Solution Of Initial Value Problem Research Articles

Self-similar evolution of Alfven wave turbulence

We study self-similar solutions of the kinetic equation for MHD wave turbulence derived in (Galtier S et al 2000 J. Plasma Phys. 63 447–88). Motivated by finding the asymptotic behaviour of solutions for initial value problems, we formulate a nonlinear eigenvalue problem comprising in finding a number such that the self-similar shape function would have a power-law asymptotic at low values of the self-similar variable η and would be the fastest decaying positive solution at . We prove that the solution of this problem has a tail decaying as a power-law, and not exponentially or super-exponentially. We present a relationship between the power-law exponents in the regions and , and an integral relation for and . We confirm these relationships by solving numerically the nonlinear eigenvalue problem, and find that .

Axisymmetric viscous interfacial oscillations – theory and simulations

We study axisymmetric, free oscillations driven by gravity and surface tension at the interface of two viscous, immiscible, radially unbounded fluids, analytically and numerically. The interface is perturbed as a zeroth-order Bessel function (in space) and its evolution is obtained as a function of time. In the linearised approximation, we solve the initial value problem (IVP) to obtain an analytic expression for the time evolution of wave amplitude. It is shown that a linearised Bessel mode temporally evolves in exactly the same manner as a Fourier mode in planar geometry. We obtain novel analytical expressions for the time varying vorticity and pressure fields in both fluids. For small initial amplitudes, our analytical results show excellent agreement with those obtained from solving the axisymmetric Navier–Stokes equations numerically. We also compare our results with the normal mode approximation and find the latter to be an accurate representation at very early and late times. The deviation between the normal mode approximation and the IVP solution is found to increase as a function of viscosity ratio. The vorticity field has a jump discontinuity at the interface and we find that this jump depends on the viscosity and the density ratio of the two fluids. Upon increasing the initial perturbation amplitude in the simulations, nonlinearity produces qualitatively new features not present in the analytical IVP solution. Notably, a jet is found to emerge at the axis of symmetry rising to a height greater than the initial perturbation amplitude. Increasing the perturbation amplitude further causes the jet to undergo end pinch off, giving birth to a daughter droplet. This can happen either for an advancing or a receding jet, depending on the viscosity ratio. A relation is found between the maximum jet height and the perturbation amplitude. Hankel transform of the interface demonstrates that at large perturbation amplitudes higher wavenumbers emerge, sharing some of the energy of the lowest mode. When these additional higher modes are present, the interface has pointed crests and rounded troughs.

Stability and boundedness of solutions of the initial value problem for a class of time-fractional diffusion equations

The aim of this paper is to study the stability and boundedness of solutions of the initial value problem for a class of time-fractional diffusion equations. We first establish a fractional Duhamel principle for the nonhomogeneous time-fractional diffusion equation. Then based on it and the superposition principle, the solution of the above initial value problem is represented. Finally, we obtain the stability and boundedness of the solution and present an illustrative example.

Exact optimal values of step-size coefficients for boundedness of linear multistep methods

Linear multistep methods (LMMs) applied to approximate the solution of initial value problems—typically arising from method-of-lines semidiscretizations of partial differential equations—are often required to have certain monotonicity or boundedness properties (e.g., strong-stability-preserving, total-variation-diminishing or total-variation-boundedness properties). These properties can be guaranteed by imposing step-size restrictions on the methods. To qualitatively describe the step-size restrictions, one introduces the concept of step-size coefficient for monotonicity (SCM, also referred to as the strong-stability-preserving (SSP) coefficient) or its generalization, the step-size coefficient for boundedness (SCB). An LMM with larger SCM or SCB is more efficient, and the computation of the maximum SCM for a particular LMM is now straightforward. However, it is more challenging to decide whether a positive SCB exists, or determine if a given positive number is a SCB. Theorems involving sign conditions on certain linear recursions associated to the LMM have been proposed in the literature that allow us to answer the above questions: the difficulty with these theorems is that there are in general infinitely many sign conditions to be verified. In this work, we present methods to rigorously check the sign conditions. As an illustration, we confirm some recent numerical investigations concerning the existence of positive SCBs in the BDF and in the extrapolated BDF (EBDF) families. As a stronger result, we determine the optimal values of the SCBs as exact algebraic numbers in the BDF family (with 1 ≤ k ≤ 6 steps) and in the Adams–Bashforth family (with 1 ≤ k ≤ 3 steps).

