Equations, Formulas and Algorithms - Revision history

JonoThora: /* Differential Equations */

2024-02-22T19:20:24Z

Differential Equations

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	=== Differential Equations ===		= Differential Equations =

	==== Finite Difference Methods ====		== Finite Difference Methods ==
	Finite difference methods discretize differential equations by approximating derivatives using finite differences. These methods are widely used for solving ordinary and partial differential equations in various domains, including fluid dynamics, heat transfer, and structural mechanics.		Finite difference methods discretize differential equations by approximating derivatives using finite differences. These methods are widely used for solving ordinary and partial differential equations in various domains, including fluid dynamics, heat transfer, and structural mechanics.

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	==== Finite Element Methods (FEM) ====		== Finite Element Methods (FEM) ==
	Finite element methods divide the domain of a differential equation into smaller elements, allowing for the approximation of the solution within each element. FEM is employed in structural analysis, electromagnetics, and other fields to solve partial differential equations and boundary value problems.		Finite element methods divide the domain of a differential equation into smaller elements, allowing for the approximation of the solution within each element. FEM is employed in structural analysis, electromagnetics, and other fields to solve partial differential equations and boundary value problems.

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	==== Finite Volume Methods (FVM) ====		== Finite Volume Methods (FVM) ==
	Finite volume methods discretize differential equations by integrating them over control volumes. FVM is commonly used in computational fluid dynamics and heat transfer simulations to solve conservation equations governing fluid flow and heat transfer phenomena.		Finite volume methods discretize differential equations by integrating them over control volumes. FVM is commonly used in computational fluid dynamics and heat transfer simulations to solve conservation equations governing fluid flow and heat transfer phenomena.

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	==== Boundary Element Methods (BEM) ====		== Boundary Element Methods (BEM) ==
	Boundary element methods reduce differential equations to integral equations over the boundary of the domain. BEM is useful for problems with infinite domains or complex geometries, such as acoustic and electromagnetic scattering problems.		Boundary element methods reduce differential equations to integral equations over the boundary of the domain. BEM is useful for problems with infinite domains or complex geometries, such as acoustic and electromagnetic scattering problems.

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	==== Spectral Methods ====		== Spectral Methods ==
	Spectral methods approximate the solution of differential equations using expansions in terms of basis functions, such as Fourier series or Chebyshev polynomials. These methods offer high accuracy and convergence rates and are used in various scientific and engineering applications.		Spectral methods approximate the solution of differential equations using expansions in terms of basis functions, such as Fourier series or Chebyshev polynomials. These methods offer high accuracy and convergence rates and are used in various scientific and engineering applications.

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	==== Runge-Kutta Methods ====		== Runge-Kutta Methods ==
	Runge-Kutta methods are numerical techniques for solving ordinary differential equations. They involve iterative procedures to approximate the solution trajectory of the differential equation over a specified time interval. Runge-Kutta methods are widely used in numerical simulations and scientific computing.		Runge-Kutta methods are numerical techniques for solving ordinary differential equations. They involve iterative procedures to approximate the solution trajectory of the differential equation over a specified time interval. Runge-Kutta methods are widely used in numerical simulations and scientific computing.

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	==== Adaptive Methods ====		== Adaptive Methods ==
	Adaptive algorithms adjust the step size or mesh resolution dynamically based on solution properties to achieve a desired level of accuracy. These methods are employed to solve differential equations efficiently while minimizing computational cost and resource usage.		Adaptive algorithms adjust the step size or mesh resolution dynamically based on solution properties to achieve a desired level of accuracy. These methods are employed to solve differential equations efficiently while minimizing computational cost and resource usage.

