Partial Differential Equations

Partial Differential Equations

• A partial differential equation (PDE) contains unknown multivariable functions and their partial derivatives.

• General form: $F(x, y, u, u_x, u_y, u_{xx}, u_{yy}, u_{xy},...) = 0$, where $x, y$ are independent variables, $u = u(x, y)$ is a dependent variable, $u_x, u_y$ are first order partial derivatives, $u_{xx}, u_{yy}, u_{xy}$ are second order partial derivatives, and $F$ is a function.

• Examples:

• Advection Equation: $\frac{\partial u}{\partial t} + c \frac{\partial u}{\partial x} = 0$

• Heat Equation: $\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$

• Wave Equation: $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$

• Laplace Equation: $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$

• Poisson Equation: $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = f(x, y)$

• The order of a PDE is determined by the highest order derivative in the equation.

• Examples:

• $\frac{\partial u}{\partial t} + c \frac{\partial u}{\partial x} = 0$ (First Order)

• $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$ (Second Order)

• A PDE is linear if the dependent variable and its derivatives appear linearly; otherwise, it is nonlinear.

• Examples:

• Linear PDE: $\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$

• Nonlinear PDE: $\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = 0$ (Burger's Equation)

• A solution to a PDE is a function that satisfies the equation, and can be general or particular.

• To obtain a unique solution, boundary conditions (BCs) and initial conditions (ICs) are often required.

• Types of Boundary Conditions:

• Dirichlet BC: Specifies the value of the solution; $u(x, t) = f(t)$ on boundary

• Neumann BC: Specifies the value of the normal derivative; $\frac{\partial u}{\partial x}(x, t) = g(t)$ on boundary

• Robin BC: A combination of Dirichlet and Neumann conditions; $\alpha u(x, t) + \beta \frac{\partial u}{\partial x}(x, t) = h(t)$ on boundary

• Methods for Solving PDEs:

• Analytical Methods: Separation of variables, method of characteristics, etc.

• Numerical Methods: Finite difference method, finite element method, finite volume method, etc.

• A general second-order linear PDE in two independent variables: $Au_{xx} + Bu_{xy} + Cu_{yy} + Du_x + Eu_y + Fu = G$, where $A, B, C, D, E, F, G$ are functions of $x$ and $y$.

• The discriminant $\Delta = B^2 - 4AC$ is for classification.

• PDE Classification based on Discriminant:

• Hyperbolic: $\Delta > 0$ (e.g., Wave Equation)

• Parabolic: $\Delta = 0$ (e.g., Heat Equation)

• Elliptic: $\Delta < 0$ (e.g., Laplace Equation)

• Applications of PDEs include: heat transfer, fluid dynamics, electromagnetism, quantum mechanics, wave propogation and financial modeling.

Numerische Integration

• The objective is to calculate definite integrals: $I[f] = \int_a^b f(x) dx$
• Achieved through quadrature formulas: $Q[f] = \sum_{i=0}^n w_i f(x_i)$
• $x_i$: Nodes
• $w_i$: Weights
• Newton-Cotes Formulas: $f$ is replaced by an interpolation polynomial $p_n$ and integrated
• $I[f] = \int_a^b f(x) dx \approx \int_a^b p_n(x) dx =: Q[f]$
• Derivation:
• $x_0, \dots, x_n \in [a, b]$ are pairwise distinct, and $p_n$ is the interpolation polynomial for $f$, i.e., $p_n(x) = \sum_{i=0}^n f(x_i) L_i(x)$
• $L_i$ are the Lagrange fundamental polynomials: $L_i(x) = \prod_{\substack{k=0 \ k \neq i}}^n \frac{x - x_k}{x_i - x_k}$
• $Q[f] = \int_a^b p_n(x) dx = \int_a^b \sum_{i=0}^n f(x_i) L_i(x) dx = \sum_{i=0}^n \underbrace{\int_a^b L_i(x) dx}_{=: w_i} f(x_i)$
• The weights, $w_i = \int_a^b L_i(x) dx$, do not depend on $f$!
• Knots, $x_i$, are fixed.
• For the Newton-Cotes formulas, the nodes $x_i$ are equidistant: $x_i = a + i \cdot h, \quad i = 0, \dots, n \quad \text{with} \quad h = \frac{b-a}{n}$.
• Examples of Newton-Cotes Formulas:
• $n = 0: \quad x_0 = a + \frac{h}{2} = \frac{a + b}{2}$ (Midpoint Rule)
• $Q[f] = (b - a) f(\frac{a + b}{2})$
• $n = 1: \quad x_0 = a, \quad x_1 = b$ (Trapezoidal Rule)
• $Q[f] = \frac{b - a}{2} (f(a) + f(b))$
• $n = 2: \quad x_0 = a, \quad x_1 = \frac{a + b}{2}, \quad x_2 = b$ (Simpson's Rule)
• $Q[f] = \frac{b - a}{6} (f(a) + 4f(\frac{a + b}{2}) + f(b))$

