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18.303 Linear Partial Differential Equations (MIT) 18.303 Linear Partial Differential Equations (MIT)

Description

This course covers the classical partial differential equations of applied mathematics: diffusion, Laplace/Poisson, and wave equations. It also includes methods of solution, such as separation of variables, Fourier series and transforms, eigenvalue problems. Green's function methods are emphasized.Technical RequirementsSpecial software is required to use some of the files in this course: .m. This course covers the classical partial differential equations of applied mathematics: diffusion, Laplace/Poisson, and wave equations. It also includes methods of solution, such as separation of variables, Fourier series and transforms, eigenvalue problems. Green's function methods are emphasized.Technical RequirementsSpecial software is required to use some of the files in this course: .m.

Subjects

diffusion | diffusion | Laplace equations | Laplace equations | Poisson | Poisson | wave equations | wave equations | separation of variables | separation of variables | Fourier series | Fourier series | Fourier transforms | Fourier transforms | eigenvalue problems | eigenvalue problems | Green's function | Green's function | Heat Equation | Heat Equation | Sturm-Liouville Eigenvalue problems | Sturm-Liouville Eigenvalue problems | quasilinear PDEs | quasilinear PDEs | Bessel functions | Bessel functions

License

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18.303 Linear Partial Differential Equations (MIT) 18.303 Linear Partial Differential Equations (MIT)

Description

This course covers the classical partial differential equations of applied mathematics: diffusion, Laplace/Poisson, and wave equations. It also includes methods of solution, such as separation of variables, Fourier series and transforms, eigenvalue problems. Green's function methods are emphasized.Technical RequirementsMATLAB® software is required to run the .m files found on this course site. This course covers the classical partial differential equations of applied mathematics: diffusion, Laplace/Poisson, and wave equations. It also includes methods of solution, such as separation of variables, Fourier series and transforms, eigenvalue problems. Green's function methods are emphasized.Technical RequirementsMATLAB® software is required to run the .m files found on this course site.

Subjects

diffusion | diffusion | Laplace equations | Laplace equations | Poisson | Poisson | wave equations | wave equations | separation of variables | separation of variables | Fourier series | Fourier series | Fourier transforms | Fourier transforms | eigenvalue problems | eigenvalue problems | Green's function | Green's function | Heat Equation | Heat Equation | Sturm-Liouville Eigenvalue problems | Sturm-Liouville Eigenvalue problems | quasilinear PDEs | quasilinear PDEs | Bessel functions | Bessel functions

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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16.901 Computational Methods in Aerospace Engineering (MIT) 16.901 Computational Methods in Aerospace Engineering (MIT)

Description

This course serves as an introduction to computational techniques arising in aerospace engineering. Applications are drawn from aerospace structures, aerodynamics, dynamics and control, and aerospace systems. Techniques include: numerical integration of systems of ordinary differential equations; finite-difference, finite-volume, and finite-element discretization of partial differential equations; numerical linear algebra; eigenvalue problems; and optimization with constraints.Technical RequirementsMATLAB® software is required to run the .m and .mat files found on this course site.MATLAB® is a trademark of The MathWorks, Inc. This course serves as an introduction to computational techniques arising in aerospace engineering. Applications are drawn from aerospace structures, aerodynamics, dynamics and control, and aerospace systems. Techniques include: numerical integration of systems of ordinary differential equations; finite-difference, finite-volume, and finite-element discretization of partial differential equations; numerical linear algebra; eigenvalue problems; and optimization with constraints.Technical RequirementsMATLAB® software is required to run the .m and .mat files found on this course site.MATLAB® is a trademark of The MathWorks, Inc.

