points of very close proximity which would not be possible with numerical solutions. This is particularly useful near heated surfaces and near corners. Similarly, sensitivity coefficients for parameter estimation problems can be calculated with extreme precision using this same technique. Another contribution of these solutions is the insight that they can bring. Important dimensionless groups are identified and their influence can be more readily seen than with numerical results. For linear problems, basic heating elements on plates, for example, can be solved to aid in understanding more complex cases. Furthermore these basic solutions can be superimposed both in time and space to obtain solutions for numerous other problems. This paper provides an analytical two-dimensional, transient solution for heating over a rectangular region on a homogeneous square plate. Several methods are available for the solution of such problems. One of the most common is the separation of variables (SOV) method. In the standard implementation of the SOV method, convergence can be slow and accuracy lacking. Another method of generating a solution to this problem makes use of time-partitioning which can produce accurate results. However, numerical integration may be required in these cases, which, in some ways, negates the advantages offered by the analytical solutions. The method given herein requires no numerical integration; it also exhibits exponential series convergence and can provide excellent accuracy. The procedure involves the derivation of previously-unknown simpler forms for the summations, in some cases by virtue of the use of algebraic components. Also, a mathematical identity given in this paper can be used for a variety of related problems.
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Surana, K. S., e M. A. Bona. "Computations of Higher Class Solutions of Partial Differential Equations". In ASME 2001 Engineering Technology Conference on Energy. American Society of Mechanical Engineers, 2001. http://dx.doi.org/10.1115/etce2001-17142.
Abstract This paper presents a new computational strategy, computational framework and mathematical framework for numerical computations of higher class solutions of differential and partial differential equations. The approach presented here utilizes ‘strong forms’ of the governing differential equations (GDE’s) and least squares approach in constructing the integral form. The conventional, or currently used, approaches seek the convergence of a solution in a fixed (order) space by h, p or hp-adaptive processes. The fundamental point of departure in the proposed approach is that we seek convergence of the computed solution by changing the orders of the spaces of the basis functions. With this approach convergence rates much higher than those from h,p–processes are achievable and the progressively computed solutions converge to the ‘strong’ i.e. ‘theoretical’ solutions of the GDE’s. Many other benefits of this approach are discussed and demonstrated. Stationary and time-dependant convection-diffusion and Burgers equations are used as model problems.
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Prasanna, D., e K. Aung. "Numerical Solutions of Single and Multiple Laminar Jets". In ASME 2005 Fluids Engineering Division Summer Meeting. ASMEDC, 2005. http://dx.doi.org/10.1115/fedsm2005-77079.
Modern power plants discharge approximately 1.5 to 2kWhr of waste heat for every kWhr of electrical energy produced. Modern power plants discharge approximately 1.5 to 2kWhr of waste heat for every kWhr of electrical energy produced. Usually this heat is discharged to an adjacent water body which increases the water temperature near the outfall. In order to assess the ecological consequences of waste heat discharge one must first know the physical changes (temperature, velocity, salinity) induced by these discharges. It is with this later aspect, prediction of physical properties, that the current work is primarily concerned. Existing theoretical work on axisymmetric buoyant jets is confined to integral techniques developed by Morton in the early 1950’s. From these techniques only centerline velocities and temperatures can be calculated. Experimental data for this type of flow are essentially confined to centerline temperature measurements except for pure jet or plume data which constitute the extremes for a buoyant jet. The present work addresses the problem of developing a theoretical model for an axisymmetric laminar buoyant jet. The governing equations for an axisymmetric buoyant jet in rectangular co-ordinates are transformed into an orthogonal curvilinear co-ordinate system which moves along the length of the jet axis. The complete partial differential equations governing steady, incompressible laminar flow are solved in the new curvilinear co-ordinates using finite-difference techniques. This method is applicable to a much wider range of jet flows issued at arbitrary angles into quiescent or flowing ambience. This method is also applied to the case for multiple jets spaced by a finite distance apart. Results for the momentum jet, axial and radial distribution of velocity and temperature, show good agreement with published data.
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Kasharin, Alexander V., e Jens O. M. Karlsson. "Diffusion-Limited Cell Dehydration: Analytical and Numerical Solutions for a Planar Model". In ASME 1999 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 1999. http://dx.doi.org/10.1115/imece1999-0600.
