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We describe an algorithm for the numerical solution of second order linear differential equations in the highly-oscillatory regime. It is founded on the recent observation that the solutions of equations of this type can be accurately represented using nonoscillatory phase functions. Unlike standard solvers for ordinary differential equations, the running time of our algorithm is independent of the frequency of oscillation of the solutions. We illustrate the performance of the method with several numerical experiments.

In the present study, a numerical method, perturbation-iteration algorithm (shortly PIA), have been employed to give approximate solutions of nonlinear fractional-integro differential equations (FIDEs). Comparing with the exact solution, the PIA produces reliable and accurate results for FIDEs.

In the present study, a numerical method, perturbation-iteration algorithm (shortly PIA), have been employed to give approximate solutions of nonlinear fractional-integro differential equations (FIDEs). Comparing with the exact solution, the PIA produces reliable and accurate results for FIDEs.

In the present study, a numerical method, perturbation-iteration algorithm (shortly PIA), have been employed to give approximate solutions of nonlinear fractional-integro differential equations (FIDEs). Comparing with the exact solution, the PIA produces reliable and accurate results for FIDEs.

This project is concerned with several questions conern ing the existence, uniqueness, continuous data dependence and numerical computation of solutions of various ill posed problems in partial differential equations. Several problems involving reaction diffusion equations with and without convection terms present were studied. In the latter case the ability of finite element approximate solutions to reproduce the continuous time dynamics was investigated. In the former case, a convective diffusion equation with a semilinear source in the boundary conditions was analyzed. A fairly complete picture of the dynamics was obtained. With the source term in the equation, computations revealed a rich structure which has been partially analyzed theoretically. Several problems for reaction diffusion equations in unbounded regimes were also investigated. It was shown that under certain conditions in the rate law all nonzero solutions blow up in finite time, while for other conditions in the rate law, solutions damp out. It was shown that a potential well theory is possible for certain hyperbolic problems in which a nonlinear boundary condition is prescribed and not possible in certain cases when forcing term in the differential equation is singular. Numerical experiments performed on the wave equation with a singular forcing term have down that when quenching occurs, the time and exact derivatives blow up in finite time. The nature of the blowup was studied computationally.

A method is developed for using the integrals of systems of nonlinear, ordinary differential equations in a numerical integration process to control the local errors in these integrals and reduce the global errors of the solution. The method is general and can be applied to either scaler or vector integrals. A number of example problems, with accompanying numerical results, are used to verify the analysis and support the conjecture of global error reduction.

Recently the determination of the best technique for numerically solving systems of ordinary differential equations on a digital computer has received much attention. The use of these formulas in conjunction with a stepsize control developed is explained, and one of the formulas is chosen for comparison with other integration techniques. This comparison of one of the best of Fehlberg's formulas with the different numerical techniques described in previous studies on a variety of test problems clearly shows the superiority of Fehlberg's formula. That is, on each of the test problems, the chosen Fehlberg formula is able to achieve a given accuracy in less computer time than any of the other techniques tested. Also, the computer program for the chosen Fehlberg formula is less complex and easier to use than the computer programs for most of the other techniques. To illustrate the use of the chosen Fehlberg formula, a computer listing of its application to several example problems is included along with a sample of the computer output from these applications.

Recently the determination of the best technique for numerically solving systems of ordinary differential equations on a digital computer has received much attention. The use of these formulas in conjunction with a stepsize control developed is explained, and one of the formulas is chosen for comparison with other integration techniques. This comparison of one of the best of Fehlberg's formulas with the different numerical techniques described in previous studies on a variety of test problems clearly shows the superiority of Fehlberg's formula. That is, on each of the test problems, the chosen Fehlberg formula is able to achieve a given accuracy in less computer time than any of the other techniques tested. Also, the computer program for the chosen Fehlberg formula is less complex and easier to use than the computer programs for most of the other techniques. To illustrate the use of the chosen Fehlberg formula, a computer listing of its application to several example problems is included along with a sample of the computer output from these applications.