Subjects

numerical integration | numerical integration | ODEs | ODEs | ordinary differential equations | ordinary differential equations | finite difference | finite difference | finite volume | finite volume | finite element | finite element | discretization | discretization | PDEs | PDEs | partial differential equations | partial differential equations | numerical linear algebra | numerical linear algebra | probabilistic methods | probabilistic methods | optimization | optimization | omputational methods | omputational methods | aerospace engineering | aerospace engineering | computational methods | computational methods

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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16.901 Computational Methods in Aerospace Engineering (MIT) 16.901 Computational Methods in Aerospace Engineering (MIT)

Description

This course serves as an introduction to computational techniques arising in aerospace engineering. Applications are drawn from aerospace structures, aerodynamics, dynamics and control, and aerospace systems. Techniques include: numerical integration of systems of ordinary differential equations; finite-difference, finite-volume, and finite-element discretization of partial differential equations; numerical linear algebra; eigenvalue problems; and optimization with constraints. This course serves as an introduction to computational techniques arising in aerospace engineering. Applications are drawn from aerospace structures, aerodynamics, dynamics and control, and aerospace systems. Techniques include: numerical integration of systems of ordinary differential equations; finite-difference, finite-volume, and finite-element discretization of partial differential equations; numerical linear algebra; eigenvalue problems; and optimization with constraints.

Subjects

numerical integration | numerical integration | ODEs | ODEs | ordinary differential equations | ordinary differential equations | finite difference | finite difference | finite volume | finite volume | finite element | finite element | discretization | discretization | PDEs | PDEs | partial differential equations | partial differential equations | numerical linear algebra | numerical linear algebra | probabilistic methods | probabilistic methods | optimization | optimization | omputational methods | omputational methods | aerospace engineering | aerospace engineering | computational methods | computational methods

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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18.303 Linear Partial Differential Equations: Analysis and Numerics (MIT) 18.303 Linear Partial Differential Equations: Analysis and Numerics (MIT)

Description

This course provides students with the basic analytical and computational tools of linear partial differential equations (PDEs) for practical applications in science engineering, including heat/diffusion, wave, and Poisson equations. Analytics emphasize the viewpoint of linear algebra and the analogy with finite matrix problems. Numerics focus on finite-difference and finite-element techniques to reduce PDEs to matrix problems. This course provides students with the basic analytical and computational tools of linear partial differential equations (PDEs) for practical applications in science engineering, including heat/diffusion, wave, and Poisson equations. Analytics emphasize the viewpoint of linear algebra and the analogy with finite matrix problems. Numerics focus on finite-difference and finite-element techniques to reduce PDEs to matrix problems.

Subjects

diffusion | diffusion | Laplace equations | Laplace equations | Poisson | Poisson | wave equations | wave equations | separation of variables | separation of variables | Fourier series | Fourier series | Fourier transforms | Fourier transforms | eigenvalue problems | eigenvalue problems | Green's function | Green's function | Heat Equation | Heat Equation | Sturm-Liouville Eigenvalue problems | Sturm-Liouville Eigenvalue problems | quasilinear PDEs | quasilinear PDEs | Bessel functionsORDS | Bessel functionsORDS

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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18.303 Linear Partial Differential Equations (MIT) 18.303 Linear Partial Differential Equations (MIT)

Description

This course covers the classical partial differential equations of applied mathematics: diffusion, Laplace/Poisson, and wave equations. It also includes methods and tools for solving these PDEs, such as separation of variables, Fourier series and transforms, eigenvalue problems, and Green's functions. This course covers the classical partial differential equations of applied mathematics: diffusion, Laplace/Poisson, and wave equations. It also includes methods and tools for solving these PDEs, such as separation of variables, Fourier series and transforms, eigenvalue problems, and Green's functions.

Subjects

diffusion | diffusion | Laplace equations | Laplace equations | Poisson | Poisson | wave equations | wave equations | separation of variables | separation of variables | Fourier series | Fourier series | Fourier transforms | Fourier transforms | eigenvalue problems | eigenvalue problems | Green's function | Green's function | Heat Equation | Heat Equation | Sturm-Liouville Eigenvalue problems | Sturm-Liouville Eigenvalue problems | quasilinear PDEs | quasilinear PDEs | Bessel functions | Bessel functions

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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18.327 Wavelets, Filter Banks and Applications (MIT) 18.327 Wavelets, Filter Banks and Applications (MIT)

Description

Wavelets are localized basis functions, good for representing short-time events. The coefficients at each scale are filtered and subsampled to give coefficients at the next scale. This is Mallat's pyramid algorithm for multiresolution, connecting wavelets to filter banks. Wavelets and multiscale algorithms for compression and signal/image processing are developed. Subject is project-based for engineering and scientific applications. Wavelets are localized basis functions, good for representing short-time events. The coefficients at each scale are filtered and subsampled to give coefficients at the next scale. This is Mallat's pyramid algorithm for multiresolution, connecting wavelets to filter banks. Wavelets and multiscale algorithms for compression and signal/image processing are developed. Subject is project-based for engineering and scientific applications.

