Math-4

Partial Differential Equations (PDEs) – Complete Formulas and Concepts with Clear Explanation

1. Origin of Partial Differential Equations PDEs arise when a physical phenomenon depends on more than one independent variable (e.g., time t …

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Origin of PDEs + First-Order Linear & Non-Linear PDEs with Examples

Origin of Partial Differential Equations (PDEs) + First-Order Linear & Non-Linear PDEs 1. Origin of Partial Differential Equations Partial Differential Equations arise …

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Applications-of-Partial-Differential-Equations

1. Classification of Second-Order Linear PDEs (in two independent variables) General form: \[ a(x,y) u_{xx} + 2h(x,y) u_{xy} + b(x,y) u_{yy} + …

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Detailed Solved Example – Laplace Equation in Two Dimensions

Detailed Solved Example for Laplace Equation in Two Dimensions (with Full Steps) Problem (Standard University Level) Solve the Laplace equation \[ \frac{\partial^2 …

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Module III: Statistical Techniques I – Complete Formulas with Clear Explanation

1. Measures of Central Tendency Measure Formula (Raw Data) Formula (Grouped Data) Remarks Arithmetic Mean (\(\bar{x}\)) \(\bar{x} = \frac{\Sigma x_i}{n}\) \(\bar{x} = …

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Module IV: Statistical Techniques II – Complete Formulas with Clear Explanation

1. Basic Probability Concepts Concept Formula / Definition Remarks Probability of an event A P(A) = Number of favourable outcomes / Total …

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Module V: Statistical Techniques III – Complete Formulas with Clear Explanation

1. Sampling Theory Type Condition Standard Error (SE) Formulas Large Sample (n ≥ 30 or np, nq > 5) Population variance known/unknown …

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