Nonlinear Initial Value Problems Research Articles

A class of linearization-based collocation methods for initial value and boundary value engineering problems

Nonlinear initial value and boundary value problems are generally classified as two different categories of problems and the methods utilized to solve them are also different. This paper proposed a uniform linearization-collocation framework through which the initial value and boundary value problems can be solved in a similar way. An optional pre-conditioning procedure is provided to guarantee the computational accuracy. Rigorous analysis about the convergence rate is given. Different from the commonly seen convergence rate analysis based on one-dimensional system and mean-value theorem, this analysis is based on multidimensional system and polynomial approximation. A perturbed orbit propagation problem, a radiation heat transfer problem and an orbit optimization problem are solved to verify the validity and efficiency of the proposed methods. The proposed methods not only outperform the highly optimized MATLAB built-in integrator ODE45 in orbit propagation, but also outperform the bvp4c and bvp5c in solving the boundary value problem. Most importantly, the proposed methods are capable of solving some problems where the bvp4c and bvp5c are invalid. The quasi-linearization-collocation method proved to be a highly accurate, efficient and stable solver for ordinary differential equations with various types of constraints.

Comparative study of Adomian decomposition method and Clique polynomial method

This paper generated the novel approach called the Clique polynomial method (CPM) using the Clique polynomials raised in graph theory. Nonlinear initial value problems are converted into nonlinear algebraic equations by discretization with suitable grid points in the current approach. We solved highly nonlinear initial value problems using the Adomian decomposition method (ADM) and CPM. Obtained results reveal that the present technique is better than ADM that is discussed through tables and simulations.

Forecasting the behavior of fractional order Bloch equations appearing in NMR flow via a hybrid computational technique

In this paper, we present an efficient computational approach named as Sumudu residual power series method (SRPSM) to solve fractional Bloch equations appearing in an NMR flow. This method is a copulation of the residual power series method (RPSM) and the Sumudu transform to construct approximate solution in shapes of sharp convergent series by adopting the notion of limit with a view of smooth calculations and time saving strategy as compared to the classical RPSM in which computations of fractional derivatives are required. This work also verifies and compares the solution obtained by the proposed technique with the classical RPSM. To assure the applicability, performance, and reliability of the introduced method, a system of fractional Bloch equations are examined along with numerical simulations. The aspect of application supported with computer simulations demonstrates that the suggested approach is precise, and appropriate to explore the solutions of linear & nonlinear initial value problems with fractional-order derivatives.

Accuracy Improvement of Block Backward Differentiation Formulas for Solving Stiff Ordinary Differential Equations Using Modified Versions of Euler's Method

In this study, the fully implicit 2-point block backward differentiation formulas (BBDF) method has been successfully utilized for solving stiff ordinary differential equations (ODEs) by taking into account the uses of new starting methods namely, modified Euler's method (MEM), improved modified Euler's method (IMEM), and new Euler's method (NEM). The reason of proposing the BBDF is that the method has been proven useful for stiff ODEs due to its A-stable properties. Furthermore, the method is able to approximate the solutions at two points simultaneously at each step. The proposed method is also implemented through Newton's iteration procedure, which involves the calculation of the Jacobian matrix. Accuracy of the method is evaluated based on its performance in solving linear and non-linear initial value problems (IVPs) of first order stiff ODEs with transient and steady-state solutions. Some comparisons are made with the conventional BBDF approach for indicating the reliability of the proposed method. Numerical results indicate that not only classical Euler's method provides accurate solutions for BBDF, but also the numerous modified versions of Euler's methods improve the accuracy of BBDF, in terms of absolute error at certain step size and stage of iteration.

CMMSE: Analysis of order reduction when Lawson methods integrate nonlinear initial boundary value problems

In this paper a thorough analysis is carried out of the type of order reduction that Lawson methods exhibit when used to integrate nonlinear initial boundary value problems. In particular, we focus on nonlinear reaction‐diffusion problems, and therefore, this study is important in a large number of practical applications modeled by this type of nonlinear equations. A theoretical study of the local and global error of the total discretization of the problem is carried out, taking into account both, the error coming from the space discretization and that due to the integration in time. These results are also corroborated by the numerical experiments performed in this paper.