Bibliography: p. 79-80

A mathematical equation which involves a function and its derivatives is called a differential equation. We consider a real life situation, from this form a mathematical model, solve that model using some mathematical concepts and take interpretation of solution. It is a well known and popular concept in mathematics because of its massive application in real world problems. Differential equations are one of the most important mathematical tools used in modeling problems in Physics, Biology, Economics, Chemistry, Engineering and medical Sciences. Differential equation can describe many situations viz exponential growth and de cay, the population growth of species, the change in investment return over time. We can solve differential equations using classical as well as numerical methods, In this paper we compare numerical methods of solving initial valued first order ordinary differential equations namely Euler method, Improved Euler method, Runge Kutta method and their accuracy level. We use here Scilab Software to obtain direct solution for these methods. Vibahvari Tukaram Dhokrat "Comparative Analysis of Different Numerical Methods for the Solution of Initial Value Problems in First Order Ordinary Differential Equations" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-5 , August 2021, URL: https://www.ijtsrd.com/papers/ijtsrd45066.pdf Paper URL: https://www.ijtsrd.com/mathemetics/applied-mathematics/45066/comparative-analysis-of-different-numerical-methods-for-the-solution-of-initial-value-problems-in-first-order-ordinary-differential-equations/vibahvari-tukaram-dhokrat

This volume was digitized and made accessible online due to deterioration of the original print copy.

This volume was digitized and made accessible online due to deterioration of the original print copy.

This volume was digitized and made accessible online due to deterioration of the original print copy.

Technical report for period October 1975 - April 1976.

Technical report for period October 1975 - April 1976.

Technical report for period October 1975 - April 1976.

16 p. 28 cm

This paper presents the method of solving one-dimensional differential equations through the weighted residual technique, employing Bernoulli wavelets as the basis functions. These wavelets serve as the foundation for the calculation of numerical solutions for one-dimensional differential equations. The numerical outcomes are contrasted with those from current techniques and the precise solution. A selection of numerical test problems is included to demonstrate the practicality and efficiency of the proposed approach.

Starting from Neeteson's approximation for the flux switching characteristic of a magnetic core, the differential equation describing the case when such a core is terminated in series inductance and resistance was derived. By means of successive transformations, the equation was reduced to a form which could be solved by an iterative procedure. A family of solutions was computed and plotted. To date, no attempt has been made to check these curves against experimental results, although verification is planned for the future. Since Neeteson's approximation is admittedly a compromise between exact fit to the flux characteristic and mathematical simplicity, only a fair agreement is to be expected. It is believed that the differential equation and solutions are applicable to flux steering through multipath magnetic structures. The mathematical methods employed are themselves of some interest.

This report documents a program for the numerical solution of large sparse systems of algebraic and implicitly defined stiff differential equations. The principal use is intended to be the solution of differential equations arising from time dependent partial differential equations when the finite element method is used to discretize the space domain. The use of compact matrix storage techniques and iteration for the solution of the quasi-Newton iterates in Gear's method makes the program extremely efficient both in terms of storage requirements and execution times.

This report documents a program for the numerical solution of large sparse systems of algebraic and implicitly defined stiff differential equations. The principal use is intended to be the solution of differential equations arising from time dependent partial differential equations when the finite element method is used to discretize the space domain. The use of compact matrix storage techniques and iteration for the solution of the quasi-Newton iterates in Gear's method makes the program extremely efficient both in terms of storage requirements and execution times.

General differentiation formulas for successive derivatives of a function are obtained in terms of the values of the function at unequally spaced arguments and the corresponding distances between the successive arguments using Lagrangian polynomials of various degrees. The remainder term is also obtained. Tables of coefficients in the formulas for the first four derivatives are given in intervals of 0.01 for the special case where only one spacing at either and is different from the others, as is often encountered near a curved boundary, for different ratios of this spacing to the others.

