Numerical Methods

Solution of Partial Differential, Elliptical, Parabolic, Hyperbolic Equations and their relevant examples

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Many physical phenomena in applied science and engineering when formulated intomathematical models fall into a category of system known as partial differential equations. Apartial differential equation is a differential equation involving more than one in independentvariables.We can write a second order equation involving two independent variables in general form as : Where a,b,c may be … Read more

Solution of Ordinary Differential and Higher Order Equations: Overview of initial and boundary value problems & Taylor series, Euler, Huen’s and Runge-Kutta methods

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Differential equations Many of the laws in physics, chemistry, engineering, economics are based on empiricalobservations that describe changes in the state of the system. Mathematical models thatdescribe the state of such system are often expressed in terms of not only certain systemparameters but also their derivatives, such mathematical model which uses differentialcalculus to express relationship … Read more

Solution of Linear Algebraic Equations: Matrices and their properties, Elimination and Gauss Jordan methods, Method of factorization, power, iterative

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Matrices and properties Importance of linear equations First mathematical models of many of the real world problems are either linear or can beapproximated reasonably well using linear relationships. Analysis of linear relationship of variablesis generally easier than that of non-linear relationships.A linear equation involving two variables x and y has the standard form π‘Žπ‘₯ + … Read more

Numerical Differentiation and Integration

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Introduction Let us consider a set of values (π‘₯𝑖, 𝑦𝑖) of a function. The process of computing the derivative orderivatives of that function at some values of x from the given set of values is called NumericalDifferentiation. This may be done by first approximating the function by suitable interpolationformula and then differentiating. Derivatives using Newton’s … Read more

Interpolation and Approximation: Numerical Differentiation and Regression

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INTERPOLATION Introduction The process of estimating intermediate values between given data points is called Interpolation . The most common method used for this purpose is Polynomial Interpolation. Find the functional value for π‘₯ = 2.5 , i.e. 𝑓 2.5 = ? Is this a problem of Interpolation or extrapolation?? π‘₯ 1 3 4 6 9 … Read more

Solution of Non- Linear Equation

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Bisection Method: This method is based on the repeated application of intermediate value property. Let, F(x) be continuous between a and b. Consider, F(a) be negative and F(b) be positive. Then, the first approximation of the root is x1= (a+b)/2 , if F(x1)= 0 then x1 is root of F(x)=0. Otherwise, the root lies between … Read more