Certain Investigation on Euler Matrix Method for Linear Second-Order Partial Differential Equations with Various Conditions

There are some well-known numerical methods such as finite difference methods, finite element methods, polynomial approximate methods, spectral methods, Galerkin, and collocation methods to numerically solve PDEs [1-2]. However, recently various approximate methods are discussed in the literature which as the Transform differential process, Legendre-wavelet method. Chebyshev-tau method and Form of Adomian breaking down. In this article, we have developed a matrix method dependent on method Euler polynomials. The method was given by error estimation and error analysis. Let Ω be a rectangular region {(x,y):0x,yb} and  is the boundary of Ω. In general form, for all , linear Partial equations differential with variable coefficients follow as, In this article, we take into account (1.1): conditions in three complicated form [12]. Case 1: Conditions defined at the points x = αk and y = βk, where

Case 2 : Conditions defined at the points y = yk, where Case 3 : Conditions defined at the points Here are functions defined in Ω.

Euler numbers and polynomials are very useful in classical analysis and numerical mathematics. In many respects, they are closely linked to theory of Bemoulli polynomials and numbers. Euler

Definition 2.1, Here M The linear is the space of n real matrices, by the identity matrix and S the subspace of all symmetric matrices in M.A. linear functional L on M It is said that it is ―positive‖ if for any . Definition 2.2: A couple real-valued functions (f, g) defined on is being named a averaging pair if. (i) f is nonnegative and locally integrable on satisfying ; (ii) g>0 is absolutely continuous on every compact subinterval of ; and (iii) for

Definition 2.3 : Let L be a positive linear functional and B = B(t) a real valued matrix function which is invertible for each . A quartet of real-valued functions (f, g, L, B) defined on is a generalized averaging quartet if the conditions (i) and (ii) in Definition 2.2 and the following condition (iii) hold (iii) for .

(I) let conditions in Definition 2.3 hold: then (II) Let, then Implies

Lemma 2.5: [36] Let L be a positive linear functional on M. Then, for any , We got

Lemma 2.6: Let L be a positive linear functional on M. For any , So for everyone

Lemma 2.7: Let X(t) be a nontrivial prepared solution of (1.1) and det . So for everyone then matrix function. Satisfies the equation

There are some well-known Numerical approaches such as methods of final differences finite element methods, polynomial approximate methods, spectral methods, Galerkin, and collocation methods to numerically solve PDEs (1.1) Theorem 3.1: Assume that all conditions stated in Section 1 are satisfied; suppose for any solution X (t) for (1.1) for t > t0, and P(t) and R(t) are commutative with Suppose further that a function occurs a and a generalized averaging quartet. Where L is positive linear functional on M, satisfying And the matrix J defined by and is the linear operator defined by Then every prepared Oscillatory solution of (1.1) no Proof: Suppose the Theorem 3.1 is not true and X (t) is any nontrivial prepared result of (1.1) in which is nonoscillatory. Suppose without lack of generality that. Then by Lemma 2.7. W(t) is symmetric and satisfies the Riccati equation (2.2). That is, Integrating both sides of (3.4) for t1 to 1, we obtain W(t) Now use of previous lemma and integrate Then we know what contradicts the fact (f, ar, L, Is a generalized averaging quartet Corollary 3.2 : in case the above conditions hold and And where A, B S are constant positive definite matrices, and A is commutative with P(t) and R(t). Suppose further that there exist an averaging pair (f, ar), where M satisfying (3.1), where and is the linear operator defined by (3.3). Then any prepared solution of oscillatory (1.1) on Remark 3.2. Theorem 3.1 and Corollary 3.2 are improvement and generalize of Theorem 3.1 and Corollary 3.1 by Yang [56]. In fact, Theorem 3.1 in [56] is not applicable if we choose such that Or Remark 3.4 : Theorem 3.1 is improvement and generalize of Theorem 3.1 by Xu and Zhu [53]. In fact, Theorem 3.1 in [53] is not applicable if we choose such that Remark 3.5 : Theorem 3.1 and Corollary 3.2 are improved and generalize to Theorem 3.1 and Corollary 3.1 by Yang and Tang[59]. In fact, Theorem 3.1 in [57] is not applicable if we choose such that P(t)≠R(t). But when P(t) = R(t), and in Theorem 3.1 and Corollary 3.2 give Theorem 3.1 and Corollary 3.2 in [57], respectively. Also, when and P(t) > 0 in Theorem 3.1 [57], the outcome of these positive definite matrices is not necessarily positive definite.

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Research Scholar, Department of Mathematics, Magadh University, Bodhgaya, Bihar