Wave polynomials, transmutations and Cauchy’s problem for the Klein–Gordon equation

Abstract

a b s t r a c tWe prove a completeness result for a class of polynomial solutions of the wave equation called wave polynomials and construct generalized wave polynomials, solutions of the Klein-Gordon equation with a variable coefficient. Using the transmutation (transformation) operators and their recently discovered mapping properties we prove the completeness of the generalized wave polynomials and use them for an explicit construction of the solution of the Cauchy problem for the Klein-Gordon equation. Based on this result we develop a numerical method for solving the Cauchy problem and test its performance.

Solution of parabolic free boundary problems using transmuted heat polynomials

“…Because we may need the values of the formal powers at arbitrary points , we approximated the formal powers by splines passing through their values on the selected mesh. Spline integration can be used as well for the construction of the formal powers; cf Khmelnytskaya et al See also Kravchenko et al for the discussion of other possible methods. Compute the coefficients B n and C n given by and and the respective functions g 1 and g 2 . Because these conditions are independent of the free boundary (ie, of the choice of ), they need to be computed only once. Construct a function that solves problem for each given set of coefficients . Construct the value function using . Solve the constrained minimization Problem using any suitable algorithm; see, eg, Nocedal and Wright .…”

Analytic Approximation of Solutions of Parabolic Partial Differential Equations with Variable Coefficients

“…The formal powers were constructed like in [6] using the spapi and fnint MATLAB routines from the spline Toolbox. Then the transmuted heat polynomials (20) were calculated.…”

“…In the paper, the Dirichlet boundary conditions are considered; however the proposed method can be easily extended onto other standard boundary conditions. The complete system of solutions is constructed with the aid of the transmutation operators relating (1) to the heat equation (see, e.g., [4][5][6]). The possibility of constructing complete systems of solutions by means of transmutation operators was proposed and explored in [4], and the approach developed in [4] requires the knowledge of the transmutation operators.…”

Generalized wave polynomials and transmutations related to perturbed Bessel equations

“…We precise several known results concerning its mapping properties and use its composition with Y l , t (same operator but with respect to the variable t ) in order to obtain a system of solutions U k of the equation as images of the wave polynomials (a family of polynomial solutions of the wave equation. ()) Second, using a recent result from regarding a mapping property of the operator T , we obtain the system of the generalized wave polynomials u k for as images under the action of T on U k obtained on the previous step.…”

“…U(x, t) = 0 ( 4 ) as images of the wave polynomials (a family of polynomial solutions of the wave equation. 5,6,9 ) Second, using a recent result from 10 regarding a mapping property of the operator T, we obtain the system of the generalized wave polynomials u k for (3) as images under the action of T on U k obtained on the previous step. In order to achieve the aim (b) we study the preimage k of the transmutation kernel K, that is k ∶ = T −1 [K] which satisfies a Goursat problem for (4).…”