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View Full Interview# What is the method of finite differences for linear equations?

Intuitive derivation Finite-difference methods approximate the solutions to differential equations by replacing derivative expressions with approximately equivalent differen…ce quotients. That is, because the first derivative of a function f is, by definition, then a reasonable approximation for that derivative would be to take for some small value of h. In fact, this is the forward difference equation for the first derivative. Using this and similar formulae to replace derivative expressions in differential equations, one can approximate their solutions without the need for calculus. [edit] Derivation from Taylor's polynomial Assuming the function whose derivatives are to be approximated is properly-behaved, by Taylor's theorem, where n! denotes the factorial of n, and Rn(x) is a remainder term, denoting the difference between the Taylor polynomial of degree n and the original function. Again using the first derivative of the function f as an example, by Taylor's theorem, f(x0 + h) = f(x0) + f'(x0)h + R1(x), which, with some minor algebraic manipulation, is equivalent to so that for R1(x) sufficiently small, [edit] Accuracy and order The error in a method's solution is defined as the difference between its approximation and the exact analytical solution. The two sources of error in finite difference methods are round-off error, the loss of precision due to computer rounding of decimal quantities, and truncation error or discretization error, the difference between the exact solution of the finite difference equation and the exact quantity assuming perfect arithmetic (that is, assuming no round-off). The finite difference method relies on discretizing a function on a grid. To use a finite difference method to attempt to solve (or, more generally, approximate the solution to) a problem, one must first discretize the problem's domain. This is usually done by dividing the domain into a uniform grid (see image to the right). Note that this means that finite-difference methods produce sets of discrete numerical approximations to the derivative, often in a "time-stepping" manner. An expression of general interest is the local truncation error of a method. Typically expressed using Big-O notation, local truncation error refers to the error from a single application of a method. That is, it is the quantity f'(xi) − f'i if f'(xi) refers to the exact value and f'i to the numerical approximation. The remainder term of a Taylor polynomial is convenient for analyzing the local truncation error. Using the Lagrange form of the remainder from the Taylor polynomial for f(x0 + h), which is , where x0 < ξ < x0 + h, the dominant term of the local truncation error can be discovered. For example, again using the forward-difference formula for the first derivative, knowing that f(xi) = f(x0 + ih), and with some algebraic manipulation, this leads to and further noting that the quantity on the left is the approximation from the finite difference method and that the quantity on the right is the exact quantity of interest plus a remainder, clearly that remainder is the local truncation error. A final expression of this example and its order is: This means that, in this case, the local truncation error is proportional to the step size. [edit] Example: ordinary differential equation For example, consider the ordinary differential equation The Euler method for solving this equation uses the finite difference quotient to approximate the differential equation by first substituting in for u'(x) and applying a little algebra to get The last equation is a finite-difference equation, and solving this equation gives an approximate solution to the differential equation. [edit] Example: The heat equation Consider the normalized heat equation in one dimension, with homogeneous Dirichlet boundary conditions (boundary condition) (initial condition) One way to numerically solve this equation is to approximate all the derivatives by finite differences. We partition the domain in space using a mesh x0,...,xJ and in time using a mesh t0,....,tN. We assume a uniform partition both in space and in time, so the difference between two consecutive space points will be h and between two consecutive time points will be k. The points will represent the numerical approximation of u(xj,tn). [edit] Explicit method The stencil for the most common explicit method for the heat equation. Using a forward difference at time tn and a second-order central difference for the space derivative at position xj ("FTCS") we get the recurrence equation: This is an explicit method for solving the one-dimensional heat equation. We can obtain from the other values this way: where r = k / h2. So, knowing the values at time n you can obtain the corresponding ones at time n+1 using this recurrence relation. and must be replaced by the boundary conditions, in this example they are both 0. This explicit method is known to be numerically stable and convergent whenever . The numerical errors are proportional to the time step and the square of the space step: [edit] Implicit method The implicit method stencil. If we use the backward difference at time ti + 1 and a second-order central difference for the space derivative at position xj ("BTCS") we get the recurrence equation: This is an implicit method for solving the one-dimensional heat equation. We can obtain from solving a system of linear equations: The scheme is always numerically stable and convergent but usually more numerically intensive than the explicit method as it requires solving a system of numerical equations on each time step. The errors are linear over the time step and quadratic over the space step. [edit] Crank-Nicolson method Finally if we use the central difference at time tn + 1 / 2 and a second-order central difference for the space derivative at position xj ("CTCS") we get the recurrence equation: This formula is known as the Crank-Nicolson method. The Crank-Nicolson stencil. We can obtain from solving a system of linear equations: The scheme is always numerically stable and convergent but usually more numerically intensive as it requires solving a system of numerical equations on each time step. The errors are quadratic over the time step and formally are of the fourth degree regarding the space step: However, near the boundaries, the error is often O(h2) instead of O(h4). Usually the Crank-Nicolson scheme is the most accurate scheme for small time steps. The explicit scheme is the least accurate and can be unstable, but is also the easiest to implement and the least numerically intensive. The implicit scheme works the best for large time steps. (MORE)

