The Experts below are selected from a list of 9558 Experts worldwide ranked by ideXlab platform
Ghasem Barid Loghmani - One of the best experts on this subject based on the ideXlab platform.
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Numerical solution of Fredholm integral equations of the second kind by using integral Mean Value Theorem II. High dimensional problems
Applied Mathematical Modelling, 2013Co-Authors: Mohammad Hossein Heydari, H. Navabpour, Zakieh Avazzadeh, Ghasem Barid LoghmaniAbstract:Abstract In this work, we generalize the numerical method discussed in [Z. Avazzadeh, M. Heydari, G.B. Loghmani, Numerical solution of Fredholm integral equations of the second kind by using integral Mean Value Theorem, Appl. math. modelling, 35 (2011) 2374–2383] for solving linear and nonlinear Fredholm integral and integro-differential equations of the second kind. The presented method can be used for solving integral equations in high dimensions. In this work, we describe the integral Mean Value method (IMVM) as the technical algorithm for solving high dimensional integral equations. The main idea in this method is applying the integral Mean Value Theorem. However the Mean Value Theorem is valid for multiple integrals, we apply one dimensional integral Mean Value Theorem directly to fulfill required linearly independent equations. We solve some examples to investigate the applicability and simplicity of the method. The numerical results confirm that the method is efficient and simple.
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numerical solution of fredholm integral equations of the second kind by using integral Mean Value Theorem
Applied Mathematical Modelling, 2011Co-Authors: Zakieh Avazzadeh, Mohammad Hossein Heydari, Ghasem Barid LoghmaniAbstract:In this paper, we present a new semi-analytical method for solving linear and nonlinear Fredholm integral and integro-differential equations of the second kind and the systems including them. The main idea in this method is applying the Mean Value Theorem for integrals. Some examples are presented to show the ability of the model. The results confirm that the method is very effective and simple.
Zakieh Avazzadeh - One of the best experts on this subject based on the ideXlab platform.
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Numerical solution of Fredholm integral equations of the second kind by using integral Mean Value Theorem II. High dimensional problems
Applied Mathematical Modelling, 2013Co-Authors: Mohammad Hossein Heydari, H. Navabpour, Zakieh Avazzadeh, Ghasem Barid LoghmaniAbstract:Abstract In this work, we generalize the numerical method discussed in [Z. Avazzadeh, M. Heydari, G.B. Loghmani, Numerical solution of Fredholm integral equations of the second kind by using integral Mean Value Theorem, Appl. math. modelling, 35 (2011) 2374–2383] for solving linear and nonlinear Fredholm integral and integro-differential equations of the second kind. The presented method can be used for solving integral equations in high dimensions. In this work, we describe the integral Mean Value method (IMVM) as the technical algorithm for solving high dimensional integral equations. The main idea in this method is applying the integral Mean Value Theorem. However the Mean Value Theorem is valid for multiple integrals, we apply one dimensional integral Mean Value Theorem directly to fulfill required linearly independent equations. We solve some examples to investigate the applicability and simplicity of the method. The numerical results confirm that the method is efficient and simple.
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numerical solution of fredholm integral equations of the second kind by using integral Mean Value Theorem
Applied Mathematical Modelling, 2011Co-Authors: Zakieh Avazzadeh, Mohammad Hossein Heydari, Ghasem Barid LoghmaniAbstract:In this paper, we present a new semi-analytical method for solving linear and nonlinear Fredholm integral and integro-differential equations of the second kind and the systems including them. The main idea in this method is applying the Mean Value Theorem for integrals. Some examples are presented to show the ability of the model. The results confirm that the method is very effective and simple.
Mohammad Hossein Heydari - One of the best experts on this subject based on the ideXlab platform.
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Numerical solution of Fredholm integral equations of the second kind by using integral Mean Value Theorem II. High dimensional problems
Applied Mathematical Modelling, 2013Co-Authors: Mohammad Hossein Heydari, H. Navabpour, Zakieh Avazzadeh, Ghasem Barid LoghmaniAbstract:Abstract In this work, we generalize the numerical method discussed in [Z. Avazzadeh, M. Heydari, G.B. Loghmani, Numerical solution of Fredholm integral equations of the second kind by using integral Mean Value Theorem, Appl. math. modelling, 35 (2011) 2374–2383] for solving linear and nonlinear Fredholm integral and integro-differential equations of the second kind. The presented method can be used for solving integral equations in high dimensions. In this work, we describe the integral Mean Value method (IMVM) as the technical algorithm for solving high dimensional integral equations. The main idea in this method is applying the integral Mean Value Theorem. However the Mean Value Theorem is valid for multiple integrals, we apply one dimensional integral Mean Value Theorem directly to fulfill required linearly independent equations. We solve some examples to investigate the applicability and simplicity of the method. The numerical results confirm that the method is efficient and simple.
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numerical solution of fredholm integral equations of the second kind by using integral Mean Value Theorem
Applied Mathematical Modelling, 2011Co-Authors: Zakieh Avazzadeh, Mohammad Hossein Heydari, Ghasem Barid LoghmaniAbstract:In this paper, we present a new semi-analytical method for solving linear and nonlinear Fredholm integral and integro-differential equations of the second kind and the systems including them. The main idea in this method is applying the Mean Value Theorem for integrals. Some examples are presented to show the ability of the model. The results confirm that the method is very effective and simple.
