Fuzzy Differential Equation

The aim of this article is to implement the Generalized Modified Adomian Decomposition Method to compute the semi-numerical solution of the linear system of intuitionistic fuzzy initial value problems. Here, we consider the initial values as generalized trapezoidal intuitionistic fuzzy numbers. The technique is applied to brine tanks problem and coupled mass spring systems.Theoretically, different approaches to solving a system of generalized trapezoidal intuitionistic fuzzy differential equations are discussed in this study under the presumption that the coefficients of the system of the differential equations are associated to generalized trapezoidal intuitionistic fuzzy numbers. The approximate results are compared with exact solutions which shows good efficiency. The corresponding graphs at different levels of uncertainty show the example's numerical outcomes. The graphical representations further demonstrate the effectiveness and accuracy of the proposed method in comparison to existing semi-numerical methods in the literature. INDEX TERMS Fuzzy set, fuzzy number, generalized trapezoidal intuitionistic fuzzy number, system of fuzzy differential equation, analytical technique, engineering applications.

In this paper, a solution procedure for the solution of the system of fuzzy differ-ential equations ˙x(t) = α(A − In)[t(A − In) + In] ... −1 x(t), x(0) = x0 where A is real ... Keywords: Fuzzy number, Fuzzy linear system, Fuzzy differential equations, Func- ... In order to solve an n×n fuzzy ...

This paper aims to develop a fourth order fuzzy finite difference scheme to solve an intuitionistic fuzzy hyperbolic partial differential equation. The initial and boundary conditions of the intuitionistic fuzzy hyperbolic partial differential equation are intuitionistic triangular fuzzy numbers. The proposed finite difference scheme is found to be conditionally stable, and convergence of the proposed fuzzy finite difference method is discussed in detail. Further, the proposed method is validated through an example. The approximate solution is compared with the exact solution at each level and these results are illustrated through graphs and tables for each space level.

Operator splitting is a powerful method for the numerical investigation of complex (physical) time-dependent models, where the stationary (elliptic) part consists of a sum of several simpler operators (processes. Some fields where different splitting methods play a crucial role include the air-pollution phenomena, Maxwell's equations or the Hamiltonian systems. These tasks are usually very complicated, and therefore, the analytical solution is impossible to find, moreover, the numerical modelling with direct discretization is also hopeless from a practical point of view. To avoid this difficulty, the operator splitting is introduced and applied.

In this paper, a one-step hybrid block method is formulated for the numerical approximation of Fuzzy Differential Equations (FDEs). In deriving the method, the techniques of interpolation and collocation were adopted using the sum of the first seven terms of Legendre polynomial as basis function. The convergence and stability properties of the method derived were analyzed and shown to be convergent and stable. The method was then applied in approximating FDEs and the results obtained show that the method competes favorably with existing ones.

The aim of this article is to implement the Generalized Modified Adomian Decomposition Method to compute the semi-numerical solution of the linear system of intuitionistic fuzzy initial value problems. Here, we consider the initial values as generalized trapezoidal intuitionistic fuzzy numbers. The technique is applied to brine tanks problem and coupled mass spring systems.Theoretically, different approaches to solving a system of generalized trapezoidal intuitionistic fuzzy differential equations are discussed in this study under the presumption that the coefficients of the system of the differential equations are associated to generalized trapezoidal intuitionistic fuzzy numbers. The approximate results are compared with exact solutions which shows good efficiency. The corresponding graphs at different levels of uncertainty show the example's numerical outcomes. The graphical representations further demonstrate the effectiveness and accuracy of the proposed method in comparison to existing semi-numerical methods in the literature. INDEX TERMS Fuzzy set, fuzzy number, generalized trapezoidal intuitionistic fuzzy number, system of fuzzy differential equation, analytical technique, engineering applications.