Subjects

Discrete-time filters | Discrete-time filters | convolution | convolution | Fourier transform | Fourier transform | owpass and highpass filters | owpass and highpass filters | Sampling rate change operations | Sampling rate change operations | upsampling and downsampling | upsampling and downsampling | ractional sampling | ractional sampling | interpolation | interpolation | Filter Banks | Filter Banks | time domain (Haar example) and frequency domain | time domain (Haar example) and frequency domain | conditions for alias cancellation and no distortion | conditions for alias cancellation and no distortion | perfect reconstruction | perfect reconstruction | halfband filters and possible factorizations | halfband filters and possible factorizations | Modulation and polyphase representations | Modulation and polyphase representations | Noble identities | Noble identities | block Toeplitz matrices and block z-transforms | block Toeplitz matrices and block z-transforms | polyphase examples | polyphase examples | Matlab wavelet toolbox | Matlab wavelet toolbox | Orthogonal filter banks | Orthogonal filter banks | paraunitary matrices | paraunitary matrices | orthogonality condition (Condition O) in the time domain | orthogonality condition (Condition O) in the time domain | modulation domain and polyphase domain | modulation domain and polyphase domain | Maxflat filters | Maxflat filters | Daubechies and Meyer formulas | Daubechies and Meyer formulas | Spectral factorization | Spectral factorization | Multiresolution Analysis (MRA) | Multiresolution Analysis (MRA) | requirements for MRA | requirements for MRA | nested spaces and complementary spaces; scaling functions and wavelets | nested spaces and complementary spaces; scaling functions and wavelets | Refinement equation | Refinement equation | iterative and recursive solution techniques | iterative and recursive solution techniques | infinite product formula | infinite product formula | filter bank approach for computing scaling functions and wavelets | filter bank approach for computing scaling functions and wavelets | Orthogonal wavelet bases | Orthogonal wavelet bases | connection to orthogonal filters | connection to orthogonal filters | orthogonality in the frequency domain | orthogonality in the frequency domain | Biorthogonal wavelet bases | Biorthogonal wavelet bases | Mallat pyramid algorithm | Mallat pyramid algorithm | Accuracy of wavelet approximations (Condition A) | Accuracy of wavelet approximations (Condition A) | vanishing moments | vanishing moments | polynomial cancellation in filter banks | polynomial cancellation in filter banks | Smoothness of wavelet bases | Smoothness of wavelet bases | convergence of the cascade algorithm (Condition E) | convergence of the cascade algorithm (Condition E) | splines | splines | Bases vs. frames | Bases vs. frames | Signal and image processing | Signal and image processing | finite length signals | finite length signals | boundary filters and boundary wavelets | boundary filters and boundary wavelets | wavelet compression algorithms | wavelet compression algorithms | Lifting | Lifting | ladder structure for filter banks | ladder structure for filter banks | factorization of polyphase matrix into lifting steps | factorization of polyphase matrix into lifting steps | lifting form of refinement equationSec | lifting form of refinement equationSec | Wavelets and subdivision | Wavelets and subdivision | nonuniform grids | nonuniform grids | multiresolution for triangular meshes | multiresolution for triangular meshes | representation and compression of surfaces | representation and compression of surfaces | Numerical solution of PDEs | Numerical solution of PDEs | Galerkin approximation | Galerkin approximation | wavelet integrals (projection coefficients | moments and connection coefficients) | wavelet integrals (projection coefficients | moments and connection coefficients) | convergence | convergence | Subdivision wavelets for integral equations | Subdivision wavelets for integral equations | Compression and convergence estimates | Compression and convergence estimates | M-band wavelets | M-band wavelets | DFT filter banks and cosine modulated filter banks | DFT filter banks and cosine modulated filter banks | Multiwavelets | Multiwavelets

