Dirac’s formalism combined with complex Fourier operational matrices to solve initial and boundary value problems

Boundary value problems (BVP) arise in many mathematical model of frontier science. In short, a BVP is a second order ordinary differential equation with boundary conditions prescribed at two distinct points. Usually, we prescribe three types of boundary conditions: Dirichlet, Neumann and Mixed. Among them mixed boundary conditions is little difficult to approach. One of the method to solve the BVP with mixed boundary conditions is the shooting method. In this method, we first convert the BVP into an initial value problems (IVP) and then solve it. We then check whether the solution of IVP is a solution of the BVP. There are standard ways of solving IVPs. Recently, some alternate methods are also used to solve them. For example, several authors use Haar wavelet. Frames are considered as an extension to wavelet. In this paper, we construct operational matrices from a frame and then apply them to solve the IVP which is obtained from the given BVP by shooting method. This method is simple and easy to use because we construct the operational matrix once, store it and then apply it any times we want. Moreover, we also show that the given operational matrix method converges quadratically.

The shifted Chebyshev polynomial approximation is employed to solve the linear, constant parameter, ordinary differential equations of initial or two-point boundary value problems. An effective recursive algorithm is developed to calculate the expansion coefficients of the shifted Chebyshev series. An effective transformation is proposed to transform the two-point boundary value problem into an initial value problem. An illustrative example is included to show that the computational results are accurate.

On initial value and terminal value problems for Hamilton–Jacobi equation

Several orthogonal polynomials have limit forms in which Hermite polynomials show up. Examples are limits with respect to certain parameters of the Jacobi and Laguerre polynomials. In this paper we are interested in more details of these limits and we give asymptotic representations of several orthogonal polynomials in terms of Hermite polynomials. In fact we give finite exact representations that have an asymptotic character. From these representations the well-known limits can be derived easily. Approximations of the zeros of the Gegenbauer polynomials $C_n^{gamma(x)$ and Laguerre polynomials $L_n^{alpha(x)$ are derived (for large values of $gamma$ and $alpha$, respectively) in terms of zeros of the Hermite polynomials and compared with numerical values. We also consider the Jacobi polynomials and the so-called Tricomi-Carlitz polynomials.

An initial boundary value problem for a quasilinear equation of pseudoparabolic type with a nonlinear boundary condition of the Neumann–Dirichlet type is investigated in this work. From a physical point of view, the initial boundary value problem considered here is a mathematical model of quasistationary processes in semiconductors and magnets, which takes into account a wide variety of physical factors. Many approximate methods are suitable for finding eigenvalues and eigenfunctions in problems where the boundary conditions are linear with respect to the desired function and its derivatives. Among these methods, the Galerkin method leads to the simplest calculations. On the basis of a priori estimates, we prove a local existence theorem and uniqueness for a weak generalized solution of the initial boundary value problem for the quasilinear pseudoparabolic equation. A special place in the theory of nonlinear equations is occupied by the study of unbounded solutions, or, as they are called in another way, blow‐up regimes. Nonlinear evolutionary problems admitting unbounded solutions are globally unsolvable. In the article, sufficient conditions for the blow‐up of a solution in a finite time in a limited area with a nonlinear Neumann–Dirichlet boundary condition are obtained.

In this paper we presents further developments regarding the enrichment of the basic Theory of Order Completion as presented in Oberguggenberger and Rosinger (Solution of continuous nonlinear PDEs through order completion, North-Holland, Amsterdam, 1994). In particular, spaces of generalized functions are constructed that contain generalized solutions to a large class of systems of continuous, nonlinear PDEs. In terms of the existence and uniqueness results previously obtained for such systems of equations (van der Walt, Acta Appl. Math. 103:1–17, 2008), one may interpret the existence of generalized solutions presented here as a regularity result. Furthermore, it is indicated how the methods developed in this paper may be adapted to solve initial and/or boundary value problems. In particular, we consider the Navier-Stokes equations in three spacial dimensions, subject to an initial condition on the velocity. In this regard, we obtain the existence of a generalized solution to a large class of such initial value problems.

