Linear Initial Value Problems Research Articles

Analysis of order reduction when integrating linear initial boundary value problems with Lawson methods

In this paper, a thorough analysis is given for the order which is observed when integrating evolutionary linear partial differential equations with Lawson methods. The analysis is performed under the general framework of C0-semigroups in Banach spaces and hence it can be applied to the numerical time integration of many initial boundary value problems which are described by linear partial differential equations. Conditions of regularity and annihilation at the boundary of these problems are then stated to justify the precise order which is observed, including fractional order of convergence.

A Petrov--Galerkin Spectral Method of Linear Complexity for Fractional Multiterm ODEs on the Half Line

We present a new tunably accurate Laguerre Petrov--Galerkin spectral method for solving linear multiterm fractional initial value problems with derivative orders at most one and constant coefficients on the half line. Our method results in a matrix equation of special structure which can be solved in $\mathcal{O}(N \log N)$ operations. We also take advantage of recurrence relations for the generalized associated Laguerre functions (GALFs) in order to derive explicit expressions for the entries of the stiffness and mass matrices, which can be factored into the product of a diagonal matrix and a lower-triangular Toeplitz matrix. The resulting spectral method is efficient for solving multiterm fractional differential equations with arbitrarily many terms, which we demonstrate by solving a fifty-term example. We apply this method to a distributed order differential equation, which is approximated by linear multiterm equations through the Gauss--Legendre quadrature rule. We provide numerical examples demonstra...

Avoiding order reduction when integrating linear initial boundary value problems with Lawson methods

Avoiding order reduction when integrating linear initial boundary value problems with Lawson methods

A Generalisation of the Cauchy–Kovalevskaïa Theorem

We prove that time evolution of a linear analytic initial value problem leads to sectorial holomorphic solutions in time.

APPLICATION OF LDTM-PAD\'{E} FOR SOLVING LINEAR INITIAL VALUE PROBLEMS

In this paper, we study the approximate analytical solutions of homogeneous lin- ear PDEs with initial conditions by using the Laplace-Differential Transform method (LDTM) and Pade approximation. It is determined that the method works very well for the wide range of parameters and an excellent agreement is demonstrated and discussed between the approx- imate solution and the exact one in two examples. This method is capable of greatly reducing the size of computational domain and a few numbers of iterations are required to reach the closed form solutions as series expansions of some known functions.

Modeling of vibration for functionally graded beams

Abstract In this study, a vibration problem of Euler-Bernoulli beam manufactured with Functionally Graded Material (FGM), which is modelled by fourth-order partial differential equations with variable coefficients, is examined by using the Adomian Decomposition Method (ADM).The method is one of the useful and powerful methods which can be easily applied to linear and nonlinear initial and boundary value problems. As to functionally graded materials, they are composites mixed by two or more materials at a certain rate. This mixture at a certain rate is expressed with an exponential function in order to try to minimize singularities from transition between different surfaces of materials as much as possible. According to the structure of the ADM in terms of initial conditions of the problem, a Fourier series expansion method is used along with the ADM for the solution of simply supported functionally graded Euler-Bernoulli beams. Finally, by choosing an appropriate mixture rate for the material, the results are shown in figures and compared with those of a standard (homogeneous) Euler-Bernoulli beam.

Tau-Collocation Approximation Approach for Solving First and Second Order Ordinary Differential Equations

This paper presents Tau-collocation approximation approach for solving first and second orders ordinary differential equations. We use the method in the stimulation of numerical techniques for the approximate solution of linear initial value problems (IVP) in first and second order ordinary differential equations. The resulting numerical evidences show the method is adequate and effective.

Some error estimates for the reproducing kernel Hilbert spaces method

In this paper we derive some effective error estimates for the reproducing kernel Hilbert space method applied to a general class of linear initial or boundary value problems. The first error estimate is computable and yields a worst case bound in the form of a percentage of the norm of the true solution which has not yet been discussed according to the knowledge of the authors. The second error estimate is a residual based error estimate, which is expressed in terms of the fill distance, so that convergence is studied for the fill distance tends to zero. This is a generalization and improvement of the existing error estimates. Some numerical results are presented to demonstrate the applicability of the estimates.

