solution methods for partial differential equations

... numerical solution is greatly desired.High-order numerical methods for solving ordinary differential equations (ODEs) are very common For instance, there are many variations of multistep methods ... financial market?Stochastic Differential Equations 0.004 10Stochastic differential equations (SDEs) are defined by a vector field that includes a random (Brownian) forcing Such equations have become ... on noise.Numerical Solution of Stochastic Differential EquationsA thorough background to SDEs and their numerical solution can be found in Kloeden and Platen [50] Although the solution of SDEs

Ngày tải lên: 02/10/2024, 01:54

... ness of fuzzy solutions for simple fuzzy heat equations and wave equations withconcrete formulation of the solutions.Arara et al.(2005) considered the local andnonlocal initial problem for some classes ... Moreover, in view of the develop-ment of calculus for fuzzy functions, the investigation of fuzzy differential equations(DEs) and fuzzy partial differential equations (PDEs) have been initiated (Buckleyand ... MakingDOI 10.1007/s10700-014-9186-0 The existence and uniqueness of fuzzy solutions for hyperbolic partial differential equations Hoang Viet Long · Nguyen Thi Kim Son · Nguyen Thi My Ha · Le Hoang

Ngày tải lên: 16/12/2017, 08:45

... meth-ods have been developed for DAEs Most numerical methods for differentialalgebraic equations based on standard methods from the theory of ordinarydifferential equations It is well known that ... thesis Therefore, this thesis concentrates on numerical methods for semi-explicitDAEs of index 2 Here, we are concerned with one-step methods for index 2 DAEs in senberg form These methods combine ... half-explicitRunge-Kutta (HERK) methods for index 2 DAEs in Hessenberg form Weconsider the numerical solution of systems of semi-explicit index 2 differential-algebraic equations (DAEs) by methods based on

Ngày tải lên: 03/01/2018, 08:38

... Kim Son and Ha Thi Thanh Tam solutions and Seikkala solutions for fuzzy partial differential equations by using crisp solution and the extension principle Some other efforts have been recently made ... recently made to find the numerical solutions for fuzzy partial differential equations by Allahviranloo [1] With regards to the fuzzy hyperbolic partial differential equations with local and nonlocal ... especially partial differential equations When databases that are transformed from real life into mathematical models are incomplete or vague, we often use fuzzy partial differential equations

Ngày tải lên: 25/11/2020, 23:24

... descriptor form.Implicit Runge-Kutta methodsFormula of implicit Runge-Kutta methodsThe s-stage Runge-Kutta method for semi-explicit index 2 differential-algebraic equations (DAEs) is defined for a ... This modification allows for the extension of well-established high-order explicit Runge-Kutta methods for ordinary differential equations (ODEs) to half-explicit forms for differential-algebraic ... implicit RKmethods it is im- portant to notice that from (2.7b) allinternal stages g(y) = 0, whereas the numerical solution does not However, for stiffly accurate RK methods, i.e.,for methods satisfying

Ngày tải lên: 23/12/2021, 21:29

... Animation of Solutions to Partial Differential Equations Introduction In the past three years the authors have developed a series of MATLABTM scripts that illustrate the solutions to partial differential ... physical phenomena governed by partial differential equations Over many years the authors have observed the difficulty students have with the solutions to partial differential equation problems ... to clarify physical problems governed by partial differential equations An early paper used MATLABTM to illustrate solutions to hyperbolic differential equations.1 Several papers at about the

Ngày tải lên: 01/11/2022, 23:30

... E S E A R C H Open AccessFeynman-Kac formula for switching diffusions: connections of systems of partial differential equations and stochastic differential equations Nicholas A Baran1, George ... reproducAttribu-tion in any Trang 2For diffusion processes, the Feynman-Kac formula provides a stochastic representation for solutions to certain second-order partial differential equations (PDEs) These repre-sentations ... numerical methods for solution of stochastic differential equations with switching, etc., have been obtained Nevertheless, certain important concepts are yet fully investigated The Feynman-Kac formula

Ngày tải lên: 04/12/2022, 10:35

... Advances in Difference Equations Volume 2011, Article ID 404917, 16 pages doi:10.1155/2011/404917 Research Article Nonlocal Boundary Value Problem for Impulsive Differential Equations of Fractional ... fractional differential equations By means of a fixed point theorem due to O’Regan, we establish sufficient conditions for the existence of at least one solution of the problem For the illustration ... differential equations also provide an excellent tool for the description of memory and hereditary properties of many materials and processes In consequence, fractional differential equations have

Ngày tải lên: 21/06/2014, 07:20

... solution for a class of second-order nonlinear impulsive differential equations on an infinite interval with an infinite number of impulsive times Consider the following boundary value problem for ... are used to investigate the minimal nonnegative solution of nonlocal boundary value problems for second-order nonlinear impulsive differential equations on an infinite interval with an infinite ... impulsive differential equations has become an important area of investigation in the recent years and is much richer than the corresponding theory of differential equations For an introduction of