Mathematical modeling and computer simulations are nowadays widely used tools to predict the behavior of problems in engineering and in the natural and social sciences. All such predictions are obtained by formulating mathematical models and then using computational methods to solve the corresponding problems. We use a probability theory approach for uncertainty quantification (UQ) since it is particularly well suited for SPDE models, and focus on the broad research areas of algorithmic development and numerical analysis for the discretization of systems of linear or nonlinear SPDEs, building upon and significantly extending our previous successful work. We conduct comprehensive theoretical and computational comparison of the efficiency, accuracy, and range of applicability of non-intrusive methods, such as stochastic collocation methods, and intrusive techniques, such as stochastic Galerkin methods, for solving SPDEs and for UQ applications. We extend the algorithmic and analysis advances wrought by these efforts to the even more challenging settings of optimal control and parameter identification problems for SPDEs. The parameter identification problem is especially important in the SPDE setting since it provides a very useful mechanism for determining statistical information about the input parameters from, e.g., measurements of output quantities. This effort builds on our previous work on adjoint and sensitivity-based methods for deterministic optimal control and parameter identification problems to develop similar methods for tracking statistical quantities of interest from the computational solutions of linear and nonlinear SPDEs driven by high-dimensional random inputs.

Mathematical modeling and computer simulations are nowadays widely used tools to predict the behavior of problems in engineering and in the natural and social sciences. All such predictions are obtained by formulating mathematical models and then using computational methods to solve the corresponding problems. We use a probability theory approach for uncertainty quantification (UQ) since it is particularly well suited for SPDE models, and focus on the broad research areas of algorithmic development and numerical analysis for the discretization of systems of linear or nonlinear SPDEs, building upon and significantly extending our previous successful work. We conduct comprehensive theoretical and computational comparison of the efficiency, accuracy, and range of applicability of non-intrusive methods, such as stochastic collocation methods, and intrusive techniques, such as stochastic Galerkin methods, for solving SPDEs and for UQ applications. We extend the algorithmic and analysis advances wrought by these efforts to the even more challenging settings of optimal control and parameter identification problems for SPDEs. The parameter identification problem is especially important in the SPDE setting since it provides a very useful mechanism for determining statistical information about the input parameters from, e.g., measurements of output quantities. This effort builds on our previous work on adjoint and sensitivity-based methods for deterministic optimal control and parameter identification problems to develop similar methods for tracking statistical quantities of interest from the computational solutions of linear and nonlinear SPDEs driven by high-dimensional random inputs.

Mathematical modeling and computer simulations are nowadays widely used tools to predict the behavior of problems in engineering and in the natural and social sciences. All such predictions are obtained by formulating mathematical models and then using computational methods to solve the corresponding problems. We use a probability theory approach for uncertainty quantification (UQ) since it is particularly well suited for SPDE models, and focus on the broad research areas of algorithmic development and numerical analysis for the discretization of systems of linear or nonlinear SPDEs, building upon and significantly extending our previous successful work. We conduct comprehensive theoretical and computational comparison of the efficiency, accuracy, and range of applicability of non-intrusive methods, such as stochastic collocation methods, and intrusive techniques, such as stochastic Galerkin methods, for solving SPDEs and for UQ applications. We extend the algorithmic and analysis advances wrought by these efforts to the even more challenging settings of optimal control and parameter identification problems for SPDEs. The parameter identification problem is especially important in the SPDE setting since it provides a very useful mechanism for determining statistical information about the input parameters from, e.g., measurements of output quantities. This effort builds on our previous work on adjoint and sensitivity-based methods for deterministic optimal control and parameter identification problems to develop similar methods for tracking statistical quantities of interest from the computational solutions of linear and nonlinear SPDEs driven by high-dimensional random inputs.

Mathematical modeling and computer simulations are nowadays widely used tools to predict the behavior of problems in engineering and in the natural and social sciences. All such predictions are obtained by formulating mathematical models and then using computational methods to solve the corresponding problems. We use a probability theory approach for uncertainty quantification (UQ) since it is particularly well suited for SPDE models, and focus on the broad research areas of algorithmic development and numerical analysis for the discretization of systems of linear or nonlinear SPDEs, building upon and significantly extending our previous successful work. We conduct comprehensive theoretical and computational comparison of the efficiency, accuracy, and range of applicability of non-intrusive methods, such as stochastic collocation methods, and intrusive techniques, such as stochastic Galerkin methods, for solving SPDEs and for UQ applications. We extend the algorithmic and analysis advances wrought by these efforts to the even more challenging settings of optimal control and parameter identification problems for SPDEs. The parameter identification problem is especially important in the SPDE setting since it provides a very useful mechanism for determining statistical information about the input parameters from, e.g., measurements of output quantities. This effort builds on our previous work on adjoint and sensitivity-based methods for deterministic optimal control and parameter identification problems to develop similar methods for tracking statistical quantities of interest from the computational solutions of linear and nonlinear SPDEs driven by high-dimensional random inputs.