Global existence of smooth solutions is proved for an effective theory of bubbly liquids for either the initial value problem or initial boundary value problem in one dimension. This shows that the theory does not describe shock waves or bubble collapse. Since the analysis is not for the steady boundary value problem, there is no discussion of resonance. The proof uses a semilinear form of the equations to get local existence. A priori bounds resulting from energy conservation and a nonlinear Gronwall‐like inequality are then derived to prove global existence.

<p>In this paper, we present a framework to determine Haar solutions of Bratu-type equations that are widely applicable in fuel ignition of the combustion theory and heat transfer. The method is proposed by applying Haar series for the highest derivatives and integrate the series. Several examples are given to confirm the efficiency and the accuracy of the proposed algorithm. The results show that the proposed way is quite reasonable when compared to exact solution.</p>

A computer program is described which has been developed for obtaining approximate solutions to linear initial and boundary value problems involving differential equations. The program can be used for ordinary curve fitting as well. For each problem, input to the program includes:1. The equations (in symbolic form) to be satisfied - the differential equations, equations describing auxiliary conditions such as boundary conditions, etc.2. A numerical description of the regions in which each of the equations are to be satisfied.3. Sets of functions (in symbolic form) to be used in linear combinations to approximate the solution functions.Given the above input, the program generates an approximation to the solutuon of the specified problem in terms of the specified functions which is optimum in the least squares sense. A number of examples illustrating the use of the program are included in the paper.

Due to the applications of Boundary Value Problems (BVPs) in real-life phenomena, the shooting method has proven itself useful and efficient in handling BVP. The shooting method works by first reducing the BVP to an Initial Value Problem (IVP), then one/two initial value guesses are made. The IVPs are then solved using an iterative solution, and this process is then repeated until the second boundary condition is reached to a satisfactory level. The iterative formula used in this study is the Euler’s method while the initial value estimation method used is the Secant method (interpolation formula).

Numerical solution of two-point boundary value problems

A rigorous methodology for the analysis of initial boundary value problems on the half-line, 0 < x < ∞, t > 0, for integrable nonlinear evolution PDEs has recently appeared in the literature. As an application of this methodology the solution q(x, t) of the sine-Gordon equation can be obtained in terms of the solution of a 2 × 2 matrix Riemann–Hilbert problem. This problem is formulated in the complex k-plane and is uniquely defined in terms of the so-called spectral functions a(k), b(k), and B(k)/A(k). The functions a(k) and b(k) can be constructed in terms of the given initial conditions q(x, 0) and qt(x, 0) via the solution of a system of two linear ODEs, while for arbitrary boundary conditions the functions A(k) and B(k) can be constructed in terms of the given boundary condition via the solution of a system of four nonlinear ODEs. In this paper, we analyse two particular boundary conditions: the case of constant Dirichlet data, q(0, t) = χ, as well as the case when qx(0, t), sin (q(0, t)/2), and cos(q(0, t)/2) are linearly related by two constants χ1 and χ2. We show that for these particular cases, the system of the above nonlinear ODEs can be avoided, and B(k)/A(k) can be computed explicitly in terms of { a(k), b(k), χ} and {a(k), b(k), χ1, χ2}, respectively. Thus, these ‘linearizable’ initial boundary value problems can be solved with absolutely the same level of efficiency as the classical initial value problem of the line.

