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List of textbooks in physics: Category:Physics textbooks; List of textbooks on classical mechanics and quantum mechanics; List of textbooks in electromagnetism; List of textbooks on relativity; List of textbooks in thermodynamics and statistical mechanics
General linear methods — a class of methods encapsulating linear multistep and Runge-Kutta methods; Bulirsch–Stoer algorithm — combines the midpoint method with Richardson extrapolation to attain arbitrary order; Exponential integrator — based on splitting ODE in a linear part, which is solved exactly, and a nonlinear part
Here we presume an understanding of basic multivariate calculus and Fourier series.If (,) is a known, complex-valued function of two real variables, and g is periodic in x and y (that is, (,) = (+,) = (, +)) then we are interested in finding a function f(x,y) so that
in which case the equation to be solved becomes the Poisson equation. A general method of solving such equations is by means of Green's functions , namely, rather than a direct attack, one first finds the function K = K ( x , y ) {\displaystyle K=K(x,y)} such that for a given pair x,y ,
In mathematics, a collocation method is a method for the numerical solution of ordinary differential equations, partial differential equations and integral equations.The idea is to choose a finite-dimensional space of candidate solutions (usually polynomials up to a certain degree) and a number of points in the domain (called collocation points), and to select that solution which satisfies the ...
The problem of two fixed centers conserves energy; in other words, the total energy is a constant of motion.The potential energy is given by =where represents the particle's position, and and are the distances between the particle and the centers of force; and are constants that measure the strength of the first and second forces, respectively.
For many problems in applied linear algebra, it is useful to adopt the perspective of a matrix as being a concatenation of column vectors. For example, when solving the linear system =, rather than understanding x as the product of with b, it is helpful to think of x as the vector of coefficients in the linear expansion of b in the basis formed by the columns of A.
Algebra is the art of manipulating equations and formulas. Diophantus (3rd century) and al-Khwarizmi (9th century) were the two main precursors of algebra. [27] [28] Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. [29]