Algèbre Linéaire

• An space vector on a field is a set E with two operations that must satisfy eight axioms.
• Addition Vectorielle
• $E \times E \rightarrow E$, notée $(u, v) \mapsto u + v$
• Multiplication scalaire
• $\mathbb{K} \times E \rightarrow E$, notée $(\lambda, u) \mapsto \lambda u$
• Associativity of Addition: $(u + v) + w = u + (v + w)$ for all $u, v, w \in E$
• Commutativity of Addition: $u + v = v + u$ for all $u, v \in E$
• Existence of Neutral Element: There is a 0 such that $u + 0 = u$ for all u in E.
• Existence of Additive Inverse: For all $u \in E$, there is $-u \in E$ such that $u + (-u) = 0$.
• Compatibility of Scalar Multiplication: $\lambda(\mu u) = (\lambda \mu)u$ for all scalars $\lambda, \mu$ and vectors $u$.
• Distributivity over Vector Addition: $\lambda(u + v) = \lambda u + \lambda v$ for all scalars $\lambda$ and vectors $u, v$.
• Distributivity over Scalar Addition: $(\lambda + \mu)u = \lambda u + \mu u$ for all scalars $\lambda, \mu$ and vectors $u$.
• Identity for Scalar Multiplication: $1u = u$ for all vectors $u$.
• Examples include:
• $\mathbb{R}^n$ is a vector space on $\mathbb{R}$.
• $\mathbb{C}^n$ is a vector space on $\mathbb{C}$ and $\mathbb{R}$.
• The set of continuous fonctions de $\mathbb{R}$ in $\mathbb{R}$, noté $C(\mathbb{R}, \mathbb{R})$
• The set of ponynomials, noté $\mathbb{R}[X]$, is a vector space on $\mathbb{R}$
• A set F of and vector space E is a subspace if: stable by addition, stable by scalar multiplication
• Combination Linéaires: $\lambda_1 u_1 + \lambda_2 u_2 +... + \lambda_n u_n$ where $\lambda_1, \lambda_2,..., \lambda_n$ are des scalaires
• Space Engendré: $Vect(S)$,is the set of all combinasion linéaires, un sous-space vectoriel of $E$
• Linear Independence: $\lambda_1 u_1 + \lambda_2 u_2 +... + \lambda_n u_n = 0$, implique $\lambda_1 = \lambda_2 =... = \lambda_n = 0$
• Base d'un espace vectoriel: linearly independent and générateur de $E$
• Dimension d'un espace vectoriel: If E admet une base finie, of dimension finie.

Corporate Social Responsibility

• The thesis examined CSR's impact on financial performance, focusing on customer satisfaction as a mediator.

• The study employed a quantitative approach, using data from various industries.

• Key findings:

• CSR initiatives positively and significantly affect financial performance.

• Customer satisfaction partially mediates the CSR-financial performance relationship.

• The impact of CSR is stronger in industries with high customer interaction.

• Theoretical Implications: The research adds to literature by providing evidence for customer satisfaction's role in CSR's impact on financial performance. It notes the importance of industry-specific contexts.