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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16.901 Computational Methods in Aerospace Engineering (MIT) 16.901 Computational Methods in Aerospace Engineering (MIT)

Description

This course serves as an introduction to computational techniques arising in aerospace engineering. Applications are drawn from aerospace structures, aerodynamics, dynamics and control, and aerospace systems. Techniques include: numerical integration of systems of ordinary differential equations; finite-difference, finite-volume, and finite-element discretization of partial differential equations; numerical linear algebra; eigenvalue problems; and optimization with constraints. This course serves as an introduction to computational techniques arising in aerospace engineering. Applications are drawn from aerospace structures, aerodynamics, dynamics and control, and aerospace systems. Techniques include: numerical integration of systems of ordinary differential equations; finite-difference, finite-volume, and finite-element discretization of partial differential equations; numerical linear algebra; eigenvalue problems; and optimization with constraints.

Subjects

numerical integration | numerical integration | ODEs | ODEs | ordinary differential equations | ordinary differential equations | finite difference | finite difference | finite volume | finite volume | finite element | finite element | discretization | discretization | PDEs | PDEs | partial differential equations | partial differential equations | numerical linear algebra | numerical linear algebra | probabilistic methods | probabilistic methods | optimization | optimization | omputational methods | omputational methods | aerospace engineering | aerospace engineering | computational methods | computational methods

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm

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18.303 Linear Partial Differential Equations: Analysis and Numerics (MIT) 18.303 Linear Partial Differential Equations: Analysis and Numerics (MIT)

Description

This course provides students with the basic analytical and computational tools of linear partial differential equations (PDEs) for practical applications in science engineering, including heat / diffusion, wave, and Poisson equations. Analytics emphasize the viewpoint of linear algebra and the analogy with finite matrix problems. Numerics focus on finite-difference and finite-element techniques to reduce PDEs to matrix problems. The Julia Language (a free, open-source environment) is introduced and used in homework for simple examples. This course provides students with the basic analytical and computational tools of linear partial differential equations (PDEs) for practical applications in science engineering, including heat / diffusion, wave, and Poisson equations. Analytics emphasize the viewpoint of linear algebra and the analogy with finite matrix problems. Numerics focus on finite-difference and finite-element techniques to reduce PDEs to matrix problems. The Julia Language (a free, open-source environment) is introduced and used in homework for simple examples.

Subjects

diffusion | diffusion | Laplace equations | Laplace equations | Poisson | Poisson | wave equations | wave equations | separation of variables | separation of variables | Fourier series | Fourier series | Fourier transforms | Fourier transforms | eigenvalue problems | eigenvalue problems | Green's function | Green's function | Heat Equation | Heat Equation | Sturm-Liouville Eigenvalue problems | Sturm-Liouville Eigenvalue problems | quasilinear PDEs | quasilinear PDEs | Bessel functionsORDS | Bessel functionsORDS

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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18.303 Linear Partial Differential Equations: Analysis and Numerics (MIT) 18.303 Linear Partial Differential Equations: Analysis and Numerics (MIT)

Description

This course provides students with the basic analytical and computational tools of linear partial differential equations (PDEs) for practical applications in science engineering, including heat/diffusion, wave, and Poisson equations. Analytics emphasize the viewpoint of linear algebra and the analogy with finite matrix problems. Numerics focus on finite-difference and finite-element techniques to reduce PDEs to matrix problems. This course provides students with the basic analytical and computational tools of linear partial differential equations (PDEs) for practical applications in science engineering, including heat/diffusion, wave, and Poisson equations. Analytics emphasize the viewpoint of linear algebra and the analogy with finite matrix problems. Numerics focus on finite-difference and finite-element techniques to reduce PDEs to matrix problems.

