# What are the differences of linear and non-linear equation?

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# 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)

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In Education

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

Materials that obey Ohm's Law are called 'linear' or 'ohmic'; those that don't are called 'non-ohmic' or 'non-linear'. Ohm's Law isn't by any means a universal law…; it doesn't apply to all conductors! Ohm's Law simply states that 'the current flowing through a wire, at constant temperature, is directly proportional to the potential difference across the ends of that wire'. This doesn't apply to, for example, a tungsten filament lamp, whose ratio between voltage and current changes as the voltage increases (due to its resistance changing as its temperature increases). The so-called 'Ohm's Law equation' (R = V/I) is, in fact, derived from the definition of the ohm, and not from Ohm's Law. For this reason, the equation applies even when Ohm's Law does not. If the ratio of voltage to current remains constant over a wide range of voltages, then Ohm's Law applies for that range of voltages. If the ratio of voltage to current changes over a range of volages, then Ohm's Law does not apply. In the case of resistors (as opposed to 'resistance'), 'linear' and 'non-linear' describes the way in which variable resistors have been wound. 'Non-linear' variable resistors are those that have been wound to produce specific characteristics, such as logarithmic values of resistance, as they are adjusted, whereas 'linear' variable resistors produce values of resistance that are directly proportional to how far the adjustment wiper has been moved. (MORE)

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# How are linear inequalities different from linear equations?

A linear equation represents a line. A linear inequality represents part of the space on one side (or the other) of the line defined by the corresponding equation.

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In Movies

# 12 Great Movies With a Nonlinear Storyline

Most stories have a beginning, middle and end, but are not necessarily told in that order. Some filmmakers find it more effective to jumble up the story. Here are a few of the… best movies with nonlinear plots. (MORE)

# How do You Balance Chemistry Equations?

Chemistry is used in everyday life, from making food to breathing air. Chemical reactions are what creates life, makes food into energy, and even what causes food to rot. When… you balance equations, you are learning the amount of elements used in the reactants, or found in the products. If you do not balance the equation, you will only know what you started with, and not the final product.Chemistry is just like math when it comes to balancing equations. You want both sides of the equation to be the same. In Chemistry, this is done by making sure all atoms in your products can be found in the reactants.In a chemistry equation, the reactant is the material you start with. For example, when making water, your reactants would be two hydrogen atoms and one oxygen atom. The more complex the equation, the more reactants will be present.The product in a chemistry equation is the final product formed when the two reactants combine through a chemical reaction. Using the same example from above, when you combine two hydrogen atoms and one oxygen atom, the product would be water, or H20. It often takes a complex amount of reactants to create a product.When you are presented with an equation and must balance it, you will first want to identify all of the elements in the equation. This will give you a quick view of what will need to be added or subtracted from the equation to make it equal. The next step is to see what the charge is, or how many electrons are on each side. Then, start with one side of the equation and change the coefficients of the atom so that there will be the same number on both sides. Continue with this until you have all of the elements accounted for. Finally, check the charge on both sides of the equation.There are a few requirements to meet when properly balancing chemical equations. The first is making sure there is the same number of atoms on both sides of the equation. Atoms do not disappear or appear in a chemical reaction, so they must all be accounted for. By the same token, atoms in an element do not change to another element. Therefore, if you have hydrogen atoms on one side of the equation, you must have hydrogen atoms on the other side of the equation. Finally, the charge should be the same on both sides of the equation. This can be the hardest part of balancing equations. Unless it is a nuclear reaction, you cannot create nor destroy electrons in a reaction.While it seems to be a complicated process, balancing chemistry equations is fairly easy once you learn how. Knowing how reactants respond to various temperatures and pressures is how warning labels are made on common products, such as hairspray. Balancing equations ensures the reaction adheres to the conservation law of matter. That law states that elements cannot be created nor destroyed in an isolated system, or reaction.It is easiest to balance equations if you isolate the chemicals and then take an inventory of the elements. Many find this easiest if they draw a chart that shows the name of the element, the number before the equal sign, and then the number after the equal sign. The equation is not balanced until the numbers are the same. (MORE)