• Managerial Implications: Managers should consider CSR investments, customer satisfaction and tailor CSR strategies to their specific industry.

• Limitations: The study had a relatively small sample size, relied on secondary data, and focused on few CSR dimensions.

• Future Research: Larger and more diverse samples, primary data collection, examining CSR dimensions, moderating role exploration (firm size, ownership structure), and longitudinal data are needed for future investigations.

Linear Programming - Duality

• Primal problem:

    max c^T x
    s.t. Ax = 0

• Dual problem:

min b^T y
s.t. A^T y >= c
     y >= 0

• Weak duality: If $x \in \mathbb{R}^n$ and $y \in \mathbb{R}^m$ are feasible solutions to the primal and dual problems respectively, then $c^T x \le b^T y$.
• Strong duality: If the primal and dual problems both have feasible solutions, they both have optimal solutions with the same objective function value.
• Complementary slackness: For feasible solutions $x \in \mathbb{R}^n$ and $y \in \mathbb{R}^m$, $x$ and $y$ are optimal if and only if:
• $y_i (b_i - (Ax)_i) = 0$ for all $i = 1,..., m$ (primal complementary slackness).
• $x_j ((A^T y)_j - c_j) = 0$ for all $j = 1,..., n$ (dual complementary slackness).
• Example:
• Primal problem:

max 5x_1 + 6x_2 + 9x_3 + 8x_4
s.t. x_1 + 2x_2 + 3x_3 + x_4 = 5
     2y_1 + y_2 >= 6
     3y_1 + 2y_2 >= 9
     y_1 + 3y_3 >= 8
     y_1, y_2 >= 0

• Solution: $x^* = (2, 0, 1, 0)$, $y^* = (3, 2)$
• Verifying complementary slackness: Primal:
• $y_1 (5 - (x_1 + 2x_2 + 3x_3 + x_4)) = 3 \cdot (5 - (2 + 0 + 3 + 0)) = 0$
• $y_2 (3 - (x_1 + x_2 + 2x_3 + 3x_4)) = 2 \cdot (3 - (2 + 0 + 2 + 0)) = 0$ Dual:
• $x_1 ((y_1 + y_2) - 5) = 2 \cdot ((3 + 2) - 5) = 0$
• $x_2 ((2y_1 + y_2) - 6) = 0 \cdot ((6 + 2) - 6) = 0$
• $x_3 ((3y_1 + 2y_2) - 9) = 1 \cdot ((9 + 4) - 9) = 0$
• $x_4 ((y_1 + 3y_2) - 8) = 0 \cdot ((3 + 6) - 8) = 0$

Cardiovascular System

• Blood Vessels
• Arteries: Carry blood away from the heart, thick walls withstand high pressure, elastic fibers allow stretching, smooth muscle controls diameter.
• Veins: Carry blood to the heart, thinner walls, lower pressure, valves prevent backflow and skeletal muscle aids flow.
• Capillaries: Microscopic vessels, thin walls for exchange, connect arteries and veins, nutrient and waste transfer occurs here.
• Blood Flow
• Pulmonary Circuit: to and from lungs: Right ventricle $\rightarrow$ pulmonary artery $\rightarrow$ lungs $\rightarrow$ pulmonary vein $\rightarrow$ left atrium $O_2$ in, $CO_2$ out.
• Systemic Circuit: To and from body: Left ventricle $\rightarrow$ aorta $\rightarrow$ body $\rightarrow$ vena cava $\rightarrow$ right atrium: Nutrient delivery, waste removal.
• Heart Anatomy
• Chambers: Atria (right and left) receive blood, ventricles (right and left) pump blood.
• Valves: Atrioventricular (AV): Between atria and ventricles: Tricuspid (right), Bicuspid/Mitral (left), semilunar: Between ventricles and arteries: Pulmonary, Aortic.
• Vessels: Aorta (from left ventricle, to body), Vena Cava (to right atrium, from body: Superior - upper body, Inferior - lower body), Pulmonary Artery (from right ventricle, to lungs), Pulmonary Vein (to left atrium, from lungs).
• Cardiac Cycle
• Systole: contraction phase: ventricles pump blood
• Diastole: relaxation phase: ventricles fill with blood
• Regulation
• Heart Rate: Autonomic nervous system: Sympathetic (increases rate), Parasympathetic (decreases rate), Hormones (e.g., epinephrine), Exercise, stress, etc.
• Blood Pressure: Force of blood on vessel walls, systolic/diastolic (e.g., 120/80 mmHg), regulated by: heart rate, blood volume, vasoconstriction/vasodilation Diagram notes blood flow direction.