Consistent discrete global dynamics of a general initial boundary value problem for hepatitis B virus infection with capsids and adaptive immunity

In this paper, we employ the nonstandard finite difference method to discretize a nonlinear initial boundary value problem that describes the global dynamics of hepatitis B virus infection with adaptive immunity. The model considered incorporates intracellular HBV DNA-containing capsids, a general incidence function and two categories of infected hepatocytes: exposed infected hepatocytes and productively infected hepatocytes. Adaptive immunity cells are recruited to attack all infected hepatocytes and free virus particles. The dynamics of fundamental properties of both discrete and continuous models are thoroughly studied, and it is shown that the discrete system is dynamically consistent with the continuous model. These properties include existence, nonnegativity and boundedness of solutions as well as the global stability of spatially homogeneous equilibria. We carry out some numerical simulations to support the theoretical findings and illustrate the behaviour of the solution in space and time.

An iterative technique based on HPM for a class of one dimensional Bratu’s type problem

An effective iterative computational technique is developed for the nonlinear initial and boundary value problems and employed on a class of one dimensional Bratu’s type problems. In the proposed technique, we use the homotopy concepts (similar as homotopy perturbation method (HPM)) for the recursive scheme which is obtained in terms of Green’s function with the support of the Picard’s iterative method and boundary/initial conditions. It generates, estimated solutions in the form of series, which is extremely useful in terms of computing. The suggested technique is also subjected to a convergence and error analysis. To highlight the method’s ease of implementation and usefulness, certain numerical examples have been provided and also compared our results with other existing results.

Solving Nonlinear COVID-19 Mathematical Model Using a Reliable Numerical Method

This research aims to numerically solve a nonlinear initial value problem presented as a system of ordinary differential equations. Our focus is on epidemiological systems in particular. The accurate numerical method that is the Runge-Kutta method of order four has been used to solve this problem that is represented in the epidemic model. The COVID-19 mathematical epidemic model in Iraq from 2020 to the next years is the application under study. Finally, the results obtained for the COVID-19 model have been discussed tabular and graphically. The spread of the COVID-19 pandemic can be observed via the behavior of the different stages of the model that approximates the behavior of actual the COVID-19 epidemic in Iraq. In our study, the COVID-19 pandemic will disappear during the next few years within about five years, through the behavior of all stages of the epidemic presented in our research.

High order approach for solving chaotic and hyperchaotic problems

In this work, the method of Taylor's decomposition on two points is suggested in order to find approximate solutions of chaotic and hyperchaotic initial value problems and to analyze the behaviors of these solutions. Unlike to the classical Taylor's method, the proposed numerical scheme is based on the application of the Taylor's decomposition on two points to the system of nonlinear initial value problems, and as a result an implicit method is obtained. Stability and error analysis of the method are presented, and its high-order accuracy and A-stability are proven. One of the advantages of the proposed method is that it is a stable and very efficient method for chaotic problems as it is an implicit one-step method. The most important advantage of the Taylor's decomposition method is that it has high order accuracy for large step sizes with a simple algorithm compared to other methods. The applicability of the proposed method has been examined in some famous chaotic systems; the Lorenz and Chen systems, and hyperchaotic systems; the Chua and Rabinovich-Fabrikant systems, to emphasize both its accuracy and effectiveness. The accuracy of the proposed method is checked by comparing the calculated results with semi-explicit Adams-Bashforth-Moulton method and ninth order Runge-Kutta method. The calculated results are also compared with multi-stage spectral relaxation method and multi-domain compact finite difference relaxation method. Comparisons have shown that the method is more accurate and efficient than the other mentioned methods for large step sizes. The obtained results are also compared with the theoretical findings and it is shown that the theoretical and numerical results are consistent.

One-step Hybrid Block Method for Directly Solving Fifth-order Initial Value Problems of Ordinary Differential Equations

An effective one-step hybrid block for getting the approximate solution of a fifth-order IVP with applications to problems in the sciences and engineering is constructed in this study. The mathematical formulation of the method is based on the principle of interpolation and collocation of the trial solution and its derivatives at the chosen equidistant grid and off-grid points. The basic properties of the derived method are examined, and it has an order greater than one, zero stable, consistent, and hence convergent. The derived method is applied to solve five different linear and nonlinear fifth-order initial value problems. Comparison of the absolute errors obtained using the derived method with a few existing ones in the literature supports its good performance.