Shooting-projection method for two-point boundary value problems

This paper presents a novel shooting method for solving two-point boundary value problems for second order ordinary differential equations. The method works as follows: first, a guess for the initial condition is made and an integration of the differential equation is performed to obtain an initial value problem solution; then, the end value of the solution is used in a simple iteration formula to correct the initial condition; the process is repeated until the second boundary condition is satisfied. The iteration formula is derived utilizing an auxiliary function that satisfies both boundary conditions and minimizes the H 1 semi-norm of the difference between itself and the initial value problem solution.

A new approach to the construction of DIMSIMs of high order and stage order

In this work we describe a new approach to the construction of diagonally implicit multistage integration methods (DIMSIMs) for the numerical solution of initial value problems for ordinary differential equations (ODEs). Differential Evolution, a very popular computational intelligence technique is employed to construct type 1 and type 2 methods with better or equivalent characteristics to the methods presented in the literature. The numerical results in selected problems justify this argument.

Neuro-heuristic computational intelligence for solving nonlinear pantograph systems

We present a neuro-heuristic computing platform for finding the solution for initial value problems (IVPs) of nonlinear pantograph systems based on functional differential equations (P-FDEs) of different orders. In this scheme, the strengths of feed-forward artificial neural networks (ANNs), the evolutionary computing technique mainly based on genetic algorithms (GAs), and the interior-point technique (IPT) are exploited. Two types of mathematical models of the systems are constructed with the help of ANNs by defining an unsupervised error with and without exactly satisfying the initial conditions. The design parameters of ANN models are optimized with a hybrid approach GA–IPT, where GA is used as a tool for effective global search, and IPT is incorporated for rapid local convergence. The proposed scheme is tested on three different types of IVPs of P-FDE with orders 1–3. The correctness of the scheme is established by comparison with the existing exact solutions. The accuracy and convergence of the proposed scheme are further validated through a large number of numerical experiments by taking different numbers of neurons in ANN models.

Stability and boundedness in the numerical solution of initial value problems

This paper concerns the theoretical analysis of step-by-step methods for solving initial value problems in ordinary and partial differential equations. The main theorem of the paper answers a natural question arising in the linear stability analysis of such methods. It guarantees a (strong) version of numerical stability—under a stepsize restriction related to the stability region of the numerical method and to a circle condition on the differential equation. The theorem also settles an open question related to the properties total-variation-diminishing, strong-stability-preserving, monotonic and (total- variation-)bounded. Under a monotonicity condition on the forward Euler method, the theorem specifies a stepsize condition guaranteeing boundedness for linear problems. The main theorem is illustrated by applying it to linear multistep methods. For important classes of these methods, conclusions are thus obtained which supplement earlier results in the literature.

Impulsive Functional-Differential Equations of Fractional Order with Variable Moments

We establish some existence results for the solutions of initial-value problems for fractional-order impulsive functional differential equations with neutral delay at variable moments.

Development of a New One-Step Scheme for the Solution of Initial Value Problem (IVP) in Ordinary Differential Equations

In this paper, a new one-step scheme was developed for the solution of initial value problems of first order in ordinary differential equations. In its development a combination of interpolating function and Taylor series were used. The method was used for the solution of initial value problems emanated from real life situations. The numerical results showed that the new scheme is consistent, robust and efficient.