JonoThora: /* Spectral Methods */

2024-02-22T19:12:41Z

Spectral Methods

← Older revision		Revision as of 12:12, 22 February 2024
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	<syntaxhighlight lang="matlab">		<syntaxhighlight lang="matlab">
	% Define domain and initial conditions		% Define domain and initial conditions
			domain = [-1, 1]; % Define the spatial domain
			initial_condition = chebfun(@(x) exp(-20*x.^2), domain); % Define initial condition

	% Define Chebfun objects for functions and derivatives		% Define Chebfun objects for functions and derivatives
			c = 1; % Wave speed
			wave_equation = @(t, u) -c^2 * diff(u, 2); % Define the wave equation
			t_span = [0, 1]; % Time span for simulation

	% Solve the wave equation using Chebfun's time-stepping capabilities		% Solve the wave equation using Chebfun's time-stepping capabilities
	% Plot solution ~~or animate wave propagation~~		u = pdepe(0, wave_equation, initial_condition, @(x, t) 0, domain, t_span);

			% Plot solution
			figure;
			plot(u(end, :));
			title('Solution of the One-dimensional Wave Equation');
			xlabel('Spatial Domain');
			ylabel('Solution');
	</syntaxhighlight>		</syntaxhighlight>

JonoThora: /* Boundary Element Methods (BEM) */

2024-02-22T19:11:08Z

Boundary Element Methods (BEM)

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	'''Source Code Example (MATLAB with BEM++):'''		'''Source Code Example (MATLAB with BEM++):'''
	<syntaxhighlight lang="matlab">		<syntaxhighlight lang="matlab">
	% Define boundary conditions ~~(if any)~~		% Define boundary conditions
	% For simplicity, let's consider Dirichlet boundary conditions ~~where the~~		% For simplicity, let's consider Dirichlet boundary conditions.
	~~% solution is known at specific boundary points~~.

	% Define geometry and boundary elements		% Define geometry and boundary elements
	~~% Let's consider a simple circular domain with a known radius.~~
	radius = 1;		radius = 1;
	~~center~~ = [0, 0]; ~~% Center of the circle~~		num_elements = 50;
			theta = linspace(0, 2*pi, num_elements+1);
			theta = theta(1:end-1);
			boundary_points = [radius * cos(theta); radius * sin(theta)];

	% ~~Define mesh~~		% Solve Laplace's equation analytically for demonstration
	~~num_elements = 50; % Number of boundary elements~~
	~~theta = linspace(0, 2*pi, num_elements+1); % Angle values for boundary points~~
	~~theta = theta(1:end-1); % Remove duplicate endpoint~~
	~~boundary_points = [radius * cos(theta); radius * sin(theta)]; % Boundary points~~

	~~% Assemble system matrix~~
	~~% For~~ Laplace's equation~~, the system matrix is typically obtained by~~
	~~% discretizing the boundary integral equation using appropriate~~
	~~% approximation methods.~~

	~~% Solve linear system~~
	~~% For~~ demonstration ~~purposes, let's assume a simple analytical solution~~
	~~% for Laplace's equation in a circular domain.~~
	analytical_solution = @(r, theta) r.^2 .* sin(2*theta);		analytical_solution = @(r, theta) r.^2 .* sin(2*theta);

	~~% Calculate the solution at boundary points~~
	boundary_solution = analytical_solution(radius, theta);		boundary_solution = analytical_solution(radius, theta);

	~~% Post-processing (e.g., visualization)~~		% Plot analytical solution and boundary points
	% Plot ~~the~~ analytical solution and ~~the solution obtained using BEM~~

	~~% Plot the analytical solution~~
	figure;		figure;
	plot(theta, boundary_solution, 'r-', 'LineWidth', 2);		plot(theta, boundary_solution, 'r-', 'LineWidth', 2);
	hold on;		hold on;
			scatter(theta, zeros(size(theta)), 'k', 'filled');
	title('Analytical Solution for Laplace''s Equation (Circular Domain)');		title('Analytical Solution for Laplace''s Equation (Circular Domain)');
	xlabel('\theta');		xlabel('\theta');
	ylabel('Solution');		ylabel('Solution');
			legend('Analytical Solution', 'Boundary Points');
	grid on;		grid on;

	~~% Plot the boundary points~~
	~~scatter(theta, zeros(size(theta)), 'k', 'filled');~~
	~~legend('Analytical Solution', 'Boundary Points');~~
	hold off;		hold off;

	</syntaxhighlight>		</syntaxhighlight>

JonoThora: /* Boundary Element Methods (BEM) */

2024-02-22T19:10:06Z

Boundary Element Methods (BEM)

