Undetermined coefficients method

One solution of the forced response xf is the undetermined coefficient method. Assuming the forced response has the same form of source function f(t) but a different coefficient, putting this trial forced response into the differential equation yields the coefficients in the forced response xf. 

This is an example using the undetermined coefficient method [1]. 

As a result, a considerable amount of effort has been expended in designing various methods for providing difference approximations of differential equations. The simplest and, in a certain sense, natural method is connected with selecting a, suitable pattern and imposing on this pattern a difference equation with undetermined coefficients which may depend on nodal points and step. Requirements of solvability and approximation of a certain order cause some limitations on a proper choice of coefficients. However, those constraints are rather mild and we get an infinite set (for instance, a multi-parameter family) of schemes. There is some consensus of opinion that this is acceptable if we wish to get more and more properties of schemes such as homogeneity, conservatism, etc., leaving us with narrower classes of admissible schemes. 

This solution also is found readily by the method of undetermined coefficients. 

Method of Undetermined Coefficients If Q x) is a product or linear combination of products of the functions a , x p a positive integer or zero) cos cx and sin cx, this method may be used. The "families [a ], [e " ], [sin cx, cos cx] and [x, xf, . .., x, 1] are defined for each of the above functions in the following way The family of a term f is the set of all functions of whichand all operations of the form cos c x + y), sin c x + y), (x + yf onf and their linear combinations result in. The technique involves the following steps (1) Solve the homogeneous system. (2) Construct the family of each term. (3) If the family has no representative in the homogeneous solution, assume i/J is a linear combination of the families of each term and determine the constants so that the equation is satisfied. (4) If a family has a representative in the homogeneous solution, multiply each member of the family by the smallest integral power of x for which all such representatives are removed and revert to step 3. 

In the T-method one admits that one cannot solve the linear differential equation exactly, and inserts an error term tF (x), where t is an a priori undetermined coefficient, and P x) is a known orthogonal polynomial of order n. Thus, one writes... 

Several methods exist for finding particular solutions. Laplace transform methods are probably the most convenient, and we use them in Part Two. Here we present the method of undetermined coefficients. It consists of assuming a particular solution... 

In 1926-27, Jeffreys (J3, J4) attempted to extend Rayleigh s result to a more realistic set of boundary conditions, first using finite differences to obtain successive approximations to the solution of Eq. (31) and later using a method of undetermined coefficients for the case corresponding to two solid conducting boundaries. In the latter manner, he computed a critical Rayleigh number of 1709.5. 

H3) obtained functional approximations for the velocity components by assuming a trial stream function in the Navier-Stokes equations and evaluating the undetermined coefficients from the boundary conditions using the method of residuals. Their relationship can be presented in the form of... 

The complete solution of this equation can be found by several methods e.g. Laplace transforms, method of undetermined coefficients, see Reference 3. The solution for i is. 

So, by inserting the constraints in the energy equation, the conditions of equilibrium can be obtained via the chain rule. However, the method of undetermined coefficients is more versatile. ... 

Method of Undetermined Coefficients This is a rather evolutionary technique, which builds on the functional form taken by fix). 

Linear Equations of Higher Order 73 1. Method of Undetermined Coefficients... 

The reader can clearly see the speed and efficiency of this method compared to the tedious treatment required by the Method of Undetermined Coefficients, as done in Example 2.19. 

We can verify this using the method of Undetermined Coefficients. Repeated... 

Had we used the Method of Undetermined Coefficients, it would have been necessary to make the first guess (to insure linear independence from the complementary solutions which are sin(jr) and cos(Af))... 

It was stated at the outset that analytical methods for linear difference equations are quite similar to those applied to linear ODE. Thus, we first find the complementary solution to the homogeneous (unforced) equation, and then add the particular solution to this. We shall use the methods of Undetermined Coefficients and Inverse Operators to find particular solutions. 

This would have required considerably more algebra using the method of undetermined coefficients. To check for linear independence, the roots of the... 

Since the residual is zero, if Uf x) is the exact solution of Eq. 9, it follows that the goal of all numerical methods is to choose the undetermined coefficients, so as to make R(x a ) as small as possible. The different kinds of spectral methods differ mainly in the minimization strategies. There are two basic types of spectral methods pseudospectral methods and the Galerkin methods. 

The function yq can be thought of as the best candidate for the particular solution, and the method of undetermined coefficients can be used to find yp. 

In Section 3.4, we employed the undetermined coefficients (equivalent— annihilation) or variation of parameters methods to solve the nonhomoge-... 

The method of undetermined coefficients is effective when the matrix, A in Equation 3.155 is constant and the vector f has a special form (exponentials, polynomials, sines, and cosines). However as we know from our experience with second-order equations, the method of variation of parameters is more general. This method is expected to work even when A depends rai t and f belongs to a much larger class of vector-valued functions. 

Use the method of undetermined coefficients to find a particular solution of... 

The particular constant C3 can be evaluated by the method of undetermined coefficient, while the arbitrary constants ki and 2 must be evaluated from the transformed boundary conditions on x. Following this procedure, u x, s) is given by... 

When the term e appears in the right-hand side of equation (A-64), then we first want to eliminate the e term using the Commuting Law and then follow the method of undetermined coefficients on the remaining polynomial. 

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