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

# 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.

# Is y equals mx plus b a non linear or linear equation?

In most situations, the answer that is looked for is linear. However, since y=mx+b does not follow superposition, it is in fact, nonlinear. Proof: y1=mx1+b y2=mx2+b x3=Ax1+Bx…2 y3=m(Ax1+Bx2)+b=mAx1+mAx2+b this does not equal Ay1+By2, making it nonlinear

# 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.

# 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.

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

A linear scale is a scale with equal divisions for equal vales, for example a ruler. A non linear scale is where the relationship between the variables is not directly proport…ional.

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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.

# 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.

# 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.

# Example equations of linear equations?

y=3x+2 y-4x=9 These are examples of linear equations which is a first degree algebraic expression with one, two or more variables equated to a constant. So x=2 is a lin…ear equation as is y=1 but x2 =1 is not since the variable, x , has degree 2.

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

Linear means in order Non-Linear means Organic Two examples of nonlinear text: a website with hyperlinks, a choose-your-own adventure book.

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# 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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# 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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# What is the difference between linear and non-linear data structures?

A linear data structure has a single path from the first element to the last. Thus arrays and lists are linear structures. A non-linear data structure has one or more "bra…nches"; forks in the road. This means there is more than one route through the structure and there may be more than one endpoint. A binary tree is an example of a non-linear data structure.

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# How are linear equations and linear inequalities similar?

A linear equation corresponds to a line, and a linear inequality corresponds to a region bounded by a line. Consider the equation y = x-5. This could be graphed as a line goin…g through (0,-5), (1,-4), (2,-3), and so on. The inequality y > x-5 would be the region above that line.