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 <syntaxhighlight lang="matlab">
 % Define boundary conditions (if any)
 % Define geometry and boundary elements
 % Define mesh
 % Assemble system matrix
 % Solve linear system
 % Post-processing (e.g., visualization)
 </syntaxhighlight>

JonoThora: /* Volterra Integral Equation Solvers */

2024-02-22T19:04:30Z

Volterra Integral Equation Solvers

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	'''Example: Volterra Integral Equation'''		'''Example: Volterra Integral Equation'''
	The Volterra integral equation of the second kind is given by:		The Volterra integral equation of the second kind is given by: <math> \phi(x) = f(x) + \lambda \int_{a}^{x} K(x, s) \phi(s) \, ds </math> where <math> \phi(x) </math> is the unknown function to be solved for, <math> f(x) </math> is a given function, <math> K(x, s) </math> is the kernel function, and <math> \lambda </math> is a parameter.

	<math> \phi(x) = f(x) + \lambda \int_{a}^{x} K(x, s) \phi(s) \, ds </math>

	where <math> \phi(x) </math> is the unknown function to be solved for, <math> f(x) </math> is a given function, <math> K(x, s) </math> is the kernel function, and <math> \lambda </math> is a parameter.

	'''Source Code Example (Julia):'''		'''Source Code Example (Julia):'''

JonoThora: /* Fields of Application */

2024-02-22T19:03:54Z

Fields of Application

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	=== Fields of Application ===		=== Fields of Application ===
	Algorithms find application in diverse fields, including:		Algorithms find application in diverse fields, including:
	* Computer Science: Designing algorithms for sorting, searching, graph traversal, and optimization problems.		* [[Computer Science]]: Designing algorithms for sorting, searching, graph traversal, and optimization problems.
	* Artificial Intelligence: Developing algorithms for machine learning, pattern recognition, and natural language processing.		* [[Artificial Intelligence]]: Developing algorithms for machine learning, pattern recognition, and natural language processing.
	* Optimization: Implementing algorithms for linear programming, integer programming, and combinatorial optimization.		* [[Optimization]]: Implementing algorithms for linear programming, integer programming, and combinatorial optimization.
	* Cryptography: Designing algorithms for encryption, decryption, digital signatures, and secure communication.		* [[Cryptography]]: Designing algorithms for encryption, decryption, digital signatures, and secure communication.
	* Bioinformatics: Applying algorithms for sequence alignment, phylogenetic analysis, and protein structure prediction.		* [[Bioinformatics]]: Applying algorithms for sequence alignment, phylogenetic analysis, and protein structure prediction.

	=== Examples ===		=== Examples ===

JonoThora: /* Fields of Application */

2024-02-22T19:03:19Z

Fields of Application

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	=== Fields of Application ===		=== Fields of Application ===
	Formulas find application in various fields, including:		Formulas find application in various fields, including:
	* Physics: Calculating forces, energy, motion, and electromagnetic phenomena.		* [[Physics]]: Calculating forces, energy, motion, and electromagnetic phenomena.
	* Finance: Estimating compound interest, present and future values, and investment returns.		* [[Finance]]: Estimating compound interest, present and future values, and investment returns.
	* Statistics: Determining probabilities, means, variances, and correlations in data analysis.		* [[Statistics]]: Determining probabilities, means, variances, and correlations in data analysis.
	* Engineering: Designing structures, circuits, systems, and control algorithms.		* [[Engineering]]: Designing structures, circuits, systems, and control algorithms.
	* Mathematics: Expressing mathematical relationships, solving equations, and deriving theorems.		* [[Mathematics]]: Expressing mathematical relationships, solving equations, and deriving theorems.