This report describes the derivation of a set of ordinary differential equations to model radical chain polymerization. These equations have the mathematical property of stiffness and are difficult to solve numerically. We show how these equations can be solved efficiently using either the Gear or Kaps Rentrop method. We also show how the kinetic scheme can be expanded to allow for the presence of contaminant scavenger molecules, and we apply these schemes to model experimental results for the polymerization of N- vinyl-2-pyrrolidone obtained from dilatometry measurements.

Long time integration of large stiff systems of initial value problems, arising from chemical reactions, demands efficient methods with good accuracy and extensive absolute stability region. In this paper, we apply second derivative general linear methods to solve some stiff chemical problems such as chemical Akzo Nobel problem, HIRES problem and OREGO problem.

In this paper, the sinc collocation method is proposed for solving linear and nonlinear multi-order fractional differential equations based on the new definition of fractional derivative which is recently presented by Khalil, R., Al Horani, M., Yousef, A. and Sababeh, M. in A new definition of fractional derivative, J. Comput. Appl. Math. 264 (2014), 65{70. The properties of sinc functions are used to reduce the fractional differential equation to a system of algebraic equations. Several numerical examples are provided to illustrate the accuracy and effectiveness of the presented method.

In this paper, the sinc collocation method is proposed for solving linear and nonlinear multi-order fractional differential equations based on the new definition of fractional derivative which is recently presented by Khalil, R., Al Horani, M., Yousef, A. and Sababeh, M. in A new definition of fractional derivative, J. Comput. Appl. Math. 264 (2014), 65{70. The properties of sinc functions are used to reduce the fractional differential equation to a system of algebraic equations. Several numerical examples are provided to illustrate the accuracy and effectiveness of the presented method.

In this paper, the sinc collocation method is proposed for solving linear and nonlinear multi-order fractional differential equations based on the new definition of fractional derivative which is recently presented by Khalil, R., Al Horani, M., Yousef, A. and Sababeh, M. in A new definition of fractional derivative, J. Comput. Appl. Math. 264 (2014), 65{70. The properties of sinc functions are used to reduce the fractional differential equation to a system of algebraic equations. Several numerical examples are provided to illustrate the accuracy and effectiveness of the presented method.

This text discusses the importance of boundary value problems for partial differential equations in modern technological theory. Although analytical methods are theoretically well-developed, they are often impractical due to complex computations and the lack of suitable formulas for real-world problems. Hence, engineers and scientists increasingly turn to approximate methods, which are divided into two groups: (a) methods yielding approximate analytical expressions (like the Ritz and Galerkin methods), and (b) numerical or graphical methods (like the method of differences), which provide numerical solutions but no analytical form. The second group, especially the method of differences, is more broadly applicable and less restricted by problem conditions, making it highly valuable in practice despite being relatively recent and not widely known among engineers. This volume aims to bridge that gap by presenting a structured account of these numerical methods, with explanations and examples, making it both a reference and a practical textbook. The primary focus is on solving boundary value problems for the Laplace equation, due to their significance, computational complexity, and the clarity with which they demonstrate the method of differences. The third edition includes updates, particularly on solving hyperbolic equations using the method of characteristics. Translated by Charles M. Stern

Technical report for period October 1975 - April 1976.

Technical report for period October 1975 - April 1976.

Technical report for period October 1975 - April 1976.

Technical report for period October 1975 - April 1976.

The numerical solution of stochastic partial differential equations (SPDE) presents challenges not encountered in the simulation of PDEs or SDEs. Indeed, the roughness of the noise in conjunction with nonlinearities in the drift typically make these equations particularly stiff. In practice, this means that it is tricky to construct, operate, and validate numerical methods for SPDEs. This is especially true if one is interested in path-dependent expected values, long-time simulations, or in the simulation of SPDEs whose solutions have constraints on their domains. To address these numerical issues, this paper introduces a Markov jump process approximation for SPDEs, which we refer to as the spectral random walk method (SPECTRWM). The accuracy and ergodicity of SPECTRWM are verified in the context of a heat and overdamped Langevin SPDE, respectively. We also apply the method to Burgers and KPZ SPDEs.