Abstract The process of diffusion-limited cell dehydration is modeled for a planar system by writing the one-dimensional diffusion-equation for a cell with moving, semipermeable boundaries. For the simplifying case of isothermal dehydration with constant diffusivity, an approximate analytical solution is obtained by linearizing the governing partial differential equations. The general problem must be solved numerically. The Forward Time Center Space (FTCS) and Crank-Nicholson differencing schemes are implemented, and evaluated by comparison with the analytical solution. Putative stability criteria for the two algorithms are proposed based on numerical experiments, and the Crank-Nicholson method is shown to be accurate for a mesh with as few as six nodes.
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Armand, J., L. Salles e C. W. Schwingshackl. "Numerical Simulation of Partial Slip Contact Using a Semi-Analytical Method". In ASME 2015 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2015. http://dx.doi.org/10.1115/detc2015-46464.
Almost all mechanical structures consist of an assembly of components that are linked together with joints. If such a structure experiences vibration during operation, micro-sliding can occur in the joint, resulting in fretting wear. Fretting wear affects the mechanical properties of the joints over their lifetime and as a result impacts the non-linear dynamic response of the system. For accurate prediction of the non-linear dynamic response over the lifetime of the structure, fretting wear should be considered in the analysis. Fretting wear studies require an accurate assessment of the stresses and strains in the contacting surfaces of the joints. To provide this information, a contact solver based on the semi-analytical method has been implemented in this study. By solving the normal and tangential contact problems between two elastic semi-infinite bodies, the contact solver allows an accurate calculation of the pressure and shear distributions as well as the relative slips in the contact area. The computed results for a smooth spherical contact between similar elastic materials are presented and validated against analytical solutions. The results are also compared with those obtained from finite element simulations to demonstrate the accuracy and computational benefits of the semi-analytical method. Its capabilities are further illustrated in a new test case of a cylinder with rounded edges on a flat surface, which is a more realistic contact representation of an industrial joint.
Rapporti di organizzazioni sul tema "Partial Numerical solutions":
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Levine, Howard A. Numerical Solution of Ill Posed Problems in Partial Differential Equations. Fort Belvoir, VA: Defense Technical Information Center, settembre 1987. http://dx.doi.org/10.21236/ada189383.
Levine, Howard A. Numerical Solution of I11 Posed Problems in Partial Differential Equations. Fort Belvoir, VA: Defense Technical Information Center, novembre 1985. http://dx.doi.org/10.21236/ada162378.
Levine, Howard A. Numerical Solution of Ill Posed Problems in Partial Differential Equations. Fort Belvoir, VA: Defense Technical Information Center, aprile 1985. http://dx.doi.org/10.21236/ada166096.
Sharan, M., E. J. Kansa e S. Gupta. Application of multiquadric method for numerical solution of elliptic partial differential equations. Office of Scientific and Technical Information (OSTI), gennaio 1994. http://dx.doi.org/10.2172/10156506.
Dupont, Todd F. Some Investigations into Variable Meshes for Numerical Solution of Partial Differential Equations. Fort Belvoir, VA: Defense Technical Information Center, maggio 1986. http://dx.doi.org/10.21236/ada168977.
Oliker, V. I., e P. Waltman. New Methods for Numerical Solution of One Class of Strongly Nonlinear Partial Differential Equations with Applications. Fort Belvoir, VA: Defense Technical Information Center, gennaio 1986. http://dx.doi.org/10.21236/ada186166.
Oliker, V. I., e P. Waltman. New Methods for Numerical Solution of One Class of Strongly Nonlinear Partial Differential Equations with Applications. Fort Belvoir, VA: Defense Technical Information Center, agosto 1987. http://dx.doi.org/10.21236/ada189945.
Trenchea, Catalin. Efficient Numerical Approximations of Tracking Statistical Quantities of Interest From the Solution of High-Dimensional Stochastic Partial Differential Equations. Fort Belvoir, VA: Defense Technical Information Center, febbraio 2012. http://dx.doi.org/10.21236/ada567709.
Trenchea, Catalin. Efficient Numerical Approximations of Tracking Statistical Quantities of Interest From the Solution of High-Dimensional Stochastic Partial Differential Equations. Fort Belvoir, VA: Defense Technical Information Center, febbraio 2012. http://dx.doi.org/10.21236/ada577122.