Subjects

diffusion | diffusion | Laplace equations | Laplace equations | Poisson | Poisson | wave equations | wave equations | separation of variables | separation of variables | Fourier series | Fourier series | Fourier transforms | Fourier transforms | eigenvalue problems | eigenvalue problems | Green's function | Green's function | Heat Equation | Heat Equation | Sturm-Liouville Eigenvalue problems | Sturm-Liouville Eigenvalue problems | quasilinear PDEs | quasilinear PDEs | Bessel functionsORDS | Bessel functionsORDS

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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16.901 Computational Methods in Aerospace Engineering (MIT) 16.901 Computational Methods in Aerospace Engineering (MIT)

Description

This course serves as an introduction to computational techniques arising in aerospace engineering. Applications are drawn from aerospace structures, aerodynamics, dynamics and control, and aerospace systems. Techniques include: numerical integration of systems of ordinary differential equations; finite-difference, finite-volume, and finite-element discretization of partial differential equations; numerical linear algebra; eigenvalue problems; and optimization with constraints. This course serves as an introduction to computational techniques arising in aerospace engineering. Applications are drawn from aerospace structures, aerodynamics, dynamics and control, and aerospace systems. Techniques include: numerical integration of systems of ordinary differential equations; finite-difference, finite-volume, and finite-element discretization of partial differential equations; numerical linear algebra; eigenvalue problems; and optimization with constraints.

Subjects

numerical integration | numerical integration | ODEs | ODEs | ordinary differential equations | ordinary differential equations | finite difference | finite difference | finite volume | finite volume | finite element | finite element | discretization | discretization | PDEs | PDEs | partial differential equations | partial differential equations | numerical linear algebra | numerical linear algebra | probabilistic methods | probabilistic methods | optimization | optimization | omputational methods | omputational methods | aerospace engineering | aerospace engineering | computational methods | computational methods

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm

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18.327 Wavelets, Filter Banks and Applications (MIT)

Description

Wavelets are localized basis functions, good for representing short-time events. The coefficients at each scale are filtered and subsampled to give coefficients at the next scale. This is Mallat's pyramid algorithm for multiresolution, connecting wavelets to filter banks. Wavelets and multiscale algorithms for compression and signal/image processing are developed. Subject is project-based for engineering and scientific applications.

Subjects

Discrete-time filters | convolution | Fourier transform | owpass and highpass filters | Sampling rate change operations | upsampling and downsampling | ractional sampling | interpolation | Filter Banks | time domain (Haar example) and frequency domain | conditions for alias cancellation and no distortion | perfect reconstruction | halfband filters and possible factorizations | Modulation and polyphase representations | Noble identities | block Toeplitz matrices and block z-transforms | polyphase examples | Matlab wavelet toolbox | Orthogonal filter banks | paraunitary matrices | orthogonality condition (Condition O) in the time domain | modulation domain and polyphase domain | Maxflat filters | Daubechies and Meyer formulas | Spectral factorization | Multiresolution Analysis (MRA) | requirements for MRA | nested spaces and complementary spaces; scaling functions and wavelets | Refinement equation | iterative and recursive solution techniques | infinite product formula | filter bank approach for computing scaling functions and wavelets | Orthogonal wavelet bases | connection to orthogonal filters | orthogonality in the frequency domain | Biorthogonal wavelet bases | Mallat pyramid algorithm | Accuracy of wavelet approximations (Condition A) | vanishing moments | polynomial cancellation in filter banks | Smoothness of wavelet bases | convergence of the cascade algorithm (Condition E) | splines | Bases vs. frames | Signal and image processing | finite length signals | boundary filters and boundary wavelets | wavelet compression algorithms | Lifting | ladder structure for filter banks | factorization of polyphase matrix into lifting steps | lifting form of refinement equationSec | Wavelets and subdivision | nonuniform grids | multiresolution for triangular meshes | representation and compression of surfaces | Numerical solution of PDEs | Galerkin approximation | wavelet integrals (projection coefficients | moments and connection coefficients) | convergence | Subdivision wavelets for integral equations | Compression and convergence estimates | M-band wavelets | DFT filter banks and cosine modulated filter banks | Multiwavelets

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm

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18.303 Linear Partial Differential Equations: Analysis and Numerics (MIT)

Description

This course provides students with the basic analytical and computational tools of linear partial differential equations (PDEs) for practical applications in science engineering, including heat/diffusion, wave, and Poisson equations. Analytics emphasize the viewpoint of linear algebra and the analogy with finite matrix problems. Numerics focus on finite-difference and finite-element techniques to reduce PDEs to matrix problems.