The purpose of these lecture notes is to provide an introduction to computational methods for the approximate solution of ordinary di_erential equations (ODEs).Only minimal prerequisites in di_erential and integral calculus, di_erential equation theory, complex analysis and linear algebra are assumed. The notes focus on the construction of numerical algorithms for ODEs and the mathematical analysis of their behaviour, covering the material taught in the M.Sc. in Mathematical Modelling and Scienti�c Computation in the eight-lecture course Numerical Solution of Ordinary Di_erential Equations.

The purpose of these lecture notes is to provide an introduction to computational methods for the approximate solution of ordinary di_erential equations (ODEs).Only minimal prerequisites in di_erential and integral calculus, di_erential equation theory, complex analysis and linear algebra are assumed. The notes focus on the construction of numerical algorithms for ODEs and the mathematical analysis of their behaviour, covering the material taught in the M.Sc. in Mathematical Modelling and Scienti�c Computation in the eight-lecture course Numerical Solution of Ordinary Di_erential Equations.

The purpose of these lecture notes is to provide an introduction to computational methods for the approximate solution of ordinary di_erential equations (ODEs).Only minimal prerequisites in di_erential and integral calculus, di_erential equation theory, complex analysis and linear algebra are assumed. The notes focus on the construction of numerical algorithms for ODEs and the mathematical analysis of their behaviour, covering the material taught in the M.Sc. in Mathematical Modelling and Scienti�c Computation in the eight-lecture course Numerical Solution of Ordinary Di_erential Equations.

The purpose of these lecture notes is to provide an introduction to computational methods for the approximate solution of ordinary di_erential equations (ODEs).Only minimal prerequisites in di_erential and integral calculus, di_erential equation theory, complex analysis and linear algebra are assumed. The notes focus on the construction of numerical algorithms for ODEs and the mathematical analysis of their behaviour, covering the material taught in the M.Sc. in Mathematical Modelling and Scienti�c Computation in the eight-lecture course Numerical Solution of Ordinary Di_erential Equations.

We derive the numerical schemes for the strong order integration of the set of the stochastic differential equations (SDEs) corresponding to the non-stationary Parker transport equation (PTE). PTE is 5-dimensional (3 spatial coordinates, particles energy and time) Fokker- Planck type equation describing the non-stationary the galactic cosmic ray (GCR) particles transport in the heliosphere. We present the formulas for the numerical solution of the obtained set of SDEs driven by a Wiener process in the case of the full three-dimensional diffusion tensor. We introduce the solution applying the strong order Euler-Maruyama, Milstein and stochastic Runge-Kutta methods. We discuss the advantages and disadvantages of the presented numerical methods in the context of increasing the accuracy of the solution of the PTE.

We derive the numerical schemes for the strong order integration of the set of the stochastic differential equations (SDEs) corresponding to the non-stationary Parker transport equation (PTE). PTE is 5-dimensional (3 spatial coordinates, particles energy and time) Fokker- Planck type equation describing the non-stationary the galactic cosmic ray (GCR) particles transport in the heliosphere. We present the formulas for the numerical solution of the obtained set of SDEs driven by a Wiener process in the case of the full three-dimensional diffusion tensor. We introduce the solution applying the strong order Euler-Maruyama, Milstein and stochastic Runge-Kutta methods. We discuss the advantages and disadvantages of the presented numerical methods in the context of increasing the accuracy of the solution of the PTE.

We derive the numerical schemes for the strong order integration of the set of the stochastic differential equations (SDEs) corresponding to the non-stationary Parker transport equation (PTE). PTE is 5-dimensional (3 spatial coordinates, particles energy and time) Fokker- Planck type equation describing the non-stationary the galactic cosmic ray (GCR) particles transport in the heliosphere. We present the formulas for the numerical solution of the obtained set of SDEs driven by a Wiener process in the case of the full three-dimensional diffusion tensor. We introduce the solution applying the strong order Euler-Maruyama, Milstein and stochastic Runge-Kutta methods. We discuss the advantages and disadvantages of the presented numerical methods in the context of increasing the accuracy of the solution of the PTE.