Solutions of linear second order initial value problems by using Bernoulli polynomials

In this paper we use Bernoulli polynomials to derive a new spectral method to find the numerical solutions of second order linear initial value problems. Stability and error analysis of this method are studied. Numerical examples are presented which support theoretical results and provide favorable comparisons with other existing methods.

Haar wavelet collocation method for the numerical solution of singular initial value problems

Abstract In this paper, numerical solutions of singular initial value problems are obtained by the Haar wavelet collocation method (HWCM). The HWCM is a numerical method for solving integral equations, ordinary and partial differential equations. To show the efficiency of the HWCM, some examples are presented. This method provides a fast convergent series of easily computable components. The errors of HWCM are also computed. Through this analysis, the solution is found on the coarse grid points and then converging toward higher accuracy by increasing the level of the Haar wavelet. Comparisons with exact and existing numerical methods (adomian decomposition method (ADM) & variational iteration method (VIM)) solutions show that the HWCM is a powerful numerical method for the solution of the linear and non-linear singular initial value problems. The Haar wavelet adaptive grid method (HWAGM) based solutions show the excellent performance for the proposed problems.

Optimal homotopy asymptotic method for solving nth order linear fuzzy initial value problems

In this paper the optimal homotopy asymptotic method (OHAM) is employed to obtain approximate analytical solution of nth order (n⩾2) linear fuzzy initial value problems (FIVPs). The convergence theorem of this method in fuzzy case is presented and proved. This method provides us with a convenient way to control the convergence of approximation series. The method is tested on nth linear FIVPs and comparisons of the exact solution that were made with numerical results showed the effectiveness and accuracy of this method.

Wavelet based quasilinearization method for semi-linear parabolic initial boundary value problems

In this paper, numerical methods based on quasilinearization and Haar and Legendre wavelets to solve a class of semi linear parabolic initial boundary value problem (SPIBVP) have been presented. The Haar and Legendre wavelet methods have been successfully combined with quasilinearization to solve SPIBVP efficiently. The presented numerical scheme has been illustrated using appropriate examples including Fisher equation and the obtained results show that the proposed numerical scheme is robust and easy to apply.

New ultraspherical wavelets collocation method for solving 2nth-order initial and boundary value problems

Abstract In this paper, a new spectral algorithm based on employing ultraspherical wavelets along with the spectral collocation method is developed. The proposed algorithm is utilized to solve linear and nonlinear even-order initial and boundary value problems. This algorithm is supported by studying the convergence analysis of the used ultraspherical wavelets expansion. The principle idea for obtaining the proposed spectral numerical solutions for the above-mentioned problems is actually based on using wavelets collocation method to reduce the linear or nonlinear differential equations with their initial or boundary conditions into systems of linear or nonlinear algebraic equations in the unknown expansion coefficients. Some specific important problems such as Lane–Emden and Burger’s type equations can be solved efficiently with the suggested algorithm. Some numerical examples are given for the sake of testing the efficiency and the applicability of the proposed algorithm.

The “good” Boussinesq equation on the half-line

The initial–boundary value problem for the “good” Boussinesq (GB) equation on the half-line with data in Sobolev spaces is analysed via Fokas' unified transform method and a contraction mapping approach. First, the basic space and time estimates for the linear GB initial value problem are derived and then the corresponding estimates for the initial–boundary value problem with zero initial data are obtained. Using these estimates, the Fokas solution formula for the linear GB on the half-line is shown to belong to appropriate Sobolev spaces. Finally, well-posedness of the nonlinear initial–boundary value problem is established by showing that the mapping defined by Fokas' formula for GB, when the forcing is replaced with the Boussinesq nonlinearity, is a contraction mapping on a ball of the space C([0,T⁎];Hxs(0,∞)), s>1/2, where the lifespan T⁎ depends on the size of the initial and boundary data. In addition, this work extends the validity of the solution formulas obtained by the unified method for the linear GB initial–boundary value problem to a broader Sobolev setting.

