Numerical solution of second-order hyperbolic telegraph equation via new cubic trigonometric B-splines approach

In this paper, a direct meshless method (DMM), which is based on the radial basis function, is developed to the numerical solution of the two-dimensional second-order hyperbolic telegraph equations. Since these hyperbolic telegraph equations are time dependent, we present two schemes for the basis functions from radial and nonradial aspects. The first scheme is fulfilled by considering time variable as normal space variable to construct an “isotropic” space-time radial basis function. The other scheme considered a realistic relationship between space variable and time variable which is not radial. The time-dependent variable is treated regularly during the whole solution process and the hyperbolic telegraph equations can be solved in a direct way. Numerical experiments performed with the proposed numerical scheme for several two-dimensional second-order hyperbolic telegraph equations are presented with some discussions, which show that the DMM solutions are converging very fast in comparison with the various existing numerical methods.

Numerical solutions of the second-order one-dimensional hyperbolic telegraph equations are presented using the radial basis functions. The purpose of this paper is to propose a simple novel direct meshless scheme for solving hyperbolic telegraph equations. This is fulfilled by considering time variable as normal space variable. Under this scheme, there is no need to remove time-dependent variable during the whole solution process. Since the numerical solution accuracy depends on the condition of coefficient matrix derived from the radial basis function method. We propose a simple shifted domain method, which can avoid the full-coefficient interpolation matrix easily. Numerical experiments performed with the proposed numerical scheme for several second-order hyperbolic telegraph equations are presented with some discussions.

In this paper, under the framework of Extended Chebyshev space, four new generalized quasi cubic trigonometric Bernstein basis functions with two shape functions α(t) and β(t) are constructed in a generalized quasi cubic trigonometric space span{1,sin2t,(1−sint)2α(t),(1−cost)2β(t)}, which includes lots of previous work as special cases. Sufficient conditions concerning the two shape functions to guarantee the new construction of Bernstein basis functions are given, and three specific examples of the shape functions and the related applications are shown. The corresponding generalized quasi cubic trigonometric Bézier curves and the corner cutting algorithm are also given. Based on the new constructed generalized quasi cubic trigonometric Bernstein basis functions, a kind of new generalized quasi cubic trigonometric B-spline basis functions with two local shape functions αi(t) and βi(t) is also constructed in detail. Some important properties of the new generalized quasi cubic trigonometric B-spline basis functions are proven, including partition of unity, nonnegativity, linear independence, total positivity and C2 continuity. The shape of the parametric curves generated by the new proposed B-spline basis functions can be adjusted flexibly.

Trigonometric B-spline functions of higher degrees have advantages over lower ones since they can be used as approxi-mate functions in the numerical methods if the differential equation include higher order derivatives. Thus the collocation method based on higher degree trigonometric B-splines is set up to get numerical solutions of the Coupled Burgers’ equation. The un-knowns of the coupled Burgers equation are integrated in time by the aid of the Crank-Nicolson method. Resulting time-integrated coupled Burgers’ equation is discretized using the trigonometric B-spline collocation method. One initial and boundary value problem of the Coupled Burgers’ equation is studied by the proposed technique.

Solution of the second-order one-dimensional hyperbolic telegraph equation by using the dual reciprocity boundary integral equation (DRBIE) method

Objectives: This paper aims to compute the approximate solution of one dimensional (1D) hyperbolic telegraph equation with appropriate primary and limiting conditions. Methods/Statistical Analysis: To find the approximate solution, two different modified spline basis function are used with the differential quadrature method and splines are used to compute the weighting coefficients and thus the equation is transformed to a set of first order conventional differential equations which is further solved by the SSP-RK43 method. Three test problems of this equation are simulated to establish the precision and usefulness of the proposed scheme. Findings: The obtained numerical results are found to be good in terms of accuracy, efficiency and simplicity. To validate the computed results using proposed scheme, various comparisons at different time levels has been done in the form of and errors. These errors are compared and enlisted in the form of tables with computed errors enlisted in literature. Application/Improvements: Being an important equation of nuclear material science, one dimensional (1D) hyperbolic telegraphs equation needs to take care in the sense of better numerical solution. In this context, a successful effort has been done in this research article by proposing a hybrid numerical scheme.

Numerical methods for nonlinear second-order hyperbolic partial differential equations. II – Rothe’s techniques for 1-D problems

In this paper, we develop a numerical solution based on nonpolynomial B-spline (trigonometric B-spline) collocation method for solving time-dependent equations involving PHI-Four and Allen–Cahn equations. A three-time-level implicit algorithm has been derived. This algorithm combines the trigonometric B-spline interpolant and the \(\theta \)-weighted scheme for space and time discretization, respectively. Convergence analysis is discussed and the accuracy of the presented method is \(O( {\tau ^{2}+h^{2}} ).\) Applying von Neumann stability analysis, the proposed technique is shown to be unconditionally stable. Three test problems are demonstrated to reveal that our method is reliable, efficient and very encouraging.

In this paper, we propose an efficient numerical scheme for the approximate solution of a time fractional diffusion-wave equation with reaction term based on cubic trigonometric basis functions. The time fractional derivative is approximated by the usual finite difference formulation, and the derivative in space is discretized using cubic trigonometric B-spline functions. A stability analysis of the scheme is conducted to confirm that the scheme does not amplify errors. Computational experiments are also performed to further establish the accuracy and validity of the proposed scheme. The results obtained are compared with finite difference schemes based on the Hermite formula and radial basis functions. It is found that our numerical approach performs superior to the existing methods due to its simple implementation, straightforward interpolation and very low computational cost. A convergence analysis of the scheme is also discussed.

We consider the Cauchy problem for nonlinear hyperbolic partial differential equations of second order. Then the Cauchy problem does not generally admit a classical solution in the large, that is to say, singularities generally appear in finite time. The typical example of singularity is “shock wave”. Our problem is to extend the solution beyond the singularities. Though there are many papers concerning this subject, the existence and uniqueness of weak solution is even now unsolved. In this talk we consider the above problem from geometrical point of view.KeywordsWeak SolutionCauchy ProblemContact StructureNonlinear Wave EquationJump DiscontinuityThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

At first hyperbolic numbers and hyperbolic complex functions are introduced. On this basis existence and uniqueness of solutions of oblique derivative problems for some hyperbolic complex equations of second order are discussed by a complex analytic method.KeywordsHyperbolic complex functionsoblique derivative problemshyperbolic equations

Numerical solution of hyperbolic telegraph equation by cubic B-spline collocation method

The application of cubic trigonometric B-spline to the numerical solution of the hyperbolic problems

A trigonometric quintic B-spline method is proposed for the solution of a class of turning point singularly perturbed boundary value problems (SP-BVPs) whose solution exhibits either twin boundary layers near both endpoints of the interval of consideration or an interior layer near the turning point. To resolve the boundary/interior layer(s) trigonometric quintic B-spline basis functions are used with a piecewise-uniform mesh generated with the help of a transition parameter that separates the layer and regular regions. The proposed method reduces the problem into a system of algebraic equations which can be written in matrix form with the penta-diagonal coefficient matrix. The well-known fast penta-diagonal system solver algorithm is used to solve the system. The method is shown almost fourth-order convergent irrespective of the size of the diffusion parameter ϵ. The theoretical error bounds are verified by taking some relevant test examples computationally.

A comparison study of meshfree techniques for solving the two-dimensional linear hyperbolic telegraph equation