Subjects

diffusion | Laplace equations | Poisson | wave equations | separation of variables | Fourier series | Fourier transforms | eigenvalue problems | Green's function | Heat Equation | Sturm-Liouville Eigenvalue problems | quasilinear PDEs | Bessel functionsORDS

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm

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18.303 Linear Partial Differential Equations (MIT)

Description

This course covers the classical partial differential equations of applied mathematics: diffusion, Laplace/Poisson, and wave equations. It also includes methods of solution, such as separation of variables, Fourier series and transforms, eigenvalue problems. Green's function methods are emphasized.Technical RequirementsSpecial software is required to use some of the files in this course: .m.

Subjects

diffusion | Laplace equations | Poisson | wave equations | separation of variables | Fourier series | Fourier transforms | eigenvalue problems | Green's function | Heat Equation | Sturm-Liouville Eigenvalue problems | quasilinear PDEs | Bessel functions

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm

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18.303 Linear Partial Differential Equations (MIT)

Description

This course covers the classical partial differential equations of applied mathematics: diffusion, Laplace/Poisson, and wave equations. It also includes methods of solution, such as separation of variables, Fourier series and transforms, eigenvalue problems. Green's function methods are emphasized.Technical RequirementsMATLAB® software is required to run the .m files found on this course site.

Subjects

diffusion | Laplace equations | Poisson | wave equations | separation of variables | Fourier series | Fourier transforms | eigenvalue problems | Green's function | Heat Equation | Sturm-Liouville Eigenvalue problems | quasilinear PDEs | Bessel functions

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm

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16.901 Computational Methods in Aerospace Engineering (MIT)

Description

This course serves as an introduction to computational techniques arising in aerospace engineering. Applications are drawn from aerospace structures, aerodynamics, dynamics and control, and aerospace systems. Techniques include: numerical integration of systems of ordinary differential equations; finite-difference, finite-volume, and finite-element discretization of partial differential equations; numerical linear algebra; eigenvalue problems; and optimization with constraints.Technical RequirementsMATLAB® software is required to run the .m and .mat files found on this course site.MATLAB® is a trademark of The MathWorks, Inc.

Subjects

numerical integration | ODEs | ordinary differential equations | finite difference | finite volume | finite element | discretization | PDEs | partial differential equations | numerical linear algebra | probabilistic methods | optimization | omputational methods | aerospace engineering | computational methods

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm

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18.303 Linear Partial Differential Equations: Analysis and Numerics (MIT)

Description

This course provides students with the basic analytical and computational tools of linear partial differential equations (PDEs) for practical applications in science engineering, including heat / diffusion, wave, and Poisson equations. Analytics emphasize the viewpoint of linear algebra and the analogy with finite matrix problems. Numerics focus on finite-difference and finite-element techniques to reduce PDEs to matrix problems. The Julia Language (a free, open-source environment) is introduced and used in homework for simple examples.

Subjects

diffusion | Laplace equations | Poisson | wave equations | separation of variables | Fourier series | Fourier transforms | eigenvalue problems | Green's function | Heat Equation | Sturm-Liouville Eigenvalue problems | quasilinear PDEs | Bessel functionsORDS

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm

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18.303 Linear Partial Differential Equations (MIT)

Description

This course covers the classical partial differential equations of applied mathematics: diffusion, Laplace/Poisson, and wave equations. It also includes methods and tools for solving these PDEs, such as separation of variables, Fourier series and transforms, eigenvalue problems, and Green's functions.

