First order partial differential equations ppt

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# First order partial differential equations ppt

In mathematicsthe method of characteristics is a technique for solving partial differential equations. Typically, it applies to first-order equationsalthough more generally the method of characteristics is valid for any hyperbolic partial differential equation.

The method is to reduce a partial differential equation to a family of ordinary differential equations along which the solution can be integrated from some initial data given on a suitable hypersurface. For a first-order PDE partial differential equationthe method of characteristics discovers curves called characteristic curves or just characteristics along which the PDE becomes an ordinary differential equation ODE.

Once the ODE is found, it can be solved along the characteristic curves and transformed into a solution for the original PDE. For the sake of simplicity, we confine our attention to the case of a function of two independent variables x and y for the moment.

Consider a quasilinear PDE of the form.

### Partial Differential Equation.ppt

A normal vector to this surface is given by. As a result,  equation 1 is equivalent to the geometrical statement that the vector field. In other words, the graph of the solution must be a union of integral curves of this vector field.

These integral curves are called the characteristic curves of the original partial differential equation and are given by the Lagrange-Charpit equations .

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A parametrization invariant form of the Lagrange-Charpit equations  is:. For this PDE to be linearthe coefficients a i may be functions of the spatial variables only, and independent of u. For it to be quasilineara i may also depend on the value of the function, but not on any derivatives. The distinction between these two cases is inessential for the discussion here.

Equations 2 and 3 give the characteristics of the PDE. The above equation may be written as. We must distinguish between the solutions to the ODE and the solutions to the PDE, which we do not know are equal a priori. Letting capital letters be the solutions to the ODE we find. From this, we find the above is bounded as. Suppose that u is any solution, and that. Along a solution, differentiating 4 with respect to s gives. Manipulating these equations gives. Writing these equations more symmetrically, one obtains the Lagrange-Charpit equations for the characteristic.

Geometrically, the method of characteristics in the fully nonlinear case can be interpreted as requiring that the Monge cone of the differential equation should everywhere be tangent to the graph of the solution.

For a pedagogical way of deriving the Lagrange-Charpit equations see the chapter 4 at . As an example, consider the advection equation this example assumes familiarity with PDE notation, and solutions to basic ODEs. First, we find.The method for solving such equations is similar to the one used to solve nonexact equations. There, the nonexact equation was multiplied by an integrating factor, which then made it easy to solve because the equation became exact.

Multiplying both sides by.

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Integrating both sides gives the solution:. Note that the differential equation is already in standard form. Integrating both sides yields the general solution:. Thus the general solution of the differential equation can be expressed explicitly as. Example 4: Find the general solution of each of the following equations:. Figure 1.

### FIRST ORDER ODE.ppt

Rather than having x as the independent variable and y as the dependent one, in this problem t is the independent variable and x is the dependent one. Multiplying both sides of the differential equation by this integrating factor transforms it into. Thus, the position x of the object as a function of time t is given by the equation. Previous First Order Homogeneous Equations. Next Bernoullis Equation. Removing book from your Reading List will also remove any bookmarked pages associated with this title.

Are you sure you want to remove bookConfirmation and any corresponding bookmarks? My Preferences My Reading List. Differential Equations. First-Order Linear Equations. Adam Bede has been added to your Reading List!A PDE is an equation with derivatives of at least two variables in it.

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Let u be a function of x and y. There are several ways to write a PDE, e. The order is determined by the maximum number of derivatives of any term. A nonlinear PDE has the solution times a partial derivative or a partial derivative raised to some power in it. Most interesting problems are nonlinear and time dependent. Eig A are real.

Let P and Z denote the number of positive and zero eigenvalues of A. Well posed PDE if and only if A solution to the problem exists.

The solution is unique. The solution depends continuously on the problem data. Finite Whosiwhatsit Methods There are three common methods of producing a finite dimensional problem whose solution can be computed, which approximates the solution of the original, infinite dimensional problem: Finite elements Finite differences Finite volumes Each has its place, supporters, and detractors.

There are also other methods, e. Finite Differences Assume we have a uniform mesh with a point x in the interior. Taylor Series and Truncation Error Look at the difference between the approximation and the Taylor series.

When they do not match, there is a remainder, which is known as the truncation error. It is usually specified as O hp. There are n rows of blocks in A i. This is known as a 5 point operator. Choosing the right finite element method on a square right triangles with piecewise linear elements leads to the same matrix problem.Orthogonal trajectories.

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The term orthogonal means perpendicularand trajectory means path or cruve. Orthogonal trajectories, therefore, are two families of curves that always intersect perpendicularly. Example 1 : The electrostatic field created by a positive point charge is pictured as a collection of straight lines which radiate away from the charge Figure.

Using the fact that the equipotentials surfaces of constant electric potential are orthogonal the electric field lines, determine the geometry of the equipotenitials of a point charge. If the origin of an xy coordinate system is placed at the charge, then the electric field lines can be described by the family. The first step in determining the orthogonal trajectories is to obtain an expression for the slope of the curves in this family that does not involve the parameter c.