Ngày tải lên: 21/06/2014, 11:20

... various alternative numerical methods should be used for a specific problem, or even for a large class of problems. 56 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS Table 201(II) Euler ... 62 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS 0123 0.05 0.10 0.15 x h Figure 203(iii) Stepsize h against x for the ‘mildly stiff’ problem (203a) with variable stepsize for T =0.02 The ... 60 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS 10 0 10 −1 10 −2 10 −3 10 −4 10 −5 10 8 10 6 10 4 10 2 10 −0 10 −2 10 −4 n −1 E Figure 203(i) Norm error against n −1 for the ‘mildly

Ngày tải lên: 13/08/2014, 05:21

... persisted for a larger range of stepsizes for PEC methods than was the case for PECE methods NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS 10−6 10 4 1 14 10−8 E 2 10−10 ... main reason for this is that the work devoted to each step, 92 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS and the accuracy that is achieved in the step, should be balanced for overall ... METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS 10−6 10 4 1 04 −8 E 2 10 3 5 10 −10 4 10 4 Figure 239(ii) 10−3 h 10−2 Runge–Kutta methods with cost corrections methods However, for the

Ngày tải lên: 13/08/2014, 05:21

... 0 . 124 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS 262 Generalized linear multistep methods These methods, known also as hybrid methods or modified linear multistep methods, generalize ... Numerical Methods for Ordinary Differential Equations, Second Edition J C Butcher © 2008 John Wiley & Sons, Ltd ISBN: 978-0-470-723 35- 7 138 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL ... interpolation formula in the form y(x n−1 + ht) ≈ (1 + 2t)(1 −t) 2 y(x n−1) +(3−2t)t 2 y(x n ) + t(1 −t) 2 hy  (x n−1 ) − t 2 (1 − t)hy  (x n ). 132 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS

Ngày tải lên: 13/08/2014, 05:21

... expansions for Y 1 ,forhf (Y 1 ), for Y 2 ,forhf (Y 2 ),andsoonuntilwereachY s , hf(Y s )and finally y 1 . 160 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS Theorem 313B The Taylor expansions for ... ; for. .. A For i corresponding to a member of row k for k = 1, 2, , m, the only non-zero 190 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS aij are for j = 1 and for ... Φ(t)=φ(t), for all t ∈ T 0 . 164 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS Proof. Let #T 0 = n. The set of possible values that can be taken by the vector of Φ(t)values,forallt ∈ T

Ngày tải lên: 13/08/2014, 05:21

... quadrature formula, respectively Then: I For the Radau I formula, c1 = 0 This formula is exact for polynomials of degree up to 2s − 2 II For the Radau II formula, cs = 1 This formula is exact for. ... differentialalgebraic equations, for which implicit Runge–Kutta methods also have a vital role 214 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS 341 Solvability of implicit equations As ... (336a)...204 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS Proof For a given tree t, let Φ(t) denote the elementary weight for (333a) and Φ(t) the elementary weight for (333b) Because

Ngày tải lên: 13/08/2014, 05:21

... 34.3 Find the tableau for the Gauss method with s =4andp =8. 34.4 Show that Gauss methods are invariant under reflection. 230 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS 35 Stability ... METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS Lemma 359A Let XG = W BAW, where A and B = diag(b) are as for the Gauss method of order 2s Also let 1 ξk = √ , 2 4k2 −... · , by the formula ... −t diag(ej ), for t positive The value of R(Z) becomes R(Z) = 1 − tbj + O(t2 ), 2 48 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS which is greater than 1 for t sufficiently

Ngày tải lên: 13/08/2014, 05:21

... sub-forest induced by V is the forest (V , E), where E is the intersection of V × V and E A special 288 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS case is when a sub-forest... ... Ei ), for i = 1, 2, , n, where V is the... bi bj (370a) Now consider a problem for which y Qf (y) = 0, (370b) 276 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS for all ... acceptable for many problems. In each of the values of ξ for which there is a single underline, the method is A(α)-stable with α ≥ 1.55 ≈ 89 ◦ . 268 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS

Ngày tải lên: 13/08/2014, 05:21

... well-posed These include boundary value problems for (stationary) elliptic partial di erential equations and initial-boundary value problems for (time-dependent) equations of parabolic, hyperbolic, and ... Semigroups Parabolic Equations V Implicit Evolution Equations Introduction Regular Equations Pseudoparabolic Equations Degenerate Equations Examples ... so that j C0 j (x) for all x Rn , supp( j ) Gj , and j (x) = for x Fj LetS C0 (Rn ) be chosen with (x) n , supp( ) G for all x R fFj : j N g, and (x) P for x G = Finally, for each j , j N ,...

Ngày tải lên: 31/03/2014, 15:56

... For example, Lh is the diagonal part of Lh for Jacobi iteration, or the lower triangle for Gauss-Seidel iteration The next approximation is generated by 874 Chapter 19 Partial Differential Equations ... Jespersen, D 1984, Multrigrid Methods for Partial Differential Equations (Washington: Mathematical Association of America) McCormick, S.F (ed.) 1988, Multigrid Methods: Theory, Applications, ... approximate solution is uH Then the coarse-grid correction is 884 Chapter 19 Partial Differential Equations • Fine grids are used to compute correction terms to the coarse-grid equations, yielding...

Ngày tải lên: 24/12/2013, 12:16