تقدم هذه الورقة حلاً عملي ا لمشكلة حل المعادلات التفاضلية الخطية في وجود عدم الدقة .تعتمد الطريقة المقترحة
وطرح الأعداد الضبابية لتحويل المعادلة الضبابية إلى معادلة يمكن حلها باستخدام الأساليب التقليدية. الدراسة تغطي ثلاث حالات مختلفة من المعاملات
الضبابية، بناءً على إشارات معاملات المعادلة. تم توضيح الطريقة من خلال أمثلة عملية، مما يوسع نطاق تطبيقاتها. النتائج التجريبية تؤكد فعالية هذه
الطريقة وقابليتها للتطبيق

A class of splitting iterative methods are considered for solving fuzzy system of linear equations, which covers Jacobi, Gauss Seidel, SOR, SSOR and their block variance proposed. Theoretical analysis showed that for a regular splitting, the corresponding iterative method con-verge to the unique fuzzy solution for any initial vector and fuzzy right hand side vector. Finally, we illustrate our approach by some numerical examples.

In this work, we consider special problem consisting of twelfth order two-point boundary value by using the Optimal Homotopy Asymptotic Method (OHAM). The proposed method has been thoroughly tested on problems of all kinds and shows very accurate results. Furthermore, we also compared the results obtained from this technique with the available exact solution in different literatures. This method provides easy tools to control the convergence region of approximating solution series where necessary. In order to show this, we will illustrate the method by given examples with numerical results.

Fuzzy Fixed Point Theory has emerged as a powerful tool in addressing uncertainties in numerical methods for solving fuzzy equations. This theory extends classical fixed point concepts to fuzzy environments, enabling the resolution of equations where parameters and solutions are expressed as fuzzy sets rather than crisp numbers. The paper explores the application of fuzzy fixed point theory in developing numerical methods for solving fuzzy equations, with a focus on its convergence properties, stability, and computational efficiency. Techniques such as iterative methods and fuzzy differential equations are examined, demonstrating the utility of fuzzy fixed point theory in handling imprecise or vague data. The results reveal that fuzzy fixed point-based numerical methods offer robust solutions in various fields, including engineering, finance, and decision-making, where uncertainty is prevalent.

In the present paper, two methods for the solution of an initial valued first ordered fuzzy differential equation are presented and applied in a fuzzy EOQ model. The constructed model is a bi‐level inventory problem involving wholesaler‐retailers‐customers. The wholesaler buys and sells the item instantaneously to several retailers. In the next level, the retailers sell the units to customers with a time dependent imprecise demand, which introduce the fuzzy nature in the differential equation. The selling price of the item is a step‐wise time dependent decreasing function. The fuzzy objectives are transformed into crisp one following fuzzy extension principle and centroid formula. The model is illustrated through Interactive Fuzzy Decision Making (IFDM) and Multi Objective Genetic Algorithm (MOGA) and the results from two methods are compared.

In this paper, a reliable numerical technique is proposed for solving a class of singular fractional differential equations involving Fredholm and Volterra operators subjected to suitable three-point boundary conditions. The solution methodology is presented based on reproducing-kernel method (RKM), which is used directly without employing linearization and perturbation. However, a favorable Hilbert spaces are construcred, and then the orthonormal function system is generated by using GramSchmidt orthogonalization process. Error analysis is given in Sobolev space. Numerical example is tested to multipoint singular fractional differential problems with Fredholm and Volterra operators to show the theoretical statements of the RKHS method. The results obtained indicate that the RKHS method is easy to implement, reliability and capability with a great potential of such singular problems.

Chemostat is a continuous stirred tank reactor used for continuous microbial biomass production in commercial, medical and other research problems. While modeling real world phenomena through differential equations as backbone of practical problems, we need to introduce various parameters. These parameters may be vague, imprecise and uncertain. To incorporate these uncertainties, the notion of fuzzy differential equations is used in chemostat model as one of the tool. In this paper, we discuss some new results for the stability analysis of chemostat model and the results so obtained are justifiable analytically and verified graphically in fuzzy environment.

