Scilab Programs For Undergraduate Students (Mathematical Physics)

Differential Equation Second Orde ( Runge Kutta Method ) CONCEPT  (a)(Differential Equation - Second Order) In many problems, the direct functional relation between the dependent variable y and the independent variable x is not known. However, the second rate of change in y with respect to x is known and is given by a function f(x, y). The idea is to deduce the function y(x) from the rate equation                                                           d^2y/dx^2 = f(x, y(x))   (b)(Initial Value Problem) Any DE represent a family of curves/surfaces, however most problems pertains to picking out a particular curve which satisfies the given conditions. In nutshell, we seek a solution y(x) which satisfies the initial conditions                        ...

      Differential Equation First Order ( Euler Method ) CONCEPT:- (a)(Differential Equation - First Order) In many problems, the direct functional relation between the dependent variable y and the independent variable x is not known. However, the rate of change in y with respect to x is known and is given by a function f(x, y). The idea is to deduce the function y(x) from the rate equation                                                                                           dy/dx = f(x, y(x))  (b)(Initial Value Problem) Any DE represent a family of curves/surfaces, however most problems pertains to picking out a particular curve which satisfies the given conditions. In nutshell, we seek a solution y(x) which satisfies t...

User Defined Function Legendre Function ( Series Expansion ) CONCEPT    (a)(Legendre Differential Equation)                                      The Legendre differential equation is                                   (1 − x^2 )y"− 2xy' + n(n + 1)y = 0      where n is a +ve integer.  (b)(Legendre Function)                                    The series solution Pn(x) to this equation                           Pn(x) = X(m,i=0)(((−1)^i*(2n − 2i)!)/( 2^n *i!* (n − i)!*(n − 2i)!))*x^(n−2i)           is known as the Legendre function. Here m = n/2 if n is even and m = (n...

User Defined Function Bessel Function ( Series Expansion ) The Bessel functions have been known since the 18th century when mathematicians and scientists started to describe physical processes through differential equations. Many different looking processes satisfy the same partial differential equations. These equations were named Laplace, d‘Alembert (wave), Poisson, Helmholtz, and heat (diffusion) equations. Different methods were used to investigate these equations. The most powerful was the separation of variables method, which in polar coordinates often leads to ordinary differential equations of special structure:                                 x^2*y" + xy' + (x^2 − n^2 )y = 0                             where n is an integer or a half-integer.                   ...

Simultaneous Equations Matrix Method ( Gauss-Jordan Gauss Siedel Method ) CONCEPT :-          (a) (Gauss Elimination)  Briefly explain the theory of Gauss Elimination method for solving simultaneous equations.  (b)(Gauss Seidel)  Briefly explain the theory of Gauss Seidel method for solving simultaneous equations. for theory click below;- Theory of Gauss Seidal Method METHOD /CODE :- (a) (Gauss Elimination)  Taking aii as the pivot element corresponding to i th row, we reduce all the rows Rk below the i th row by applying the row operation                                                         Rk → Rk − (aik)/( aii) *Ri  It is to be noted that pivoting may be required in cases where some of the diagonal elements may assume zero value.  (b) (Back...

Interpolation of Data, Polynomial Functions ( Lagrange Interpolation ) CONCEPT:-  (a)  (Function Fit) Suppose the given data set contains n + 1 elements (xi , yi). The idea is to look for a linear combination of exactly n + 1 set of functions fk(x) = x k for all k = 0 to n that can best describe the given data. In nutshell, for all i = [0, n]                                               yi = X(n, k=0 )αk(xi)^k   (where X is sigma and limits k=0 to n) where coefficients αk decides the unique linear combination is required.  (b)  (Interpolation)  Now for any arbitrary x, the estimate y can be obtained by the relation                                                        ...

             Scilab Program For Quadratic Regression  CONCEPT :-  (a)  (Quadratic Regression) The theoretical model proposes a quadratic relationship the dependent y and the independent x variable                                                           y(x) = a0 + a1x + a2x 2 where a0, a1 and a2 are constants (b) (Goodness of Fit)                               Goodness of Fit g of the above line to a given data set of n number of data points and the standard deviation σ are given by                                                        g = X(vi)^2 = X...

        Experiment No.1 - Linear Data Fitting using Method of  Least Squares Using scilab CONCEPT - (a)  (Linear Regression) The theoretical model proposes a linear relationship between the dependent y and the independent x variable-                                                                    y = a0 + a1x where a0 is the intercept and a1 is the slope.   (b)  (Goodness of Fit) Goodness of Fit (g) is indicated by the value of the sum of squares of the deviation (νi) of the dependent variable i.e., difference between the experimental value yi and the expected value y(xi)                                                 ...