We derive the numerical schemes for the strong order integration of the set of the stochastic differential equations (SDEs) corresponding to the non-stationary Parker transport equation (PTE). PTE is 5-dimensional (3 spatial coordinates, particles energy and time) Fokker- Planck type equation describing the non-stationary the galactic cosmic ray (GCR) particles transport in the heliosphere. We present the formulas for the numerical solution of the obtained set of SDEs driven by a Wiener process in the case of the full three-dimensional diffusion tensor. We introduce the solution applying the strong order Euler-Maruyama, Milstein and stochastic Runge-Kutta methods. We discuss the advantages and disadvantages of the presented numerical methods in the context of increasing the accuracy of the solution of the PTE.

The variational grid generation method is a powerful tool for generating structured convex grids on irregular simply connected domains whose boundary is a polygonal Jordan curve. Several examples that show the accuracy of a difference approximation to the solution of a Poisson equation using these kind of structured grids have been recently reported. In this paper, we compare the accuracy of the numerical solution calculated by applying those structured grids with finite differences against the the solution obtained with Delaunay-like triangulations on irregular regions.

This paper introduces time-continuous numerical schemes to simulate stochastic differential equations (SDEs) arising in mathematical finance, population dynamics, chemical kinetics, epidemiology, biophysics, and polymeric fluids. These schemes are obtained by spatially discretizing the Kolmogorov equation associated with the SDE in such a way that the resulting semi-discrete equation generates a Markov jump process that can be realized exactly using a Monte Carlo method. In this construction the spatial increment of the approximation can be bounded uniformly in space, which guarantees that the schemes are numerically stable for both finite and long time simulation of SDEs. By directly analyzing the generator of the approximation, we prove that the approximation has a sharp stochastic Lyapunov function when applied to an SDE with a drift field that is locally Lipschitz continuous and weakly dissipative. We use this stochastic Lyapunov function to extend a local semimartingale representation of the approximation. This extension permits to analyze the complexity of the approximation. Using the theory of semigroups of linear operators on Banach spaces, we show that the approximation is (weakly) accurate in representing finite and infinite-time statistics, with an order of accuracy identical to that of its generator. The proofs are carried out in the context of both fixed and variable spatial step sizes. Theoretical and numerical studies confirm these statements, and provide evidence that these schemes have several advantages over standard methods based on time-discretization. In particular, they are accurate, eliminate nonphysical moves in simulating SDEs with boundaries (or confined domains), prevent exploding trajectories from occurring when simulating stiff SDEs, and solve first exit problems without time-interpolation errors.

This paper introduces time-continuous numerical schemes to simulate stochastic differential equations (SDEs) arising in mathematical finance, population dynamics, chemical kinetics, epidemiology, biophysics, and polymeric fluids. These schemes are obtained by spatially discretizing the Kolmogorov equation associated with the SDE in such a way that the resulting semi-discrete equation generates a Markov jump process that can be realized exactly using a Monte Carlo method. In this construction the spatial increment of the approximation can be bounded uniformly in space, which guarantees that the schemes are numerically stable for both finite and long time simulation of SDEs. By directly analyzing the generator of the approximation, we prove that the approximation has a sharp stochastic Lyapunov function when applied to an SDE with a drift field that is locally Lipschitz continuous and weakly dissipative. We use this stochastic Lyapunov function to extend a local semimartingale representation of the approximation. This extension permits to analyze the complexity of the approximation. Using the theory of semigroups of linear operators on Banach spaces, we show that the approximation is (weakly) accurate in representing finite and infinite-time statistics, with an order of accuracy identical to that of its generator. The proofs are carried out in the context of both fixed and variable spatial step sizes. Theoretical and numerical studies confirm these statements, and provide evidence that these schemes have several advantages over standard methods based on time-discretization. In particular, they are accurate, eliminate nonphysical moves in simulating SDEs with boundaries (or confined domains), prevent exploding trajectories from occurring when simulating stiff SDEs, and solve first exit problems without time-interpolation errors.