wath is the mathematical definition of modes for a system of partial differential equations?
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Typically, the modes are the solutions of the system.
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i just learned this in class... \nafter you enter your mass matix M,\nyour Dampening Matrix C\nand your Stiffness Matrix K\nyou take [K] - lamda [M]= (b) \nyou can type \n[Evals]=eigs(new)\nThese will also be the square of your natural frequencies.\nyour mode shapes are the eigvectors! fyi
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Most introductory books on differential geometry discuss this. Looks at 'From Calculus to Cohomology' and 'Introduction to Smooth Manifolds' as references.
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Spectral methods are algorithms to solve certain kinds of partial differential equations numerically using some sort of Fast Fourier Transform.\nDetails with example can be found at\nhttp://en.wikipedia.org/wiki/Spectral_method
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The goal of modal analysis in structural mechanics is to determine the natural shapes and frequencies of an object or structure during free vibration. The types of equations which arise from modal analysis are those seen in eigensystems. The physical interpretation of the eigenvalues and eigenvectors resulting from solving equations of these types of system are that they represent the natural or eigen frequencies and their corresponding mode shapes.\n\nPS - in quantum mechanics, the eigenvalues corresponds to the natural energy states of the quantum system. The mathematics involved in eigenvalue analysis and quantum mechanics is nearly identical, and the physical interprestation is very similar.
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the best is you get a good book about initial value problems... go und get one at your closest library...\n1.- A First Course in Partial Differential Equations With Complex Variables and Transform Methods by Hans F Weinberger\n2.- A First Course In Differential Equations With Modeling Applications by Dennis G Zill\n3.- Calculus: Concepts and Contexts (with Tools for Enriching Calculus, Interactive Video Skillbuild... by James Stewart\n----------------------\nmany sites haves definitions about the topic:\nhttp://mathworld.wolfram.com/InitialValueProblem.html\nhttp://planetmath.org/encyclopedia/InitialValueProblem.html\nhttp://laps.fsl.noaa.gov/~jedwards/thesis/sectionA_2.html
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(v - v0) / (t - t0)\n\nv - final velocity\nv0 - initial velocity\n\nt - final time\nt0 - initial time
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Actually, you get X''/X=C and Y''/Y=-C where C does not have to be 0. Also, remember that sums of solutions are solutions, so the solutions you get from separation of variables only give a basis for the vector space of solutions, not all the solutions. The specifics are difficult in this forum, but you have to set up the series and use the boundary conditions to find the coefficients of the series.
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an energy oscillation
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CONFORMAL MAPPING\nIn a limited group of problems one can use a short cut to the solution of the\nLaplace's equation, conformal mapping. The problem has to be 2D (two\ndimensional), i.e., the boundaries, boundary conditions and sources contain\nonly two variables, x and y say.\nSome examples where this method can be used is: the electrostatic potential;\nheat conduction; flow of fluids.\n\nThe conformal mapping relies on the properties of analytic functions. \n\nAny function that has continuous partial derivatives of the second order and\nthat satisfies the Laplace's equation is called a Harmonic function.\nThus both the real part, u, and imaginary part, v, of f are harmonic\nfunctions.\nIf the function f = u+iv is analytic, u and v are conjugate harmonic\nfunctions. Given one of two harmonic functions, the Cauchy-Riemann\nequations can be used to find the other.\nThat u and v are harmonic functions can be used to find the potential. Both\nu and v are candidates for the potential since they satisfy Laplace's\nequation. Alternatively the function f can be viewed as a change of\nvariables, a transformation from the complex z-plane to the complex w-\nplane.\nTransformation of boundary conditions.\nTwo types of boundary conditions remain unchanged by the transformation\nand problems with these boundary conditions are best suited for conformal\nmapping. The types are:\n1) The value of a harmonic function is constant on the boundary.\n(one type of Dirichlet problem)\n2) The normal derivative of a harmonic function is zero on the boundary.\n(one type of Neumann problem)\n(The conditions are that f and its inverse are analytical)\nThe capacitance between two conductors\nIf Q is the electrical charge on either conductor and D V is the\ndifference in potential between one conductor and the other, then the\ncapacitance C is defined to be c=q/v
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http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
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go to www.google.com and type in \n\nriemann spaces  3 4 dimensions\n\nit will bring up alot of information for you to look at
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I'm a third year physics student and I know what a Laplacian differential operator is, I just don't understand your question. Could you clarify?
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the mode of a set of numbers is the number that occurs most often.  for example, in the set 2, 4, 5,8, 8, 9, the mode is 8, as it occurs twice and the others only once.  another example: 7, 9, 9, 2, 5, 5, what is the mode?  there are actually two, 5 and 9, because the both occur the most.  if all numbers occur the same number of times, then there is no mode.\ncheck your math book and ask your teacher for more explanation.\nall you have to do is count the numbers, some people find that it helps to put them in order, so that they can be sure not to miss any. (instead of 7, 9, 9..., rewrite as 2, 5, 5, 7, 9, 9, this is also helpful when doing mean and median)\ngood luck on your work!  i like that you didn't include the actual problem, that shows that you really want to learn, not just take the easy way out.
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Define all the known and unknown quantities.\nExpress the relationships between these quantities in equations.\nResolve the equations.
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Differential equations.\nLinear algebra.\nNumerical analysis.
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You can refer this book:\n\nDifferential equations : linear, nonlinear, ordinary, parial - A.C. King, J. Billingham, and S.R. Otto.