An existence result for a strongly nonlinear parabolic equations with variable nonlinearity

We prove the existence of a solution for the strongly nonlinear parabolic initial boundary value problem associated to the equation ut − div a(x, t, ∇u) + g(x, t, u, ∇u) = f, where the vector field a(x, t, ξ) exhibits non-standard growth conditions.

Nonlinear and linearised primal and dual initial boundary value problems: When are they bounded? How are they connected?

Linearisation is often used as a first step in the analysis of nonlinear initial boundary value problems. The linearisation procedure frequently results in a confusing contradiction where the nonlinear problem conserves energy and has an energy bound but the linearised version does not (or vice versa). In this paper we attempt to resolve that contradiction and relate nonlinear energy conserving and bounded initial boundary value problems to their linearised versions and the related dual problems. We start by showing that a specific skew-symmetric form of the primal nonlinear problem leads to energy conservation and a bound. Next, we show that this specific form together with a non-standard linearisation procedure preserves these properties for the new slightly modified linearised problem. We proceed to show that the corresponding linear and nonlinear dual (or self-adjoint) problems also have bounds and conserve energy due to this specific formulation. Next, the implication of the new formulation on the choice of boundary conditions is discussed. A straightforward nonlinear and linear analysis may lead to a different number and type of boundary conditions required for an energy bound. We show that the new formulation sheds some light on this contradiction. We conclude by illustrating that the new continuous formulation automatically leads to energy stable and energy conserving numerical approximations for both linear and nonlinear primal and dual problems if the approximations are formulated on summation-by-parts form.

NUMERICAL COMPUTATIONS OF GENERAL NON-LINEAR SECOND ORDER INITIAL VALUE PROBLEMS BY USING MODIFIED RUNGE-KUTTA METHOD

Numerical solution of ordinary differential equations is the most important technique which is widely used for mathematical modelling in science and engineering. The differential equation that describes the problem is typically too complex to precisely solve in real-world circumstances. Since most ordinary differential equations are not solvable analytically, numerical computations are the only way to obtain information about the solution. Many different methods have been proposed and used is an attempt to solve accurately various types of ordinary differential equations. Among them, Runge-Kutta is a well-known and popular method because of their good efficiency. This paper contains an analysis for the computations of the modified Runge-Kutta method for nonlinear second order initial value problems. This method is wide quite efficient and practically well suited for solving linear and non-linear problems. In order to verify the accuracy, we compare numerical solution with the exact solution. We also compare the performance and the computational effort of this method. In order to achieve higher accuracy in the solution, the step size needs to be small. Finally, we take some examples of non-linear initial value problems (IVPs) to verify proposed method. The results of that example indicate that the convergence, stability analysis, and error analysis which are discussed to determine the efficiency of the method.

Existence theory and generalized Mittag-Leffler stability for a nonlinear Caputo-Hadamard FIVP via the Lyapunov method

<abstract><p>This paper discusses the existence, uniqueness and stability of solutions for a nonlinear fractional differential system consisting of a nonlinear Caputo-Hadamard fractional initial value problem (FIVP). By using some properties of the modified Laplace transform, we derive an equivalent Hadamard integral equation with respect to one-parametric and two-parametric Mittag-Leffer functions. The Banach contraction principle is used to give the existence of the corresponding solution and its uniqueness. Then, based on a Lyapunov-like function and a $ \mathcal{K} $-class function, the generalized Mittag-Leffler stability is discussed to solve a nonlinear Caputo-Hadamard FIVP. The findings are validated by giving an example.</p></abstract>

THREE-DIMENSIONAL NONSTATIONARY FLOW OF TWO IMMISCIBLE FLUIDS IN A ROTATING HORIZONTAL CYLINDER

The three-dimensional nonstationary axisymmetric flow of two immiscible viscous fluids in a rotating horizontal cylinder is studied. The interface between the fluids is assumed to be flat and undeformable. The full energy condition at the interface is taken into account. The mass forces are neglected. The velocity field of the Himenz type is considered. For the solution of the nonlinear initial boundary value problem, the modified Galerkin method and the Runge-Kutta method have been applied. It is shown that, with an increase of time, the numerical solution of the nonstationary problem tends to the stationary regime.