Erwartungswert und Varianz Zufälliger Variablen

• Zufällige Variablen are a tool for modelling experiments.
• Erwartungswert and Varianz help to characterize the distribution.
• Erwartungswert
• Describes the expected value of the variable $E(X)$.
• Discreet: $E(X)=\sum_{x} x \cdot p(x)$
• Fair dice: $E(X)=1 \cdot \frac{1}{6}+2 \cdot \frac{1}{6}+3 \cdot \frac{1}{6}+4 \cdot \frac{1}{6}+5 \cdot \frac{1}{6}+6 \cdot \frac{1}{6}=3.5$
• Continuous: $E(X)=\int_{-\infty}^{\infty} x \cdot f(x) d x$
• Varianz
• Varianz measures how spread out a variable is around its mean
• Defined: $Var(X)=E\left((X-\mu)^{2}\right)$
• Diskrete Varianz: $Var(X)=\sum_{x}(x-\mu)^{2} \cdot p(x)$
• Continuous Varianz: $Var(X)=\int_{-\infty}^{\infty}(x-\mu)^{2} \cdot f(x) d x$
• Alternate form: $Var(X)=E\left(X^{2}\right)-(E(X))^{2}$
• Standard Abweichung: $\sigma=\sqrt{Var(X)}$
• Rechenregeln for Erwartungswert and Varianz
• $E(aX+b)=aE(X)+b$
• $Var(aX+b)=a^{2}Var(X)$
• IF X and Y are independant variables.
• $E(X+Y)=E(X)+E(Y)$
• $Var(X+Y)=Var(X)+Var(Y)$
• Bernoulli-verteilte $E(X)=p$ and $Var(X)=p(1-p)$
• $E(X)=\mu$ und die Varianz $Var(X)=\sigma^{2}$

Logique

• A proposition, or assertion, is a statement that can be either true or false.
• Examples: «Aujourd'hui, il pleut.» or «2 + 2 = 4.» or «$\forall x \in \mathbb{R}, x^2 \geq 0$.»
• Logical Operators: From one or two propositions, a new one using a logical operator is constructed. Some main operators:
• Négation
• Conjonction
• Disjonction
• Implication
• Équivalence.
• Négation: The negating of proposition, noted non $P$ (ou $\neg P$), est vraie si $P$ is fausse est fausse si $P$ este vraie.
• Conjonction: The conjonction of the proposition, notée $P$ et $Q$ (ou $P \wedge Q$), est vraie si $P$ et $Q$ sont vraies, et fausse dans les autres cas.
• Disjonction: The disjonction of the proposition , notée $P$ ou $Q$ (ou $P \vee Q$), est vraie si l'une au moins des deux propositions est vraie, et fausse si les deux propositions sont fausses.
• Implication: “$P$ implique $Q$”. $P$ est called l'hypothèse et $Q$ la conclusion. $P \Rightarrow Q$ est fausse uniquement lorsque $P$ est vraie et $Q$ est fausse
• $P \Rightarrow Q$ est équivalente à «non $P$ ou $Q$».
• The réciproque de $P \Rightarrow Q$ est $Q \Rightarrow P.
• The contraposée de $P \Rightarrow Q$ est «non $Q \Rightarrow$ non $P$».
• $P \Rightarrow Q$ est équivalente à sa contraposée
• ÉquivalenceL “$P$ est équivalente à $Q$” ou “$P$ si et seulement si $Q$” (souvent abrégé en “$P$ ssi $Q$”). $P \Leftrightarrow Q$ est vraie lorsque $P$ et $Q$ ont la même valeur de vérité.
• Quantificateurs :
• Quantificateur: noté $\forall$ means “pour tout”.
• Quantificateur: noté $\exists$ means “il existe”.
• The addition of vectors, with its graphical methods.

