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

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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.

# 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.

# 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 (), 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.

# What is the difference between a linear function and a quadratic equation?

A linear equation describes a line like 2x+1=y. If you were to graph that equation, then it would give you a line. A quadratic equation is like x^2+2x+1=y. Graphing this equat…ion would give you a U shaped graph called a parabola.

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# What are linear equations?

Linear equations are equations whose only terms are constants and/or single variables raised to the first power. More than one variable is allowed in a linear equation, but it… is not allowed to be multiplied with another variable. Constants are allowed to be multiplied to variables in linear equations. These equations are called "linear" due to the fact that their solution set forms a line when represented in classic Euclidean space, e.g. when graphed on the mutually perpendicular x, y, and z axes of the Cartesian coordinate system. Here are three examples of linear equations: Slope-intercept form: y = mx + b, where x is the independent variable, y is the dependent variable, and m and b are constants. This representation of a linear equation is useful because the slope of the line formed by its solution set is m. Point-slope form: y - y1 = m(x - x1), where x is the independent variable, y is the dependent variable, and m is the constant slope. The point (x1,y1) is included in this form to explicitly show that the independent distances of x and y between two points are proportional to each other by the proportionality constant, m, the slope. Intercept form: x/a + y/b = 1, where x and y are variables and a and b are non-zero constants. This form is useful because the x and y intercepts, i.e. the points on a graph where this line crosses the x and y axes, are a and b, respectively.

# Whats Difference between linear equation to linear function?

A linear equation contains only the first power of the unknown quantity. Thus, 5x - 3 = 7 and x/6 = 4 are both linear equations. Linear equations have only one solution which …is the value of the unknown that when substituted in the equation , makes the left hand side equal to the right hand side. Linear functions have the same limitation in terms of only containing the first power of the unknown quantity. They yield graphs that are straight lines and thus the name 'linear' is used. A simple linear function is f:x →2x + 1. This can also be written as f(x) = 2x + 1 or another identifying letter used such as y = 2x + 1. Consequently, for different values of the unknown quantity (in this case 'x') then the function also yields a different value.

# What are the difference between linear equation and quadratic equation?

A quadratic equation must be able to be written in the form: y = ax2 + bx + c where a is not equal to zero. The graph will be a parabola. There must be a "squared" …term and no larger exponent than "2". A linear equation will consist of variables only to the first power and the graph will be a straight line. Y = mx + b is an example of a liear equation where m will represent the slope and b will represent the y-intercept.

# How does solving a literal equation differ from solving a linear equation?

Because linear equations are based on algebra equal to each other whereas literal equations are based on solving for one variable.

# What is an linear equation?

It is a certain type of equation used in Algebra. It uses letters to replace numbers and the aim of cracking the equation is to find the value of the letter. Such as: 4b+2=26… ....you would need to find the value of 'b'. -2( 4b= 24 )-2 ( b=6 )divided by 4 I've just solved the equation.

# What are the different kinds of systems of linear equations?

Standard form: Ax + By = C, where A and B are non-zero constants. Slope-intercept form: y = mx + b, where m is the slope, and b is the y-intercept.

# Why are linear equations called linear equations?

Because, if plotted on a Cartesian plane, all solutions to the equation would lie on a straight line.

# Differences between linear and non linear?

If you graph a Linear equation it will be a strait line. If it doesn't come out strait, its not linear. Also a linear equation can be put into y=mx+b, with mx meaning the slop…e and b meaning Y-intersept.

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# How are linear inequalitys differant from linear equations?

Inequalities have greater than, less than, greater than or equal to, or less than or equal to signs. Equations have an equal sign.

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In Science

# Why are linear equations called linear?

It deals with lines on a graph, part of an ordered pair ,a steady increase in resultant answer

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# Why are linear equation named linear equation?

Y = 5X - 3 It form a linear function; a line.

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In Science

# 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 satisfie…s 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.

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In Algebra

# What is the difference of Linear Equation?

Difference in math means to subtract. To subtract two equations match up the variable and do subtraction. This is a method of solving systems of equations. Example: … 2x+3y=8 5x+3y=11 If you subtract the two you are left with -3x+0y=-3 Therefore x must equal 1 and through substitution you 2(1)+3y=8 which is y=2.