The interior-boundary Strichartz estimate for the Schrödinger equation on the half-line revisited

In recent papers, it was shown for the biharmonic Schrödinger equation and 2D Schrödinger equation that Fokas method-based formulas are capable of defining weak solutions of associated nonlinear initial boundary value problems (ibvps) below the Banach algebra threshold. In view of these results, we revisit the theory of interior-boundary Strichartz estimates for the Schrödinger equation posed on the right half line, considering both Dirichlet and Neumann cases. Finally, we apply these estimates to obtain low regularity solutions for the nonlinear Schrödinger equation (NLS) with Neumann boundary condition and a coupled system of NLS equations defined on the half line with Dirichlet/Neumann boundary conditions.

Optimized Hybrid Block Adams Method for Solving First Order Ordinary Differential Equations

Multistep integration methods are being extensively used in the simulations of high dimensional systems due to their lower computational cost. The block methods were developed with the intent of obtaining numerical results on numerous points at a time and improving computational efficiency. Hybrid block methods for instance are specifically used in numerical integration of initial value problems. In this paper, an optimized hybrid block Adams block method is designed for the solutions of linear and nonlinear first-order initial value problems in ordinary differential equations (ODEs). In deriving the method, the Lagrange interpolation polynomial was employed based on some data points to replace the differential equation function and it was integrated over a specified interval. Furthermore, the convergence properties along with the region of stability of the method were examined. It was concluded that the newly derived method is convergent, consistent, and zero-stable. The method was also found to be A-stable implying that it covers the whole of the left/negative half plane. From the numerical computations of absolute errors carried out using the newly derived method, it was found that the method performed better than the ones with which we compared our results with. The method also showed its superiority over the existing methods in terms of stability and convergence.

Wavelet Solution of Non-Linear Singular Initial Value Problem

Wavelet Solution of Non-Linear Singular Initial Value Problem

Application of reaction-diffusion equations for modeling human and breeding site attraction movement behavior of Aedes aegypti mosquito.

This paper shows how biological population dynamic models in the form of coupled reaction-diffusion equations with nonlinear reaction terms can be applied to heterogeneous landscapes. The presented systems of coupled partial differential equations (PDEs) combine the dispersal of disease-vector mosquitoes and the spread of the disease in a human population. Realistic biological dispersal behavior is taken into account by applying chemotaxis terms for the attraction to the human host and the attraction of suitable breeding sites. These terms are capable of generating the complex active movement patterns of mosquitoes along the gradients of the attractants. The nonlinear initial boundary value problems are solved numerically for geometries of heterogeneous landscapes, which have been imported from geographic information system data to construct a general-purpose finite-element solver for systems of coupled PDEs. The method is applied to the dispersal of the dengue disease vector for Aedes aegypti in a small-scale rural setting consisting of small houses and different breeding sites, and to a large-scale section of the suburban zone of a metropolitan area in Vietnam. Numerical simulations illustrate how the setup of model equations and geographic information can be used for the assessment of control measures, including the spraying patterns of pesticides and biological control by inducing male sterility.

Nonlinear systems of parabolic IBVP: A stable super-supraconvergent fully discrete piecewise linear FEM

The main purpose of this paper is the design of discretizations for second order nonlinear parabolic initial boundary value problems which are stable and present second order of convergence with respect to H1-discrete norms. The convergence results are established assuming that the solutions are in H3. The stability analysis of numerical methods around a numerical solution requires the uniform boundness of such solution. Although such bounds are usually taken as an assumption, in this paper these will be deduced as a corollary of suitable error estimates. As the methods can be simultaneously seen as piecewise linear finite element methods and finite difference methods, the convergence results can be seen simultaneously as superconvergence results and supraconvergence results. Numerical results illustrating the sharpness of the smoothness assumptions and an application to simulation of the solar magnetic field in the umbra (the central zone of a sunspot) are also included.