Implicit One-Step Block Hybrid Third-Derivative Method for the Direct Solution of Initial Value Problems of Second-Order Ordinary Differential Equations

A new one-step block method with generalized three hybrid points for solving initial value problems of second-order ordinary differential equations directly is proposed. In deriving this method, a power series approximate function is interpolated at {xn,xn+r} while its second and third derivatives are collocated at all points {xn,xn+r,xn+s,xn+t,xn+1} in the given interval. The proposed method is then tested on initial value problems of second-order ordinary differential equations solved by other methods previously. The numerical results confirm the superiority of the new method to the existing methods in terms of accuracy.

Galerkin and collocation methods for solution of initial value problem of generalised van der Pol equation

In this paper, we proposed a van der Pol oscillator with delay feed-back including duffing oscillators and a periodic forcing term model. For solution of proposed model, using collocation and Galerkin method with Legendre wavelet as a basis functions. Convergence and stability analysis of the method are discussed. An algorithm provided for computing numerical data. The solution obtained by both Legendre wavelet collocation method and Legendre wavelet Galerkin method is exactly same as exact solution and those obtained by method of averaging (Atay, 1998), PEM (Kimiaeifar et al., 2010). The Legendre wavelet collocation method for different M and k provides better results in lesser time than Legendre wavelet Galerkin method. It has been observed that the displacement have cyclic behaviour with respect to velocity for different feedback gain and angular frequency of the driving force. The displacement decreases as delay parameter increases in small domain. The displacement also decreases as duffing parameter increases and angular frequency of the driving force decreases.

Galerkin and collocation methods for solution of initial value problem of generalised van der Pol equation

In this paper, we proposed a van der Pol oscillator with delay feed-back including duffing oscillators and a periodic forcing term model. For solution of proposed model, using collocation and Galerkin method with Legendre wavelet as a basis functions. Convergence and stability analysis of the method are discussed. An algorithm provided for computing numerical data. The solution obtained by both Legendre wavelet collocation method and Legendre wavelet Galerkin method is exactly same as exact solution and those obtained by method of averaging (Atay, 1998), PEM (Kimiaeifar et al., 2010). The Legendre wavelet collocation method for different M and k provides better results in lesser time than Legendre wavelet Galerkin method. It has been observed that the displacement have cyclic behaviour with respect to velocity for different feedback gain and angular frequency of the driving force. The displacement decreases as delay parameter increases in small domain. The displacement also decreases as duffing parameter increases and angular frequency of the driving force decreases.

Derivation of Adams Method for the Numerical Solution of Ordinary Differential Equations Via Computer Algebra

In this paper, a well-known computer algebra system (CAS) was considered for the derivation of Numerical method for the solution of initial value problems. This was achieved by the use of maple software. Numerical methods were derived through Lagrange interpolation method. Both the implicit and explicit method was derived with the help of the Computer algebra system. In particular, a review of Maple’s functional role in the derivation of numerical methods was also presented. The main challenge was that the efficient handling and simplifying of very long expressions, which was met by the power of Maple’s build-in functionality. The use of the maple procedure had significantly reduced the errors and hence improved efficiency in derivation of higher order Adams Methods.

On generalized fractional operators and a gronwall type inequality with applications

In this paper, we obtain the Gronwall type inequality for generalized fractional operators unifying Riemann-Liouville and Hadamard fractional operators. We apply this inequality to the dependence of the solution of differential equations, involving generalized fractional derivatives, on both the order and the initial conditions. More properties for the generalized fractional operators are formulated and the solutions of initial value problems in certain new weighted spaces of functions are established as well.