In this article, we apply Homotopy Perturbation Method (HPM) for solving three coupled non-linear equations which play an important role in biosystems. To illustrate the capability and reliability of this method. Numerical example is given which confirms our analytical findings.

The mathematical structure of some natural phenomena of nonlinear physical and engineering systems can be described by a combination of fuzzy differential equations that often behave in a way that cannot be fully understood. In this work, an accurate numeric-analytic algorithm is proposed, based upon the use of the residual power series, to investigate the fuzzy approximate solution for a nonlinear fuzzy Duffing oscillator, along with suitable uncertain guesses under strongly generalized differentiability. The proposed approach optimizes the approximate solution by minimizing a residual function to generate r-level representation with a rapidly convergent series solution. The influence, capacity, and feasibility of the method are verified by testing some applications. Level effects of the parameter r are given graphically and quantitatively, showing good agreement between the fuzzy approximate solutions of upper and lower bounds, that together form an almost symmetric triangular str...

In order to obtain sufficient solutions for fuzzy differential equations (FDEs), reliable and efficient approximation methods are necessary. Approximate numerical methods can not directly solve fuzzy HIV models. Meanwhile, the approximate analytical methods can potentially provide more straightforward solutions without the need for extensive numerical computations or linearization and discretization techniques, which may be challenging to apply to fuzzy models. One significant advantage of approximate analytical methods is their ability to provide insights into solution accuracy without requiring an exact solution for comparison, where an exact solution may not be readily available. In this work, the fuzzy nonlinear HIV infection model is analyzed and solved using the new fuzzy form of an approximate analytical method. Fuzzy set theory mixed with standard fuzzy variational iteration method (FVIM) properties is utilized to produce a new formulation denoted by the multistage fuzzy variational iteration method (MFVIM) to process and solve a fuzzy nonlinear HIV infection model. MFVIM offers an effective method for attaining convergence in the series solution presented as a polynomial function. This approach enables efficient solutions to diverse mathematical challenges. The solution methodology is reliant on fuzzy differential equations conversion into systems of ordinary differential equations, utilizing the parametric form regarding its r-level representations, and considering the approximate solution of the system in a sequence of intervals. Subsequently, the equivalent classical systems are resolved by applying FVIM algorithms in each subinterval. Also, the existence and unique solution analysis of the proposed problem have been described, along with a fuzzy optimal control analysis. A tabular and graphical representation of the MFVIM of the proposed models is presented and analyzed in comparison with the numerical method and FVIM. The new method produces better performance in terms of solutions than a numerical method with a simple implementation for solving fuzzy nonlinear HIV infection model associated with FIVPs. The ability to better comprehend the behavior of the system under investigation can enable researchers and scientists to work on models incorporating systems with long memories and ill-defined notions to make more effective design and decision-making.

The paper represents a variation of the national income determination model with discrete and continuous process in fuzzy environment, a significant implication in economics planning, by means of fuzzy assumptions. This model is re-recognized and deliberated with fuzzy numbers to estimate its uncertain parameters whose values are not precisely known. Exhibition of imprecise solutions of the concerned model is carried out by using the proposed two methods: generalized Hukuhara difference and generalized Hukuhara derivative (gH-derivative) approaches. Moreover, the stability analysis of the model in two different systems in fuzzy environment is illustrated. Additionally, different illustrative examples for optimization of national income determination model are undertaken with the constructive graph and table for convenience for clarity of the projected approaches.

In this article, we present a one-step hybrid block method for approximating the solutions of second-order fuzzy initial value problems. We prove the stability and convergence results of the method and present several examples to illustrate the efficiency and accuracy of the proposed method. The numerical results are compared with the existing ones in the literature.