In the present case. Because this equation is separable, the solution can proceed as follows:. The equipotential and electric field lines for a point charge are shown in Figure 2. The first step is to determine an expression for the slope of the curves in this family that does not involve the parameter c. By implicit differentiation.

However, because. With a little algebra, the equation for this family may be rewritten:. This shows that the orthogonal trajectories of the circles tangent to the x axis at the origin are the circles tangent to the y axis at the origin! See Figure 3.

Radioactive decay. Some nuclei are energetically unstable and can spontaneously transform into more stable forms by various processes known collectively as radioactive decay. The rate at which a particular radioactive sample will decay depends on the identity of the sample. The rate at which a sample decays is proportional to the amount of the sample present.

Therefore, if x t denotes the amount of a radioactive substance present at time tthen. The positive constant k is called the rate constant for the particular radioisotope. The graph of this equation Figure 4 is known as the exponential decay curve:.After you enable Flash, refresh this page and the presentation should play. Get the plugin now. Toggle navigation.

## Partial Differential Equations - PowerPoint PPT Presentation

View by Category Toggle navigation. Products Sold on our sister site CrystalGraphics. Title: Partial Differential Equations. Description: www. Tags: chapman differential equations leland partial. Latest Highest Rated. In general, partial differential equations are much more difficult to solve analytically than are ordinary differential equations 3 What Does a PDE Look Like Let u be a function of x and y.

There are several ways to write a PDE, e. Eig A are real. Let P and Z denote the number of positive and zero eigenvalues of A. Parabolic Z gt 0 det A 0. Hyperbolic Z0 and P 1 or Z 0 and P n Ultra hyperbolic Z 0 and 1 lt P lt n The basic example of an elliptic partial differential equation is Laplaces Equation uxx - uyy 0 8 The Others The heat equation is the basic Hyperbolic ut - uxx - uyy 0 The wave equations are the basic Parabolic ut - ux - uy 0 utt - uxx - uyy 0 Theoretically, all problems can be mapped to one of these 9 What Happens Where You Cant Tell What Will Happen Types of boundary conditions Dirichlet specify the value of the function on a surface Neumann specify the normal derivative of the function on a surface Robin a linear combination of both Initial Conditions 10 Is It Worth the Effort?

Basically, is it well-posed?Copy embed code:. Automatically changes to Flash or non-Flash embed. WordPress Embed Customize Embed. URL: Copy. By: yumna. PowerPoint Presentation: There are six types of non-linear partial differential equations of first order as given below. Solve this differential equation and finally substitute gives the required solution. Note: This method is used only when it is possible to separate variables i. Directly substitute a in place of p and b in place of q in the given equation. Procedure: For the given PDE, let us consider the solution to be Substitute these values in the given equation, from which we can separate variables. Write the equation such that ,X and terms are on one side and similarlyY and terms are on the other side.

## Method of characteristics

PowerPoint Presentation: Let it be and Solve these equations; finally substitute in which gives the required solution.

Follow us on:. Go to Application. US Go Premium. PowerPoint Templates. Upload from Desktop Single File Upload. Post to :. URL :. Related Presentations :. Add to Channel. The presentation is successfully added In Your Favorites. Views: Like it 0. Dislike it 0. Added: November 22, Posting comment Post Reply Close. Edit Comment Close. Premium member. Presentation Transcript. PowerPoint Presentation: Solve this, get the result which will be in terms of X and Y, and the substitutewhich is the required solution.The order of an equation is not affected by any power to which the derivatives may be raised.

Now, do this exercises. Linear equation as those in which the dependent variable or variables and their derivatives do not occur as products, raised to power or in nonlinear function.

PDE - Canonical Forms of First Order Linear PDE

Solve the initial value problems of differential equations. Introduction When we solve algebraic eqn, we expect the solution to be a number eg. For DE, the solution of a DE is therefore, not a single value Or one from a set of values but a function or a family of function. In the special case in which all the boundary conditions are given at the same value of the independent variables the boundary conditions are called initial conditions. Determine and find the solutions for case non initial value problems of separable equations.

It is so-called because we rearrange the equation to be solved such that all terms involving the dependent variable appear on one side of the equation, and all terms involving the independent variable appear on the other.

Integration completes the solution. Elementary Analytical Solution Methods : But, some DE, while not being in separable form, can be Separable Equations transformed, by means of a substitution, into separable equations. For example : dx 2 t. Differentiate equation 2. Determine and find the solutions for case initial or non initial value problems of linear equations.

Rearrange the equation to be in the form of. A chemical reaction is governed by the differential equation dx 2 K 5 x dt.

The initial concentration is zero and the concentration at time 5s is found to be 2. Determine the reaction rate constant K and find the concentration at time 10s and 50s. What is the ultimate value of the concentration?

A skydivers vertical velocity is governed by the differential equation d m mg K 2 dt. If the skydiver leaves her aeroplane at time t 0 with zero vertical velocity find at what time she reaches half her final velocity. Answer: t. A chemical A is formed by an irreversible reaction from chemicals B and C. 