Highly accurate numerical schemes for multi-dimensional space variable-order fractional Schrödinger equations

As a natural generalization of the fractional Schrödinger equation, the variable-order fractional Schrödinger equation has been exploited to study fractional quantum phenomena. In this paper, we develop an exponentially accurate Jacobi–Gauss–Lobatto collocation (J–GL-C) method to solve the variable-order fractional Schrödinger equations in one dimension (1D) and two dimensions (2D). In this method, the aforementioned problem is reduced to a system of ordinary differential equations (ODEs) in the time variable. As a result, we propose two efficient schemes for dealing with the numerical solutions of initial value problems for nonlinear system of ordinary differential equations, one based on the implicit Runge–Kutta (IRK) method of fourth order and the other based on Jacobi–Gauss–Radau collocation (J–GR-C) method. The validity and effectiveness of the two methods are demonstrated by solving three numerical examples in 1D and 2D. The convergence of the methods is graphically analyzed. The results demonstrate that the proposed methods are powerful algorithms with high accuracy for solving the variable-order nonlinear partial differential equations.

APPROXIMATION OF THE DISTANCE IN THE SPHERE THROUGH OF THE NUMERICAL SOLUTION OF INITIAL VALUE PROBLEM ASSOCIATED TO GEODESICS

In this paper, we proposes an algorithm to approximate the Geodesic distance between points p and q of sphere, through the numerical solution of initial value problem associated with the system of ordinary differential equations of the geodesics; for this an appropriate direction is obtained.

On the General Solution of Initial Value Problems of Ordinary Differential Equations Using the Method of Iterated Integrals

Our goal is to give a very simple, effective and intuitive algorithm for the solution of initial value problem of ODEs of 1st and arbitrary higher order with general i.e. constant, variable or nonlinear coefficients. We find a local expansion of the differential equation/function that employs the higher derivatives (in the sense of total derivatives) of the differential equation/function with respect to independent variable to find an approximation which can be successively improved to desired accuracy by adding extra terms. The method can also be used easily for general 2nd and higher order ODEs. We explain with examples the steps to solution of initial value problems of 1st order ODEs and later follow with more examples for linear and non-linear 2nd order ODEs.

A new stepsize change technique for Adams methods

Abstract In this paper a new technique for stepsize changing in the numerical solution of Initial Value Problems for ODEs by means of Adams type methods is considered. The computational cost of the new technique is equivalent to those of the well known interpolation technique (IT). It is seen that the new technique has better stability properties than the IT and moreover, its leading error term is smaller. These facts imply that the new technique can outperform the IT.

Sufficient Conditions for First-Order Differential Operators to be Associated with a q-Metamonogenic Operator in a Clifford Type Algebra

Consider the initial value problem 0.1 $$\begin{aligned} \partial _{t}u= & {} {\mathcal L}(t,x,u,\partial _{x_{i}}u),\nonumber \\ u(0,x)= & {} \varphi (x), \end{aligned}$$ where t is the time, $${\mathcal L}$$ is a linear first-order differential operator and $$\varphi $$ is a generalized q-metamonogenic function. This problem can be solved by applying the method of associated spaces which is constructed by Tutschke (see Solution of initial value problems in classes of generalized analytic functions, Teubner Leipzig and Springer, New York, 1989). In this work, we formulate sufficient conditions on the coefficients of the operator $${\mathcal L}$$ under which this operator is associated to the space of generalized q-metamonogenic functions satisfying a differential equation with anti-q-metamonogenic right-hand side, when q and $$\lambda $$ are constant Clifford vectors. We also build a computational algorithm to check the computations in the cases $${\mathcal A}^{*}_{2,2}$$ and $${\mathcal A}^{*}_{3,2}$$ . In conical domains, the initial value problem (0.1) is uniquely solvable for an operator $${\mathcal L}$$ and for any generalized q-metamonogenic initial function $$\varphi $$ , provided an interior estimate holds for generalized q-metamonogenic functions satisfying a differential equation with anti-q-metamonogenic right-hand side. The solution is also a generalized q-metamonogenic function for each fixed t. This work generalizes the results given in Di Teodoro and Sapian (Adv. Appl. Clifford Algebras, 25:283–301, 2015) and Van (Differential operator in a Clifford analysis associated to differential equations with anti-monogenic right hand side, IC/2006/134, 2016).