Theory of fuzzy differential equations is the important new developments to model various science and engineering problems of uncertain nature because this theory represents a natural way to model dynamical systems under uncertainty. Since, it is too difficult to obtain the exact solution of fuzzy differential equations so one may need reliable and efficient numerical techniques for the solution of fuzzy differential equations. In this chapter we have presented various numerical techniques viz. Euler and improved Euler type methods and Homotopy Perturbation Method (HPM) to solve fuzzy differential equations. Also application problems such as fuzzy continuum reaction diffusion model to analyse the dynamical behaviour of the fire with fuzzy initial condition is investigated. To analyse the fire propagation, the complex fuzzy arithmetic and computation are used to solve hyperbolic reaction diffusion equation. This analysis finds the rate of burning number of trees in bounds where wave variable/ time are defined in terms of fuzzy. Obtained results are compared with the existing solution to show the efficiency of the applied methods.

Present paper proposes a new technique based on double parametric form of fuzzy numbers to solve an uncertain beam equation using Adomian decomposition method subject to unit step and impulse loads. Uncertainties appear in the initial conditions are considered in terms of triangular convex normalized fuzzy sets. Using the single parametric form viz. ␣-cut form of fuzzy numbers, the fuzzy beam equation is converted first to an interval based fuzzy differential equation. Next this differential equation is transformed to crisp form by applying double parametric form of fuzzy numbers. Finally the same is solved by Adomian decomposition method symbolically to obtain the uncertain bounds of the dynamic response. Obtained results are depicted in term of plots to show the efficiency and powerfulness of the present analysis.

This paper proposes a new technique based on double parametric form of fuzzy numbers to handle the uncertain vibration equation for very large membrane for different particular cases. Uncertainties present in the initial condition and the wave velocity of free vibration are modelled through Gaussian convex normalised fuzzy set. Using the single parametric form of fuzzy number, the original fuzzy vibration equation is converted first to an interval fuzzy vibration equation. Next this equation is transformed to crisp form by applying double parametric form of fuzzy numbers. Finally the same governing equation is solved by Adomian decomposition method (ADM) symbolically to obtain the uncertain bounds. The present methods are very simple and effective. Obtained results are depicted in terms of plots to show the efficiency and powerfulness of the present analysis. Results obtained by the methods with new techniques are compared with existing results in special cases.

Chemostat is a continuous stirred tank reactor used for continuous microbial biomass production in commercial, medical and other research problems. While modeling real world phenomena through differential equations as backbone of practical problems, we need to introduce various parameters. These parameters may be vague, imprecise and uncertain. To incorporate these uncertainties, the notion of fuzzy differential equations is used in chemostat model as one of the tool. In this paper, we discuss some new results for the stability analysis of chemostat model and the results so obtained are justifiable analytically and verified graphically in fuzzy environment.

The critical slowing down (CSD) phenomenon of the switching time in response to perturbation β (0 < β < 1) of the control parameters at the critical points of the steady state bistable curves, associated with two biological models (the spruce budworm outbreak model and the Thomas reaction model for enzyme membrane) is investigated within fractional derivative forms of order α (0 < α < 1) that allows for memory mechanism. We use two definitions of fractional derivative, namely, Caputo’s and Caputo-Fabrizio’s fractional derivatives. Both definitions of fractional derivative yield the same qualitative results. The interplay of the two parameters α (as memory index) and β shows that the time delay τD can be reduced or increased, compared with the ordinary derivative case (α = 1). Further, τD fits: (i) as function of β the scaling inverse square root formula 1/β at fixed fractional derivative index (α < 1) and, (ii) as a function of α (0 < α < 1) an exponentially inc...

Chemostat is a continuous stirred tank reactor used for continuous microbial biomass production in commercial, medical and other research problems. While modeling real world phenomena through differential equations as backbone of practical problems, we need to introduce various parameters. These parameters may be vague, imprecise and uncertain. To incorporate these uncertainties, the notion of fuzzy differential equations is used in chemostat model as one of the tool. In this paper, we discuss some new results for the stability analysis of chemostat model and the results so obtained are justifiable analytically and verified graphically in fuzzy environment.

In this paper, we discuss the stability analysis of logistic growth model with immigration function in fuzzy environment.
The notion of generalized Hukuhara (gH) differentiability is used for the analysis when the immigration function is proportional to
the population and some new results have been established with numerical verification.