Subjects

diffusion | Laplace equations | Poisson | wave equations | separation of variables | Fourier series | Fourier transforms | eigenvalue problems | Green's function | Heat Equation | Sturm-Liouville Eigenvalue problems | quasilinear PDEs | Bessel functions

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm

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18.327 Wavelets, Filter Banks and Applications (MIT)

Description

Wavelets are localized basis functions, good for representing short-time events. The coefficients at each scale are filtered and subsampled to give coefficients at the next scale. This is Mallat's pyramid algorithm for multiresolution, connecting wavelets to filter banks. Wavelets and multiscale algorithms for compression and signal/image processing are developed. Subject is project-based for engineering and scientific applications.

Subjects

Discrete-time filters | convolution | Fourier transform | owpass and highpass filters | Sampling rate change operations | upsampling and downsampling | ractional sampling | interpolation | Filter Banks | time domain (Haar example) and frequency domain | conditions for alias cancellation and no distortion | perfect reconstruction | halfband filters and possible factorizations | Modulation and polyphase representations | Noble identities | block Toeplitz matrices and block z-transforms | polyphase examples | Matlab wavelet toolbox | Orthogonal filter banks | paraunitary matrices | orthogonality condition (Condition O) in the time domain | modulation domain and polyphase domain | Maxflat filters | Daubechies and Meyer formulas | Spectral factorization | Multiresolution Analysis (MRA) | requirements for MRA | nested spaces and complementary spaces; scaling functions and wavelets | Refinement equation | iterative and recursive solution techniques | infinite product formula | filter bank approach for computing scaling functions and wavelets | Orthogonal wavelet bases | connection to orthogonal filters | orthogonality in the frequency domain | Biorthogonal wavelet bases | Mallat pyramid algorithm | Accuracy of wavelet approximations (Condition A) | vanishing moments | polynomial cancellation in filter banks | Smoothness of wavelet bases | convergence of the cascade algorithm (Condition E) | splines | Bases vs. frames | Signal and image processing | finite length signals | boundary filters and boundary wavelets | wavelet compression algorithms | Lifting | ladder structure for filter banks | factorization of polyphase matrix into lifting steps | lifting form of refinement equationSec | Wavelets and subdivision | nonuniform grids | multiresolution for triangular meshes | representation and compression of surfaces | Numerical solution of PDEs | Galerkin approximation | wavelet integrals (projection coefficients | moments and connection coefficients) | convergence | Subdivision wavelets for integral equations | Compression and convergence estimates | M-band wavelets | DFT filter banks and cosine modulated filter banks | Multiwavelets

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16.90 Computational Methods in Aerospace Engineering (MIT)

Description

This course provides an introduction to numerical methods and computational techniques arising in aerospace engineering. Applications are drawn from aerospace structures, aerodynamics, dynamics and control, and aerospace systems. Techniques covered include numerical integration of systems of ordinary differential equations; numerical discretization of partial differential equations; and probabilistic methods for quantifying the impact of variability. Specific emphasis is given to finite volume methods in fluid mechanics, and finite element methods in structural mechanics.Acknowledgement: Prof. David Darmofal taught this course in prior years, and created some of the materials found in this OCW site.

Subjects

numerical integration | ODEs | ordinary differential equations | finite difference | finite volume | finite element | discretization | PDEs | partial differential equations | numerical linear algebra | probabilistic methods | optimization | computational methods | aerospace engineering | Monte Carlo | Fourier stability analysis | Matrix stability analysis | Runge-Kutta | convergence | accuracy | stiffness | weighted residual | statistical sampling | sensitivity analysis

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16.901 Computational Methods in Aerospace Engineering (MIT)

Description

This course serves as an introduction to computational techniques arising in aerospace engineering. Applications are drawn from aerospace structures, aerodynamics, dynamics and control, and aerospace systems. Techniques include: numerical integration of systems of ordinary differential equations; finite-difference, finite-volume, and finite-element discretization of partial differential equations; numerical linear algebra; eigenvalue problems; and optimization with constraints.

Subjects

numerical integration | ODEs | ordinary differential equations | finite difference | finite volume | finite element | discretization | PDEs | partial differential equations | numerical linear algebra | probabilistic methods | optimization | omputational methods | aerospace engineering | computational methods

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm

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