Física

Sum of Vectors

• Graphical Method

• Add two vectors by placing them one after the other, preserving their magnitude, direction, and sense.

• The resultant vector joins the origin of the first vector to the end of the last vector.

  $$\overrightarrow{R} = \overrightarrow{A} + \overrightarrow{B}$$

• Sum of More Than Two Vectors

• The procedure is similar to that for two vectors.

• Place the vectors one after the other, maintaining their characteristics.

• The resultant vector joins the origin of the first vector to the end of the last vector.

  $$\overrightarrow{R} = \overrightarrow{A} + \overrightarrow{B} + \overrightarrow{C}$$

• Analytical Method, showing components of a vector

• Any vector can be expressed as the sum of two perpendicular vectors called components.

  $$A_x = A \cdot \cos{\theta}$$ $$A_y = A \cdot \sin{\theta}$$

• Vector Sum by Components

• Decompose vectors into their components and add the components of each axis.

  $$\overrightarrow{R} = \overrightarrow{A} + \overrightarrow{B} + \overrightarrow{C} = (A_x + B_x + C_x) \hat{i} + (A_y + B_y + C_y) \hat{j}$$ $$R_x = A_x + B_x + C_x$$ $$R_y = A_y + B_y + C_y$$ $$R = \sqrt{R_x^2 + R_y^2}$$ $$\theta = \arctan{\frac{R_y}{R_x}}$$

Algèbre linéaire et géométrie II

• Spaces vectoriels on $\mathbb{K}$ is set $E$ nom vide muni de deux opération :
• Addition vectorielle : $(u, v) \mapsto u+v$
• Exterme multiplication : $(\lambda, u) \mapsto \lambda \cdot u = \lambda u$

1. Associativité: $(u+v)+w = u+(v+w)$
2. Commutativité: $u+v=v+u$
3. Existence d'un élément neutre pour l'addition: $u+0_E = 0_E + u = u$ element $0_E$ is the same
4. Existence d'un inverse additif: $u+(-u) = (-u)+u = 0_E$
5. Distributivité: $\lambda(u+v) = \lambda u + \lambda v$
6. Distributivité: $(\lambda + \mu)u = \lambda u + \mu u$
7. Compatibilité: $(\lambda \mu)u = \lambda (\mu u)$
8. Élément neutre pour la multiplication scalaire: $1_{\mathbb{K}}u = u$

• Subset F that $E$ that is itself de $E$ with the operation.
• $F$ est non vide.
• $F$ est stable par addition.
• $F$ est stable par multiplication scalaire.

Corollaire 1.1:

Soit $E$ of subset $F$ is a subspace vectoriel de $E$ and $F$ est un non vide $\forall u \in E, \exists (-u) \in E, u+(-u) = (-u)+u = 0_E$ 2.$\forall \lambda \in \mathbb{K}, \forall u, v \in F$ then $\lambda u + v \in F$.

• In $\mathbb{R}^2$, les droites of the origine $0_E$ et $E$ sont des sous-espaces vectoriels of $E$

2. Soit $E$ un espace vectoriel of $A$ de $E$

• Subset vectoriel engendré par and noted Vect.
• Linear combination (n $\mathbb{N}, \lambda i \in \mathbb{K}, u_i \in A$
• Vector sum of subset
• $F+G$, est définie par $F+G = {u+v \mid u \in F, v \in G }$ Vect + Vect = vect Vect + vect * Linear is the set
• The somme is is direct of if $F \cap G = {0_E }$. On note alors $F \oplus G
• Vectors with F+G vectors with $F$ and $G$ are supplemental if Vect(vectors in $F$) = linvect(vector)$G$
• the vecter of and are supplemental if $E = F \oplus G$, written comme a sum