Janusz Matkowski - One of the best experts on this subject based on the ideXlab platform.
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Mean-Value Theorem for vector-Valued functions
2020Co-Authors: Janusz MatkowskiAbstract:For a differentiable functionf : I ! R k , where I is a real interval and k 2 N, a counterpart of the Lagrange Mean-Value Theorem is presented. Necessary and sufficient conditions for the existence of a Mean M : I 2 ! I such that f(x) f(y) = (x y)f ' (M(x, y)), x, y 2 I, are given. Similar considerations for a Theorem accompanying the Lagrange Mean-Value Theorem are presented.
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a Mean Value Theorem and its applications
Journal of Mathematical Analysis and Applications, 2011Co-Authors: Janusz MatkowskiAbstract:Abstract For a function f defined in an interval I, satisfying the conditions ensuring the existence and uniqueness of the Lagrange Mean L [ f ] , we prove that there exists a unique two variable Mean M [ f ] such that f ( x ) − f ( y ) x − y = M [ f ] ( f ′ ( x ) , f ′ ( y ) ) for all x , y ∈ I , x ≠ y . The Mean M [ f ] is closely related L [ f ] . Necessary and sufficient condition for the equality M [ f ] = M [ g ] is given. A family of Means { M [ t ] : t ∈ R } relevant to the logarithmic Means is introduced. The invariance of geometric Mean with respect to Mean-type mappings of this type is considered. A result on convergence of the sequences of iterates of some Mean-type mappings and its application in solving some functional equations is given. A counterpart of the Cauchy Mean-Value Theorem is presented. Some relations between Stolarsky Means and M [ t ] Means are discussed.
Trevor D. Wooley - One of the best experts on this subject based on the ideXlab platform.
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approximating the main conjecture in vinogradov s Mean Value Theorem
Mathematika, 2017Co-Authors: Trevor D. WooleyAbstract:We apply multigrade efficient congruencing to estimate Vino- gradov's integral of degree k for moments of order 2s, establishing strongly diagonal behaviour for 1 6 s 6 1 k(k + 1) − 1 k + o(k). In particular, as k → ∞, we confirm the main conjecture in Vinogradov's Mean Value Theorem for 100% of the critical interval 1 6 s 6 1 k(k + 1).
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vinogradov s Mean Value Theorem via efficient congruencing ii
Duke Mathematical Journal, 2013Co-Authors: Trevor D. WooleyAbstract:We apply the efficient congruencing method to estimate Vino- gradov's integral for moments of order 2s, with 1 6 s 6 k 2 − 1. Thereby, we show that quasi-diagonal behaviour holds when s = o(k 2 ), we obtain near-optimal estimates for 1 6 s 6 1 k 2 + k, and optimal estimates for s > k 2 − 1. In this way we come half way to proving the main conjecture in two different directions. There are consequences for estimates of Weyl type, and in several allied applications. Thus, for example, the anticipated asymptotic formula in Waring's problem is established for sums of s kth powers of natural numbers whenever s > 2k 2 − 2k − 8 (k > 6). Estimates stemming from Vinogradov's Mean Value Theorem deliver bounds for exponential sums of large degree, both in Mean and pointwise, beyond the competence of alternate approaches. The ubiquity of such exponential sums in analytic number theory, in the analysis for example of the Riemann zeta function, in Waring's problem, and beyond, accounts for the high pro- file of Vinogradov's methods in the associated literature. In recent work, we established a version of Vinogradov's Mean Value Theorem which achieves an essentially optimal upper bound with a number of variables only twice the number conjectured to be best possible (see (19)). For systems of degree k, previous estimates missed such a bound by a factor of order logk. Our earlier approach provides no upper bounds when the number of variables is smaller, precluding the possibility of applications involving the finer features of these Mean Values. Our goal in this paper is to remedy this deficiency, at the same time strengthening our previous conclusions. It transpires that we are able to come within a hair's breadth of proving the main conjecture concerning Vino- gradov's Mean Value Theorem in half of the basic interval of relevant moments. Such developments illustrate the flexibility of the new efficient congruencing method introduced in (19). We now introduce some notation. When k ∈ N and � ∈ R k , define
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ON VINOGRADOV'S Mean Value Theorem
Mathematika, 1992Co-Authors: Trevor D. WooleyAbstract:The object of this paper is to obtain improvements in Vinogradov's Mean Value Theorem widely applicable in additive number theory. Let J s,k (P) denote the number of solutions of the simultaneous diophantine equations with 1 ≥ x i , y i ≥ P for 1 ≥ i ≥ s . In the mid-thirties Vinogradov developed a new method (now known as Vinogradov's Mean Value Theorem ) which enabled him to obtain fairly strong bounds for J s,k (P) . On writing in which e (α) denotes e 2πiα , we observe that where T k denotes the k -dimensional unit cube, and α = (α 1 ,…,α k ).