The main focus of this paper is to develop an efficient analytical method to obtain the traveling wave fuzzy solution for the fuzzy generalized Hukuhara conformable fractional equations by considering the type of generalized Hukuhara conformable fractional differentiability of the solution. To achieve this, the fuzzy conformable fractional derivative based on the generalized Hukuhara differentiability is defined, and several properties are brought on the topic, such as switching points and the fuzzy chain rule. After that, a new analytical method is applied to find the exact solutions for two famous mathematical equations: the fuzzy fractional Wave equation and the fuzzy fractional Diffusion equation. The present work is the first report in which the fuzzy traveling wave method is used to design an analytical method to solve these fuzzy problems. The final examples are asserted that our new method is applicable and efficient.

The main purpose of this paper is to obtain an analytical solution for the time-fractional fuzzy equation. To do this, the time-fractional equation is  transformed into an algebraic equation using the fuzzy Laplace and Fourier transforms. The fractional derivatives are described in the Caputo gH-differentiability. In addition, to demonstrate the efficiency of the method some various examples are solved.

Bearing in mind the considerable importance of fuzzy differential equations (FDEs) in different fields of science and engineering, in this paper, nonlinear nth order FDEs are approximated, heuristically. The analysis is carried out on using Chebyshev neural network (ChNN), which is a type of single layer functional link artificial neural network (FLANN). Besides, explication of generalized Hukuhara differentiability (gH-differentiability) is also added for the nth order differentiability of fuzzy-valued functions. Moreover, general formulation of the structure of ChNN for the governing problem is described and assessed on some examples of nonlinear FDEs. In addition, comparison analysis of the proposed method with Runge-Kutta method is added and also portrayed the error bars that clarify the feasibility of attained solutions and validity of the method.

This paper presents the fifth order Runge-Kutta method (RK5) to find the numerical solution of the second order initial value problems of Bratu-type ordinary differential equations. In order to justify the validity and effectiveness of the method, we solve three model examples and compare the exact solutions to numerical solutions. The numerical results in terms of point wise absolute errors presented in tables and the plotted graphs show that the present method approximates the exact solution very well. Besides, the stability of the method had been checked and verified.

In this paper we propose a new analytical approach to the study of human gait dynamics. A new and reliable method, namely the Optimal Auxiliary Functions Method (OAFM) is employed to obtain explicit and accurate analytical solutions. The capabilities of this new method are successfully tested in the studyof human gait dynamics and an excellent agreement between analytical and numerical solutions is demonstrated. The accuracy of the analytical results is assured by the so-called convergence-control parameters, whose optimal values are rigorously identified in order to provide a fast convergence to the exact solution.

In this paper, we consider the following class of singular two-point boundary value problem posed on the interval x (0, 1] (g(x)y) = g(x)f (x, y), y (0) = 0, μy(1) + σy (1) = B. A recursive scheme is developed, and its convergence properties are studied. Further, the error estimation of the method is discussed. The proposed scheme is based on the integral equation formalism and optimal homotopy analysis method in which a recursive scheme is established without any undetermined coefficients. The original differential equation is transformed into an equivalent integral equation to remove the singularity. The integral equation is then made free of undetermined coefficients by imposing the boundary conditions on it. Finally, the integral equation without any undetermined coefficients is efficiently treated by using optimal homotopy analysis method for finding the numerical solution. The optimal control-convergence parameter involved in the components of the series solution is obtained by minimizing the squared residual error equation. The present method is applied to obtain numerical solution of singular boundary value problems arising in various physical models, and numerical results show the advantages of our method over the existing methods. Keywords Singular boundary value problem • Optimal homotopy analysis method • Undetermined coefficients • Oxygen-diffusion problem • Thermal-explosion problem • Shallow membrane cap problem Pradip Roul