On Multiple Integral Chebyshev Wavelets Collocation Method (MICWCM) for Solving Linear and Non-linear Second-order Differential Equations

This paper presents a new and reliable algorithm for solving linear and non-linear second-order differential equations. The new algorithm is called Multiple Integral Chebyshev Wavelets Collocation Method (MICWCM). The algorithm improves some of the earlier results obtained by some researchers except in the case of mixed boundary conditions. The method posed to be very accurate, reliable and efficient in handling linear and non-linear initial and boundary value problems. Numerical results obtained by the new method are in agreement with exact solutions available in the literature.

Potential Velocity of Water Waves Propagation with Small Bottom Undulation

Abstract Basically, when a wave meets a different depth, it will scatter into a transmitted wave and a reflected wave. In this paper, we study the relevance of sinusoidal beds as shoreline protection through the Bragg scattering mechanism. As well known, a relatively small amplitude of sinusoidal beds can reduce the amplitude of incident waves effectivelly, due to Bragg resonance. Bragg resonance will occur if the wavelength of the monochromatic wave is twice the wavelength of sinusoidal beds. We apply the multiple scale asymptotic expansion method to the linear Shallow initial value problem for sinusoidal beds. the effect of Bragg resonance pass through sinusoidal beds has been studied for surface displacement of water waves. We found that a larger amplitude disturbance leads to larger reflected wave amplitude. This result explains that the long shore sandbar indeed can reduce the amplitude of incident wave. Actually, wave's propagation not only derives from surface displacement, but also from potential velocity. At the end, we show that the sinusoidal beds can reduce the potential velocity of incident wave.

Existence of solutions for linear hyperbolic initial-boundary value problems in a rectangle

In a recent article, we achieved the well-posedness of linear hyperbolic initial and boundary value problems (IBVP) in a rectangle via semigroup method, and we found that there are only two elementary modes called hyperbolic and elliptic modes in the system. It seems that, there is only one set of boundary conditions for the hyperbolic mode, while there are infinitely many sets of boundary conditions for the elliptic mode, which can lead to well-posedness. In this article, we continue to consider linear hyperbolic IBVP in a rectangle in the constant coefficients case and we show that there are also infinitely many sets of boundary conditions for hyperbolic mode which will lead to the existence of a solution. We also have uniqueness in some special cases. The boundary conditions satisfy the reflection conditions introduced in Section 3, which turn out to be equivalent to the strictly dissipative conditions.

The Volterra integral form of the Lane–Emden equation: new derivations and solution by the Adomian decomposition method

In this paper, we present new alternate derivations for the Volterra integral forms of the Lane-Emden equation that we derived in our last work (Wazwaz et al. in Appl Math Comput 219:5004–5019, 2013). The main focus will be on Lane-Emden equations for the shape factor of \(k=1\), where an alternate derivation for L’Hospital’s formula will be developed. Following our approach, the Adomian decomposition method, which provides an efficient algorithm for analytic approximate solutions of the Lane-Emden equation, will be used. Our results are supported by investigating several numerical examples that include linear and nonlinear initial value problems.

Accelerating convergence for Backward-Euler method by adding time relaxation term

In this study, we introduce an algorithm to numerically solve any initial value problem (IVP). This algorithm depends on the Backward-Euler (BE) time discretization and time relaxation model (TRM). That is, a time relaxation term is added into IVP and discretized by the BE method. The algorithm increases the order of convergence from \(O(t)\) to \(O(t^2)\) for IVPs which admits \(c_0 + c_1.e^\alpha.x\) as exact solution where \(c_0\); \(c_1\) and constants. Hence more accurate results are obtained. To show the effectiveness, it is applied not only some stiff problems but also nonlinear and linear IVP. All results obtained by the algorithm are better than the results of the BE method.

Identifying Stiff Ordinary Differential Equations and Problem Solving Environments (PSEs)

StiffOrdinary Differential Equations (SODEs) are present in engineering, mathematics, and sciences.Identifying them foreffective simulation (or prediction) and perhaps hardware implementation in aerospace c ontrol systemsis imperative. This paper considers only linear Initial Value Problems (IVPs) and brings to light the fact that stiffness ratio or coefficient of a suspected stiff dynamic system can be elusive as regards the phenomenon of stiffness. Though,it gives the insight suggesting stiffness when the value is up to 1000 butis not necessarily so in all ODEs. Neither does a value less than 1000 imply non-stiffness.MATLAB/Simulink