In Equations

# Understanding Equations When Chemicals are Involved

Understanding chemical equations is an important part of both basic and advanced chemistry. Instructors often spend an entire semester teaching their students how to read, int…erpret, write, and balance chemical equations. Likewise, chemical equations involve many factors including chemical symbols, reactants, products, and coefficients. Read on for more information about chemical equations and learn how mathematics plays an essential role in chemistry.In short, chemical equations are used to describe what happens in a chemical reaction. The components of a chemical equation include the substances that are involved in the reaction, the formula, the product, and the phase of each substance involved. However, each component of a chemical equation has a unique and significant role to play as part of the analyzing and understanding chemical changes.Chemical symbols are one or two-letter codes that are used to identify a pure element, and these chemical symbols are used to make up the scientific periodic table. For example, He is the chemical symbol of helium, and H is the chemical symbol for hydrogen. Additionally, Na is the chemical symbol of sodium, and O is the chemical symbol of oxygen. Rather than writing the full name of the elements that are involved in a reaction, chemists identify elements using chemical symbols and codes.Chemical equations are organized in a very straightforward fashion. A chemical equation is technically an illustration of a chemical reaction. Firstly, the reactants or starting elements appear on the left hand side of the equation, and there must be at least two reactants involved in any chemical equation. Moreover, chemical equations utilize an arrow in order to symbolize the reaction that takes place, and the arrow is used to identify what the reaction will yield. In the correct form, the symbol of the arrow points from the reactants to the product or result of the chemical reaction.Chemical equations must always be balanced. The law of conservation of mass states that the quantity of each element that is present at the beginning of a reaction must be equal to the quantity of each element that is present at the end of the reaction. Thus, each side of a chemical equation must symbolize an equal amount of any element that is present in the equation. Furthermore, balancing a chemical equation is the process used to determine how many molecules of each substance are present in the reactants and products of a reaction.Chemical equations are one of the most important aspects of chemistry, as it helps chemists understand what happens during a chemical reaction. Especially for chemistry students, learning how to read, write, and balance chemical equations is the first step to understanding and predicting how various chemicals will react with one another. Therefore, it is important to memorize the chemical symbols on the scientific periodic table so that you can easily get through reading and writing chemical equations in class and when you are out working in the field as a chemist or chemical engineer.Some chemical reactions require the addition of energy such as heat. If energy is applied to the reactants in order to generate a chemical reaction, a triangle symbol will be written above the arrow to indicate that the presence of energy was needed in the chemical equation to cause a reaction. (MORE)

In Technique

# Linear and Atmospheric Perspective in Landscape Painting

Landscape painters have two powerful means of suggesting distance in painting: linear perspective and atmospheric (aerial) perspective.… (MORE)