In this paper, we have obtained an approximate solution of multi-term Caputo fractional differential equations (MFDEs) using the Variational iteration method (VIM). Further, we have obtained the convergence criteria and error approximation of VIM for solving generalized multi term fractional differential equations. The obtained results are shown using plots to demonstrated the efficiency and accuracy of the VIM

This paper targets to investigate the numerical solution of linear, non-linear and system of ordinary differential equations with fuzzy initial condition. Here, two Euler type methods have been proposed in order to obtain numerical solution of the fuzzy differential equations. Along with this, an exact solution methodology is also discussed. Obtained results are depicted in term of plots to show the efficiency of the proposed methods. The solutions are compared with the known results and are found to be in good agreement.

This paper investigates the nonlinear boundary value problem resulting from the exact reduction of the Navier-Stokes equations for unsteady magnetohydrodynamic boundary layer flow over the stretching/shrinking permeable sheet submerged in a moving fluid. To solve this equation, a numerical method is proposed based on a Laguerre functions with reproducing kernel Hilbert space method. Using the operational matrices of derivative, we reduced the problem to a set of algebraic equations. We also compare this work with some other numerical results and present a solution that proves to be highly accurate.

The point of this paper is to analyze and investigate the analytic-approximate solutions for fractional system of Volterra integro-differential equations in framework of Caputo-Fabrizio operator. The methodology relies on creating the reproducing kernel functions to gain analytical solutions in a uniform form of a rapidly convergent series in the Hilbert space. Using the Gram-Schmidt orthonomalization process, the orthonormal basis system is constructed in a dense compact domain to encompass the Fourier series expansion in view of reproducing kernel properties. Besides, convergence and error analysis of the proposed technique are discussed. For this purpose, several numerical examples are tested to demonstrate the great feasibility and efficiency of the present method and to support theoretical aspect as well. From a numerical point of view, the acquired solutions simulation indicates that the methodology used is sound, straightforward, and appropriate to deal with many physical issues in light of Caputo-Fabrizio derivatives.

Interactivity naturally occurs in many realworld problems. The main goal of this paper is to propose a first approach toward characterizing the interactivity that is inherently present in the derivative of an unknown interval or fuzzy number valued function that is subject to certain restrictions. A series of experiments regarding isothermal degradation of L-ascorbic acid tablets serves as an important case study for our approach.

In this work, variation of parameter method is applied to study two-dimensional flow of nanofluid in a porous channel through slowly deforming walls with suction or injection. The results of the developed approximate analytical solution using the variation of parameter method is verified with the results of numerical solution using fourth-order Runge-Kutta method coupled with shooing techniques. Thereafter, parametric studies are carried. The graphical illustrations of simulated results of the approximate analytical solutions show that during the expansion, the axial velocity at the center of the channel decreases as the Reynolds number increases while the axial velocity increases slightly near the surface of the channel when the wall contracts at the same rate. Also, the axial velocity decreases at the center of the channel but increases near the wall as the wall expansion ratio increases. Due to the high accuracy of the variation of parameter method, the results given in the work may be used for benchmark analysis of the subsequent studies on laminar flow behaviour of nanofluid in a porous channel through slowly deforming with injection or suction.

In this paper, we applied Optimal Homotopy Asymptotic Method (OHAM) to obtained numerical solution of nonlinear Benjamin-Bona-Mahony (BBM) and Sawada-Kotera (SK) equations. For BBM equation, solution of proposed method is compared with Homotopy Perturbation Method (HPM), Adomian Decomposition Method (ADM) and with exact solution. For SK equation comparison is made with the exact solution. The plots of exact verses approximate solutions is also drawn and it is found that approximate solution obtained by (OHAM) is much closer to the exact solution. OHAM is found efficient and fast convergent during computation by Mathematica.

In this paper, new solutions to nonlinear pseudo-hyperbolic equations with non-local conditions by residual power series (RPS) method is given. This method is based on the Taylor series formula and the residual error function. A new analytical solution to this equation is given. As an applications we have tested some examples to know the efficiency of current technique. The results obtained showed that the proposed method is effective, accurate and has fast convergent.