A closed formula for and interpretation of exact quantum mechanical off-diagonal matrix elements of powers of r is given in terms of orthogonal polynomials. The compact formulae for r-k-2 and rk-1 are in exact correspondence to the semiclassical (diagonal) expectation values and are most suitable for the practical evaluation of the matrix elements. The expressions for < nl | r-k-2 | l'n > and < nl | rk-1 | l'n > consist only of the minimum of n-l or k+1 terms. An off-diagonal generalization of the well known diagonal inversion formula is derived and the matrix elements can thus be evaluated for arbitrary (complex) powers k. Also, an analytic continuation to an arbitrary effective n* is possible. It is thus established that, complementary to the O(2,1) approach by Armstrong (1970, 1971), orthogonality properties for purely radial quantities can be obtained without the introduction of a fictitious angular variable. Using the analytical results obtained, certain deficiencies of the semiclassical Coulomb (or WKB) approximation are discussed. The normalization error involved in the WKB approximation is identified analytically and an improved version of the semiclassical limit, a correspondence limit, is derived. Furthermore, it is established that, like the exact results, the WKB matrix elements obey a three-term recursion relation and an inversion formula. For low k some explicit formulae for exact matrix elements are tabulated.

In the literature so far, for Caputo fractional boundary value problems of order 2q when 1&lt;2q&lt;2, the problems use the same boundary conditions of the integer-order differential equation of order ‘2’. In addition, they only use the left Caputo derivative in computing the solution of the Caputo boundary value problem of order 2q. Further, even the initial conditions for a Caputo fractional differential equation of order nq use the corresponding integer-order initial conditions of order ‘n’. In this work, we establish that it is more appropriate to use the Caputo fractional initial conditions and Caputo fractional boundary conditions for sequential initial value problems and sequential boundary value problems, respectively. It is to be noted that the solution of a Caputo fractional initial value problem or Caputo fractional boundary value problem of order ‘nq’ will only be a Cnq solution and not a Cn solution on its interval. In this work, we present a methodology to compute the solutions of linear sequential Caputo fractional differential equations using initial and boundary conditions of fractional order kq, k=0,1,…(n−1) when the order of the fractional derivative involved in the differential equation is nq. The Caputo left derivative can be computed only when the function can be expressed as f(x−a). Then the Caputo right derivative of the same function will be computed for the function f(b−x). Further, we establish that the relation between the Caputo left derivative and the Caputo right derivative is very essential for the study of Caputo fractional boundary value problems. We present a few numerical examples to justify that the Caputo left derivative and the Caputo right derivative are equal at any point on the Caputo function’s interval. The solution of the linear sequential Caputo fractional initial value problems and linear sequential Caputo fractional boundary value problems with fractional initial conditions and fractional boundary conditions reduces to the corresponding integer initial and boundary value problems, respectively, when q=1. Thus, we can use the value of q as a parameter to enhance the mathematical model with realistic data.

It has long been known that certain integral transforms and Fourier-type series can be used to solve many classical boundary and initial value problems in separable coordinate systems. More recently, it has been shown that these classical transforms and series are spectral representations associated with an ordinary differential system which results on applying separation of variables to the given boundary value problem. This has been the basis for recent work concerned with systematically generating the proper spectral representation needed to solve a given problem. See [1]-[4] for a list of references. &#13;\n&#13;\nWe shall consider the problem of finding the associated spectral representation when the resulting ordinary differential system has the eigenvalue parameter occurring in both the equation and one boundary condition. Moreover, the differential equation is to be satisfied on a semi-infinite interval, thus leading to a singular problem which does not seem to have been studied before. In §2 by using a transformation due to B. Friedman (which we modify appropriately for our singular case), we first give a formal derivation of the spectral representation, and then we rigorously prove the result. &#13;\n&#13;\nIn §3 our representation is applied to solve an initial-boundary value problem arising in the theory of diffusion and heat flow in one dimension. We should bear in mind that even in cases where solutions are already known, our method systematically yields alternative representations which are often more rapidly convergent and from which asymptotic expansions of solutions with respect to parameters can often be found. In the problem to be considered a representation of the solution can also be found easily by a straight-forward application of the Laplace transform in time t, while the new transform derived in §2 yields the solution when applied in space x. The new transform, however, can be applied when the coefficients in the boundary value problem are time dependent, a situation which, in general, precludes the use of the Laplace transform.. As a general rule [1] there should be a spectral representation associated with each ordinary differential system resulting from applying separation of variables to the original boundary value problem, and each spectral representation should lead to a different representation of the solution of the original problem.