Implicit Hybrid Method Research Articles

Numerical Simulation of Nonlinear Dynamics of Breast Cancer Models Using Continuous Block Implicit Hybrid Methods

In the search for causes and cures of cancer diseases, many mathematical models developed have resulted in systems of nonlinear stiff ordinary differential equations. With these models, many numerical estimates of biological knowledge of the parameters have been obtained, a number of phenomena interpreted, and predictions were made in order to gain further knowledge of cancer development and possible treatment. In this study, numerical simulations of the models were performed using continuous block implicit hybrid methods and the results obtained support the theoretical and clinical findings. We analyzed the interactions among the various tumor cell populations and present the results graphically. From the graphical representation of results, one can clearly see the effects of all the tumor cell populations involved in the competition, as well as the effects of some treatments by the applications of some therapeutic agents which have been heavily used in the clinical treatments of breast cancer. The treatments in the past were mostly conventional chemotherapies, which were used either singly (alone) or in combination with each other or other therapies, and all played vital roles, except for the side effects that these therapies incur in normal tissues and organs. Thus, from recent research works, it is now clear that in many cases they do not represent a complete cure. Therefore, the need to address not only the preventative measures of breast cancer, but also more successful treatment, is clear, and can be successfully achieved to increase the survival rate of breast cancer patients.

An explicit and implicit hybrid method for structural topology optimization

In order to improve the manufacturability of topology optimization results, this paper proposes a hybrid method based on explicit description of Moving Morphable Components (MMC) and implicit description of Solid Isotropic Material with Penalization (SIMP). The method uses the global convergence characteristics of SIMP to quickly obtain the main force transfer path of structure, and threshold processing is used to eliminate the problem of gray-scale elements generated by SIMP. Furthermore, a morphological idea is proposed to simplify the structure. Then, components are used to fit the structure and extract its geometric parameters, and transition to the MMC is further optimized. Finally, this paper studies several typical examples, and compares them with the single MMC method from three aspects of design domain components layout, structural compliance values, and structural uniformity. The results show that the structure obtained by hybrid method has smaller structural compliance value and significant increase in the uniformity of the structure size.

A reliable method for first order delay equations based on the implicit hybrid method

In this paper, a reliable approach based on the implicit hybrid method is presented to solve first order delay problems. The main difficulty in this approach is that, some points are not grid points. We use interpolation to overcome this difficulty. Theoretical results including stability and convergence results are presented. The efficiency of the proposed approach is tested through two examples.

A Fifth-fourth Continuous Block Implicit Hybrid Method for the Solution of Third Order Initial Value Problems in Ordinary Differential Equations

In this paper, block method was developed using method of collocation and interpolation of power series as approximate solution to give a system of non linear equations which is solved to give a continuous hybrid linear multistep method. The continuous hybrid linear multistep method is solved for the independent solutions to give a continuous hybrid block method which is then evaluated at some selected grid points to give a discrete block method. The basic properties of the discrete block method were investigated and found to be zero stable, consistent and convergent. The derived scheme was tested on some numerical examples and was found to give better approximation than the existing method.

Implicit hybrid methods for solving fractional Riccati equation

Implicit hybrid methods for solving fractional Riccati equation

An implicit hybrid method for the computation of rotorcraft flows

ABSTRACTThere is a wide variety of CFD grid types including Cartesian, structured, unstructured and hybrids, as well as, numerous methodologies of combining these to reduce the time required to generate high-quality grids around complex configurations. If the grid methodologies were implemented in different codes, they should be written in such a way as to obtain the maximum performance from the available computer resources. A common interface should also be required to allow for ease of use. However, it is very time consuming to develop, maintain and add extra functionally to different codes. This paper examines the possibility of taking an existing CFD solver, the Helicopter Multi-Block (HMB) CFD method, and implementing a new grid type while reusing as much as possible the original code base. The paper presents some of the challenges encountered in extending the code which was written for a single mesh type, to a more flexible solver that is still computationally efficient but can cope with a variety of grid types.

An Implicit Hybrid Method for Solving Fractional Bagley-Torvik Boundary Value Problem

In this article, a modified implicit hybrid method for solving the fractional Bagley-Torvik boundary (BTB) value problem is investigated. This approach is of a higher order. We study the convergence, zero stability, consistency, and region of absolute stability of the modified implicit hybrid method. Three of our numerical examples are presented.

Generalized one-step third derivative implicit hybrid block method for the direct solution of second order ordinary differential equation

In this article, an implicit hybrid method of order six is developed for the direct solution of second order ordinary differential equations using collocation and interpolation approach.To derive this method, the approximate solution power series is interpolated at the first and off-step points and its second and third derivatives are collocated at all points in the given interval.Besides having good numerical method properties, the new developed method is also superior to the existing methods in terms of accuracy when solving the same problems.

An Improved Self-Starting Implicit Hybrid Method

The paper presents the derivation a hybrid block method for the solutions of Initial Value Problems of Ordinary Differential Equations. This is achieved by using collocation and interpolation technique to contruct a self-starting method with continuous coefficients together with the additional methods from its first derivative which are combined to form a single block that simultaneously provide the approximate solutions for Initial Value Problem. The analysis of the properties of the method such as stability, consistency and convergence are discussed and the performance of the method is demonstrated on two test problems to show the accuracy and efficiency of the method. On comparison of the results obtained from numerical examples with some existing methods, we discovered that the method behaved favourably well.