In Population

# Studying Populations: Demographic Equation

The study of human populations is called demography, and one element of demography is Demographic Equation. Demographic Equation is a method for determining the overall change… in an area's population over time. In order to understand how this equation works, you need to know more about it. This includes what goes into the equation and who tends to use it.Demographic Equation is part of demography. It helps determine population change in an area over time. The equation is calculated by taking the number of births in a region versus the number of deaths to determine if the population increased (more births than deaths) or decreased (more deaths than births), then subtracting or adding that to the net migration of an area. The net migration is determined by looking at the number of people who have migrated to the area and the number who have migrated out of the area, and determining whether the population increased (more people migrating to the area than leaving) or decreased (more people leaving than migrating to the area) due to migration.Demographic Equation is used for a multitude of purposes. It can be used for statistical purposes in simply determining whether a region saw an overall increase or decrease in population. The equation is also used by businesses, who may look at an overall increase in population as a chance to increase marketing in a certain area, while a decrease in population may be a sign to decrease marketing expenses. People may also use the equation to determine if they want to move to a certain area. An overall increase in population may indicate that the area is experiencing growth, but it may also indicate that the area is becoming overcrowded. A decrease may indicate that the area is experiencing a slowdown, but with more people leaving new opportunities may arise.Demographic Equation can provide an idea of the overall population trend in an area. It can show which areas are experiencing growth and which may be experiencing slowdown. Of course, an overall increase in population does not always mean the area is growing, but if the equation shows that it continues to grow over a period of years, there may be something worth checking out in the area. If an area continues to see a decrease in population over time, the area may be experiencing an economic slowdown that is driving people away.Demographic Equation only shows a broad reason why a region's population is increasing or decreasing. It may show that a high number of births versus a low number of deaths, plus a large increase in migration to the area, is the cause of population increase. However, it does not show why there is an increase in births or migration. Similarly, it does not show why there are a high number of deaths or people leaving the area, only that the circumstances are occurring.Demographic Equation is a popular method for determining the change in an area's population over time. It is not without its flaws, but for showing overall population changes it works well. If you are studying demography, you are likely going to use the equation quite a bit. Even if you are not studying demography, it is a helpful equation to know.The two words that make up "demography" are "demo" and "graphy." "Demo" translates to "the people", while "graphy" means "measurement." Demography, essentially, is a measurement of people. (MORE)

# Understanding the Earth's Equator

You have probably heard the term "equator" your whole life, but you may not be familiar with the scientific and geographic definitions. Understanding the reason why The Equato…r is an important term of measurement on the earth will help you to better visualize the globe. It will not only help you with visual reference, but it will also help you to recognize temperature and weather patterns in different sections of the earth. By knowing the answers to a few basic questions, you can begin to fully comprehend the importance of the earth's equator.Most of the time, when you picture The Equator, the first visual that comes to mind is the line that is drawn around the middle of the globe. While The Equator is in the middle of the globe, it is not a physical line. Instead, the line represents the section of the earth that is exactly halfway between the north and south poles. In dealing with lines of latitude, it is set at zero degrees, which makes it the base point for the other latitude lines, as well as for measurements.Seasons and temperatures are all based on solar cycles. They depend on the placement of the earth and its rotational axis during a certain time of year in relation to the sun. The northern hemisphere will tilt toward the sun during the summer months and away during the winter months, causing the warmer and cooler temperatures, respectively. On the other hand, the southern hemisphere does the exact opposite, which is why the seasons are reversed.As discussed before, the northern and southern hemispheres experience seasons in reverse because of the way they tilt toward and away from the sun. However, The Equator is almost always an equal distance away from the sun. Because the Earth maintains an equal distance from the sun, seasonal changes happen to a much lesser degree at this global position. The temperature is normally in the higher ranges year-round. However, weather and temperatures at The Equator can vary depending on altitude and relative proximity to the ocean.When you picture the globe and The Equator you think about a ball with a line straight through the middle. Now, take that ball and tilt it at a forty-five degree angle. The points at the top and the bottom of the earth represent the north and south poles. However, they are not directly perpendicular, which means they are not one hundred percent either north or south. The poles represent the points of the earth that stand still while the rest of the sphere rotates. The earth is rotating on its side, and The Equator is the halfway measurement between the two rotational axis points.The Equator of the earth is an important tool in measuring the distance between the north and south poles. It helps to establish a line that can be set and used for mapping the globe and navigation purposes. However, this line can provide more information. It also determines how much sun certain sections of the earth receive at various times. The tilt gives the poles periods of extremes and can lead to times when during the summer the sun shines for days on end. Similarly, during the winter, some days may have no daylight at all at the poles.The earth's equator passes through the widest section of the globe. However, its presence is mostly over water and not land mass. More than 78% of the area underneath the equator is water. (MORE)

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In Education

# 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)

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In Technology

# 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 th…is 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)

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# 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)