	=== Examples ===		=== Examples ===

JonoThora: /* Volterra Integral Equation Solvers */

2024-02-22T19:02:42Z

Volterra Integral Equation Solvers

← Older revision		Revision as of 12:02, 22 February 2024
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	'''Example: Volterra Integral Equation'''		'''Example: Volterra Integral Equation'''
	The Volterra integral equation of the second kind is given by:		The Volterra integral equation of the second kind is given by:
	<math ~~mode="block"~~> \phi(x) = f(x) + \lambda \int_{a}^{x} K(x, s) \phi(s) \, ds </math>
			<math> \phi(x) = f(x) + \lambda \int_{a}^{x} K(x, s) \phi(s) \, ds </math>

	where <math> \phi(x) </math> is the unknown function to be solved for, <math> f(x) </math> is a given function, <math> K(x, s) </math> is the kernel function, and <math> \lambda </math> is a parameter.		where <math> \phi(x) </math> is the unknown function to be solved for, <math> f(x) </math> is a given function, <math> K(x, s) </math> is the kernel function, and <math> \lambda </math> is a parameter.

JonoThora: /* Volterra Integral Equation Solvers */

2024-02-22T19:02:12Z

Volterra Integral Equation Solvers

← Older revision		Revision as of 12:02, 22 February 2024
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	'''Example: Volterra Integral Equation'''		'''Example: Volterra Integral Equation'''
	The Volterra integral equation of the second kind is given by:		The Volterra integral equation of the second kind is given by:
	<math mode="~~display~~"> \phi(x) = f(x) + \lambda \int_{a}^{x} K(x, s) \phi(s) \, ds </math>		<math mode="block"> \phi(x) = f(x) + \lambda \int_{a}^{x} K(x, s) \phi(s) \, ds </math>
	where <math> \phi(x) </math> is the unknown function to be solved for, <math> f(x) </math> is a given function, <math> K(x, s) </math> is the kernel function, and <math> \lambda </math> is a parameter.		where <math> \phi(x) </math> is the unknown function to be solved for, <math> f(x) </math> is a given function, <math> K(x, s) </math> is the kernel function, and <math> \lambda </math> is a parameter.

JonoThora at 19:00, 22 February 2024

2024-02-22T19:00:52Z

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	'''Example: Trapezoidal Rule'''		'''Example: Trapezoidal Rule'''
	The trapezoidal rule approximates the integral of a function <math> f(x) </math> over the interval <math>[a, b]</math> as follows:		The trapezoidal rule approximates the integral of a function <math> f(x) </math> over the interval <math>[a, b]</math> as follows:
	\[ \int_{a}^{b} f(x) \, dx \approx \frac{h}{2} \left( f(a) + 2\sum_{i=1}^{n-1} f(x_i) + f(b) \right) \]		<math mode="display"> \int_{a}^{b} f(x) \, dx \approx \frac{h}{2} \left( f(a) + 2\sum_{i=1}^{n-1} f(x_i) + f(b) \right) </math>
	where <math> h = \frac{b - a}{n} </math> is the width of each subinterval and <math> x_i = a + ih </math> are the points at which the function is evaluated.		where <math> h = \frac{b - a}{n} </math> is the width of each subinterval and <math> x_i = a + ih </math> are the points at which the function is evaluated.

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	'''Example: Fredholm Integral Equation'''		'''Example: Fredholm Integral Equation'''
	The Fredholm integral equation of the second kind is given by:		The Fredholm integral equation of the second kind is given by:
	\[ \phi(x) = f(x) + \lambda \int_{a}^{b} K(x, s) \phi(s) \, ds \]		<math mode="display"> \phi(x) = f(x) + \lambda \int_{a}^{b} K(x, s) \phi(s) \, ds </math>
	where <math> \phi(x) </math> is the unknown function to be solved for, <math> f(x) </math> is a given function, <math> K(x, s) </math> is the kernel function, and <math> \lambda </math> is a parameter.		where <math> \phi(x) </math> is the unknown function to be solved for, <math> f(x) </math> is a given function, <math> K(x, s) </math> is the kernel function, and <math> \lambda </math> is a parameter.

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	'''Example: Volterra Integral Equation'''		'''Example: Volterra Integral Equation'''
	The Volterra integral equation of the second kind is given by:		The Volterra integral equation of the second kind is given by:
	\[ \phi(x) = f(x) + \lambda \int_{a}^{x} K(x, s) \phi(s) \, ds \]		<math mode="display"> \phi(x) = f(x) + \lambda \int_{a}^{x} K(x, s) \phi(s) \, ds </math>
	where <math> \phi(x) </math> is the unknown function to be solved for, <math> f(x) </math> is a given function, <math> K(x, s) </math> is the kernel function, and <math> \lambda </math> is a parameter.		where <math> \phi(x) </math> is the unknown function to be solved for, <math> f(x) </math> is a given function, <math> K(x, s) </math> is the kernel function, and <math> \lambda </math> is a parameter.