This paper proposes a new technique based on Galerkin method for solving nth order fuzzy boundary value problem. The proposed method has been illustrated by considering three different cases depending upon the sign of coefficients with benchmark example problems. To show the applicability of the proposed method, an application problem related to heat conduction has also been studied. The results obtained by the proposed methods are compared with the exact solution and other existing methods for demonstrating the validity and efficiency of the present method.

In this article, we propose a new method that determines an efficient numerical procedure for solving second-order fuzzy Volterra integro-differential equations in a Hilbert space. This method illustrates the ability of the reproducing kernel concept of the Hilbert space to approximate the solutions of second-order fuzzy Volterra integro-differential equations. Additionally, we discuss and derive the exact and approximate solutions in the form of Fourier series with effortlessly computable terms in the reproducing kernel Hilbert space W 3 2 [a, b] ⊕ W .3 2 [a, b]. The convergence of the method is proven and its exactness is illustrated by three numerical examples.

In this paper, we first propose the right and the left fuzzy Riemann-Liouville integrals; then the related left and right fuzzy Caputo differentiabilities are introduced. Consequently, some useful results about integration and differentiation of fractional order under uncertainty have been obtained. Moreover, some equivalent integral forms of original fuzzy fractional differential equations are derived. Finally, the fuzzy Ostrowski inequality about fractional case has been stated.

In this paper, we study the numerical method for solving hybrid fuzzy differential using Euler method under generalized Hukuhara differentiability. To this end, we determine the Euler method for both cases of H-differentiability. Also, the convergence of the proposed method is studied and the characteristic theorem is given for both cases. Finally, some numerical examples are given to illustrate the efficiency of the proposed method under generalized Hukuhara differentiability instead of suing Hukuhara differentiability.

In this paper, numerical algorithms for solving “fuzzy ordinary differential equations” are considered. A scheme based on the Taylor method of order p is discussed in detail and this is followed by a complete error analysis. The algorithm is illustrated by solving some linear and nonlinear fuzzy Cauchy problems.

There are three types of problems that arise in the numerical analysis of differential equations: (i) the initial value problem (IVP), (ii) the boundary value problem (BVP), and (iii) IVP + BVP. In general, it is natural that (i) and (ii) are thoroughly examined before the study of (iii). Thus far, it has been shown that the BVP of one-dimensional (1D) Poisson equations can be solved with extremely high accuracy and speed using the interpolation finite difference method (IFDM). Furthermore, it has been found that a high-accuracy numerical calculation system for the IVP can be constructed by applying the polynomial interpolation method, as in the case of the BVP of the 1D Poisson equation. This method is called the interpolation numerical integration method (INIM). Numerical integration is possible with an arbitrary accuracy order by specifying the degree of the interpolation polynomial for representing the integrand. The proposed calculation method enables numerical calculations that seem to have no error. IVPs are usually calculated by the fourth-order accuracy Runge-Kutta method, which is often sufficient for engineering purposes. However, in this paper, we report that a numerical algorithm that agrees with the exact solution up to 15 significant digits has theoretical value.

Functional laws may be known only at a finite number of points, and then the function is completed by interpolation techniques obeying some smoothness conditions. We rather propose here to specify constraints by means of gradual rules for delimiting areas where the function may lie between known points. The more general case where the known points of the function are imprecisely located is also dealt with. The use of gradual rules for expressing constraints on the closeness with respect to reference points leads to interpolation graphs that are imprecise but still crisp. We thus propose a refinement of the rule-based representation that enables the handling of fuzzy interpolation graphs.

We use the expansion formula for the fractional derivatives to reduce the problem of solving non-linear fractional order differential equations arising in mechanics to the problem of solving a system of integer order differential equations. We prove the convergence of the solutions to the reduced integer order systems to the solutions of the original problem. The procedure is illustrated by two specific examples.