# What is the difference between linear and non linear sensors?

In a linear sensor, the output changes evenly as the input changes. Such that, if you were to plot a graph, with the X and Y axis showing the values of input verses output, i…t would follow a straight line. Hence linear. Non linear sensors will have a greater, or lesser, output as the input changes. Plotted on a graph, the line will be curved. Our own ears are non linear sensors. For a sound to appear twice as loud, the pressure level has to increase in a logarithmic scale. (MORE)

# How does solving linear inequalities differ from solving linear equations?

Linear inequalities are equations, but instead of an equal sign, it has either a greater than, greater than or equal to, less than, or a less than or equal to sign. Both can b…e graphed. Solving linear equations mainly differs from solving linear inequalities in the form of the solution. 1. Linear equation. For each linear equation in x, there is only one value of x (solution) that makes the equation true. Example 1. The equation: x - 3 = 7 has one solution, that is x = 10. Example 2. The equation: 3x + 4 = 13 has one solution that is x = 3. 2. Linear inequality. On the contrary, a linear inequality has an infinity of solutions, meaning there is an infinity of values of x that make the inequality true. All these x values constitute the "solution set" of the inequality. The answers of a linear inequality are expressed in the form of intervals. Example 3. The linear inequality x + 5 < 9 has as solution: x < 4. The solution set of this inequality is the interval (-infinity, 4) Example 4. The inequality 4x - 3 > 5 has as solution x > 2. The solution set is the interval (2, +infinity). The intervals can be open, closed, and half closed. Example: The open interval (1, 4) ; the 2 endpoints 1 and 4 are not included in the solution set. Example: The closed interval [-2, 5] ; the 2 end points -2 and 5 are included. Example : The half-closed interval [3, +infinity) ; the end point 3 is included. (MORE)

# Algebraic Concepts

The foundation for many of the advanced mathematics is algebra. One way to learn algebra concepts is by practicing and doing homework. The importance of math cannot be oversta…ted, as all forms of it play an integral role in everyday life. Once you have the concepts down, you will have (MORE)

# Dos and Donts: Attention Deficit Hyperactivity Disorder

How do you know if your teenager is suffering from Attention Deficit Hyperactivity Disorder (ADHD)? Some symptoms appear to be the same as normal signs of puberty. However, th…ere are ways to find out using ADHD tests for teenagers and there are dos and don'ts to follow if you suspect (MORE)

# Fun iPad Games for Kids

IPads help adults manage business, email, banking and more on the go; however, iPads also have many educational apps and games for kids to enjoy. The next time you are taking …the kids on a long plane flight or visit to the doctor, pull out your iPad and let them (MORE)

# What makes linear equations different than linear inequalities?

An equation is a statement that two quantities are equal, or the same, identical, in value. It is expressed by putting an equal sign (=) between the two quantities. An inequal…ity is a statement that two quantities are not equal, or more specifically, that one is less than the other, or less than or equal to the other. It is expressed with the unequal sign (an equal sign with a slash through it), a less than sign (<), a greater than sign (>), or a less than or equal sign or greater than or equal sign. A less than or equal sign looks like a less than sign with an underscore; similarly for the greater than or equal sign. Answer 1 A linear equation may be represented by all the points on a straight line. A linear inequality would be represented by all points in the plane on one side or the other of the line which is determined by the corresponding equation. The line itself may or may not be part of the solution. (MORE)

# What is the difference between linear equations and linear inequalities?

It is easiest to describe the difference in terms of coordinate geometry. A linear equation defines a straight line in the coordinate plane. Every point on the line satisfies… the equation and no other points do. For a linear inequality, first consider the corresponding linear equality (or equation). That defines a straight line which divides the plane into two. Depending on the direction of the inequality, all points on one side of the line or the other satisfy the equation, and no point from the other side of the line does. If it is a strict inequality (< or >) then points on the line itself are excluded while if the inequality is not strict (≤or ≥) then points on the line are included. (MORE)

# What is the differences between linear and non linear optical properties?

In the case of linear optical transitions, an electron absorbs a photon from the incoming light and makes a transition to the next higher unoccupied allowed state. When this… electron relaxes it emits a photon of frequency less than or equal to the frequency of the incident light (Figure 1.3a). SHG on the other hand is a two-photon process where this excited electron absorbs another photon of same frequency and makes a transition to reach another allowed state at higher energy. This electron when falling back to its original 39 state emits a photon of a frequency which is two times that of the incident light (Figure 1.3b). This results in the frequency doubling in the output. (MORE)