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	'''Example: Finite Difference Method'''		'''Example: Finite Difference Method'''
	Consider the one-dimensional heat equation:		Consider the one-dimensional heat equation:
	\[ \frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2} \]		<math mode="display"> \frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2} </math>
	where <math> u(x, t) </math> is the temperature distribution, <math> \alpha </math> is the thermal diffusivity, <math> x </math> is the spatial coordinate, and <math> t </math> is time.		where <math> u(x, t) </math> is the temperature distribution, <math> \alpha </math> is the thermal diffusivity, <math> x </math> is the spatial coordinate, and <math> t </math> is time.

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	'''Example: Finite Element Method'''		'''Example: Finite Element Method'''
	Consider the one-dimensional Poisson equation:		Consider the one-dimensional Poisson equation:
	\[ -\frac{d^2 u}{dx^2} = f(x) \]		<math mode="display"> -\frac{d^2 u}{dx^2} = f(x) </math>
	where <math> u(x) </math> is the unknown function to be solved for and <math> f(x) </math> is a given function.		where <math> u(x) </math> is the unknown function to be solved for and <math> f(x) </math> is a given function.

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	'''Example: Finite Volume Method'''		'''Example: Finite Volume Method'''
	Consider the one-dimensional advection equation:		Consider the one-dimensional advection equation:
	\[ \frac{\partial u}{\partial t} + c \frac{\partial u}{\partial x} = 0 \]		<math mode="display"> \frac{\partial u}{\partial t} + c \frac{\partial u}{\partial x} = 0 </math>
	where <math> u(x, t) </math> is the quantity being advected, <math> c </math> is the advection velocity, <math> x </math> is the spatial coordinate, and <math> t </math> is time.		where <math> u(x, t) </math> is the quantity being advected, <math> c </math> is the advection velocity, <math> x </math> is the spatial coordinate, and <math> t </math> is time.

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	'''Example: Boundary Element Method'''		'''Example: Boundary Element Method'''
	Consider the Laplace equation in two dimensions:		Consider the Laplace equation in two dimensions:
	\[ \nabla^2 u = 0 \]		<math mode="display"> \nabla^2 u = 0 </math>
	where <math> u(x, y) </math> is the unknown function to be solved for.		where <math> u(x, y) </math> is the unknown function to be solved for.

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	'''Example: Spectral Method'''		'''Example: Spectral Method'''
	Consider the one-dimensional wave equation:		Consider the one-dimensional wave equation:
	\[ \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} \]		<math mode="display"> \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} </math>
	where <math> u(x, t) </math> is the displacement of a vibrating string, <math> c </math> is the wave speed, <math> x </math> is the spatial coordinate, and <math> t </math> is time.		where <math> u(x, t) </math> is the displacement of a vibrating string, <math> c </math> is the wave speed, <math> x </math> is the spatial coordinate, and <math> t </math> is time.

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	'''Example: Runge-Kutta Method'''		'''Example: Runge-Kutta Method'''
	Consider the following ordinary differential equation:		Consider the following ordinary differential equation:
	\[ \frac{dy}{dt} = f(t, y) \]		<math mode="display"> \frac{dy}{dt} = f(t, y) </math>
	where <math> y(t) </math> is the solution and <math> f(t, y) </math> is the derivative function.		where <math> y(t) </math> is the solution and <math> f(t, y) </math> is the derivative function.

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	'''Example: Adaptive Method'''		'''Example: Adaptive Method'''
	Consider the following ordinary differential equation:		Consider the following ordinary differential equation:
	\[ \frac{dy}{dt} = f(t, y) \]		<math mode="display"> \frac{dy}{dt} = f(t, y) </math>
	where <math> y(t) </math> is the solution and <math> f(t, y) </math> is the derivative function.		where <math> y(t) </math> is the solution and <math> f(t, y) </math> is the derivative function.