Multi-step implicit iterative methods with regularization for minimization problems and fixed point problems

Abstract In this paper we introduce a multi-step implicit iterative scheme with regularization for finding a common solution of the minimization problem (MP) for a convex and continuously Fréchet differentiable functional and the common fixed point problem of an infinite family of nonexpansive mappings in the setting of Hilbert spaces. The multi-step implicit iterative method with regularization is based on three well-known methods: the extragradient method, approximate proximal method and gradient projection algorithm with regularization. We derive a weak convergence theorem for the sequences generated by the proposed scheme. On the other hand, we also establish a strong convergence result via an implicit hybrid method with regularization for solving these two problems. This implicit hybrid method with regularization is based on the CQ method, extragradient method and gradient projection algorithm with regularization. MSC:49J30, 47H09, 47J20.

Implicit Relaxed and Hybrid Methods with Regularization for Minimization Problems and Asymptotically Strict Pseudocontractive Mappings in the Intermediate Sense

We first introduce an implicit relaxed method with regularization for finding a common element of the set of fixed points of an asymptotically strict pseudocontractive mappingSin the intermediate sense and the set of solutions of the minimization problem (MP) for a convex and continuously Frechet differentiable functional in the setting of Hilbert spaces. The implicit relaxed method with regularization is based on three well-known methods: the extragradient method, viscosity approximation method, and gradient projection algorithm with regularization. We derive a weak convergence theorem for two sequences generated by this method. On the other hand, we also prove a new strong convergence theorem by an implicit hybrid method with regularization for the MP and the mappingS. The implicit hybrid method with regularization is based on four well-known methods: the CQ method, extragradient method, viscosity approximation method, and gradient projection algorithm with regularization.

Semi Implicit Hybrid Methods with Higher Order Dispersion for Solving Oscillatory Problems

We constructed three two-step semi-implicit hybrid methods (SIHMs) for solving oscillatory second order ordinary differential equations (ODEs). The first two methods are three-stage fourth-order and three-stage fifth-order with dispersion order six and zero dissipation. The third is a four-stage fifth-order method with dispersion order eight and dissipation order five. Numerical results show that SIHMs are more accurate as compared to the existing hybrid methods, Runge-Kutta Nyström (RKN) and Runge-Kutta (RK) methods of the same order and Diagonally Implicit Runge-Kutta Nyström (DIRKN) method of the same stage. The intervals of absolute stability or periodicity of SIHM for ODE are also presented.

Chebyshev Collocation Approach for a Continuous Formulation of Implicit Hybrid Methods for Vips In Second Order Odes

Abstract: In this paper, an implicit one-step method for numerical solution of second order Initial Value Problems of Ordinary Differential Equations has been developed by collocation and interpolation technique. The one-step method was developed using Chebyshev polynomial as basis function and, the method was augmented by the introduction of offstep points in order to bring about zero stability and upgrade the order of consistency of the new method. An advantage of the derived continuous scheme is that it can produce several outputs of solution at the off-grid points without requiring additional interpolation. Numerical examples are presented to portray the applicability and the efficiency of the method.

A partially implicit hybrid method for computing interface motion in Stokes flow

We present a partially implicit hybrid method for simulating the motionof a stiff interface immersed in Stokes flow, in free space or in arectangular domain with boundary conditions.We assume the interface is aclosed curve which remains in the interior of the computational region.The implicit time integration is based on the small-scale decomposition approachand does not require the iterative solution of a system of nonlinearequations.First-order and second-order versions of the time-stepping method arederived systematically, and numerical results indicate that both methodsare substantially more stable than explicit methods.At each time level, the Stokes equations are solved using a hybrid approachcombining nearly singular integrals on a band of mesh points near theinterface and a mesh-based solver. The solutions aresecond-order accurate inspace and preserve the jump discontinuities across the interface.Finally, the hybrid method can be used as an alternative to adaptive meshrefinement to resolve boundary layers that are frequently present arounda stiff immersed interface.

Continuous block implicit hybrid one-step methods for ordinary and delay differential equations

A class of high order continuous block implicit hybrid one-step methods has been proposed to solve numerically initial value problems for ordinary and delay differential equations. The convergence and A ω -stability of the continuous block implicit hybrid methods for ordinary differential equations are studied. Alternative form of continuous extension is constructed such that the block implicit hybrid one-step methods can be used to solve delay differential equations and have same convergence order as for ordinary differential equations. Some numerical experiments are conducted to illustrate the efficiency of the continuous methods.

VARIABLE TIME-STEPPING HYBRID FINITE DIFFERENCE METHODS FOR PRICING BINARY OPTIONS

Two types of new methods with variable time steps are proposed in order to valuate binary options efficiently. Type I changes adaptively the size of the time step at each time based on the magnitude of the local error, while Type II combines two uniform meshes. The new methods are hybrid finite difference methods, namely starting the computation with a fully implicit finite difference method for a few time steps for accuracy then performing a <TEX>${\theta}$</TEX>-method during the rest of computation for efficiency. Numerical experiments for standard European vanilla, binary and American options show that both Type I and II variable time step methods are much more efficient than the fully implicit method or hybrid methods with uniform time steps.