Abstract

This paper presents the Higher-Order Haar Wavelet Collocation Method (HHWCM), an advanced numerical technique designed to address the limitations of the traditional Haar Wavelet Collocation Method (HWCM). By incorporating higher-order polynomial extensions into the Haar wavelet framework, the proposed method enhances precision and achieves faster convergence rates. The HHWCM is developed to effectively solve nonlinear ordinary differential equations (ODEs) under a wide array of conditions, including initial conditions, boundary conditions, periodic conditions, two-point conditions, integral conditions, and multi-point integral boundary conditions. The study begins with a theoretical foundation of HHWCM, demonstrating its improved approximation capabilities through convergence analysis and error estimation. This study underscores the versatility and potential of HHWCM as a robust computational tool for addressing nonlinear differential equations in scientific and engineering applications. The findings open avenues for extending the method to partial differential equations (PDEs) and exploring its integration with machine learning techniques to enhance numerical modelling and simulation in future work.

Keywords:Numerical method, Wavelet, Non-linear, Convergence, Collocation.

Introduction

First-order hyperbolic equations with two independent variables x and t are solved using a finite difference method. Typically, the first variable is space, while the second is time. For hyperbolic equations, a fundamental difficulty is constructing finite difference schemes that are both stable and do not damp out of the solution. Initial and initial-boundary value approximation algorithms are developed in this suggested study.

Motivation

First-order hyperbolic partial differential equations have a rich literature. It took a long time and a lot of work to create parameteric estimation techniques for Initial value and Initial-boundary value issues problems and initial-boundary value problems. Gottlieb et al (1987), Bo (1998), and Coulombel (2009) investigated the performance of finite – element strategies for 1st first order hyperbolic initial-boundary value problems using vectors value functions having L2(IR+, IRN). Semi-distributed approximations to the starting and boundary value issue were addressed by Warming and Beam in 1988.

(1)

Wherein L2[0, A] is “a > 0” and “v(t) =0” . Using wavelets, Sekino and Hamada in 2008 derived a numerical solution to the Advection problem ut + (a(x)u) x = 0. Despres and Teng in 2009 and 2010 respectively developed finite-element techniques for the initial value problem.Ut+aUx = 0, x ϵℝ, t ϵℝ+,

…………………(2)

For constrained initial functions u0 with discontinuous starting values. Motivated by the development of numerical methods to start initial-boundary value problems, these works are published.

Problem Formulation

Initial Value Issue (IVP) on an infinite interval is the initial model problem in this suggested study.

A(x) is greater than or equal to zero for all nonnegative values of x between IR+ and C(IR+), which is known as the initial condition. Wave propagation in homogeneous mediums is modelled by Equation (3).

IBVP (Initial-Boundary Value Issue) is an IB model problem that may be described as

Assuming “a > 0” & specification of the boundary condition v(t) at x = 0. There are no errors in this boundary condition since the information is flowing from left to right and the compatibility criterion is met (0).

With IVP (3), there is no boundary condition, and hence, no IVP (4). Many scenarios need this latter requirement.

For the issues IVP (3) and IBVP (4), the semigroup theory was heavily used in order to construct a completely discrete convergent numerical method. To solve the initial-boundary value issue, semigroup theory offers an elegant solution.

PRELIMINARIES

Theorem of Pazy (1983) was the primary tool employed in this study, and this section provides basic definitions and a specific instance.

Theorem 1. Assume X is a Banach space consisting of standard ǁ ǁ. Assuming, X has denser D(A) such that A: D(A) −→ X is a continuous (linear) projection. Further, λ is there, with

ℜ(λ) > 0 wherein the range “λI – A” is dense in X. Let Xn be the Banach spaces consisting of standards ǁ ǁn. Furthermore, there are bounded linear operators, “Pn: X → Xn and En: Xn → X” for every n ≥ 1 such that

(i) ∥Pn∥≤ C1, ∥En∥≤ C2, with C1 and C2 represented as constants that are not ependent on n.
(ii) ∥Pn x ∥n→∥x∥ as n→∞ for every xϵ X.
(iii) ∥En Pn x – x ∥→0 as n→∞ for every xϵ X.
(iv) Pn En = In (I n :identity operator on Xn)

Define F (τ n) as a series of constrained linear operations spanning Xn to Xn fulfilling ∥F(τn) k ∥ ≤ 1. (5)

In addition to this, the constrained linear maps

have the property that

and that

for all x ∈ D(A). A’s closure (A ̅) is therefore the indefinite constructor of contracting mappings “S(t) on X”. Furthermore, if knτn → t as n → ∞, then

It is permissible to refer to a generalized, solution as “solutions” in the following paragraphs.

If α = (α0, α1, …, αk), then let the notation α (i)=αi

For x ∈ℝ, [x] = sup {n ∈ Z: n ≤x}

Theorem 2 Theorem (Hille-Yosida Theorem). An infinitesimal generator of the C0 semigroup of contractions T (t), t 0 can only be generated by a linear (unbounded) operator A,

(i) D(A) = X, since A is closed

(ii) IR+ is present in ρ(A), the resolvent set of A. Further, The resolvent set ρ(A) of A for every λ > 0, ∥R(λ:A)∥≤1/λ (7)

IVP AND IBVP EXACT SOLUTIONS

For the start and initial-boundary value problems covered in this study, this section provides the precise solution.

IVP exact solution

The solution to eq (3) is provided by:

Using u (x, t) data on a bounded domain, we were able to calculate the solution of (3) that was bound not compulsorily on a bounded domain, numerically. This conclusion is made possible by the following theorem.

Theorem 1. Assuming “∈ C [0, ∞)” and “a(x) > 0” for every x ∈ IR+. Assume M > 0 and T > 0.

Define aM: [0, M] → IR as

Letting f ∈ C [0, M]. The problem’s solution is

Is existing and unique in nature. It is provided as

Further,

creates a contracting subclass on C [0, M] with the generator

V (x, t) = u (x, t), (x, t) ϵ [0, N] × [0, T] … (9)

given “f ∈ C [0, M]” as the constraint of u to [0, M].

Proof. Defining t ≥ 0, “Tt: [0, M] → [0, M]” as

It is now simple to demonstrate that Ts+t = Ts ◦ Tt.

Additionally, because “St f(x) = f (Tt x)”, it is simple to show that St is a subgraph.

Thus, according to the Hille-Yosida Theorem, Now, by Hille-Yosida Theorem, if St generator is B,

Then,

Here

Considering the differential equation,

which is equivalent to

Since, there exists unique solution for every “h ∈ X” i.e., f ∈ D(A) to the preceding differential equation that is provided as,

For the operators A and B, it can be demonstrated that (I – A) 1 = (I–B) 1.

“D(A) = D(B)” and for any “g D(A)”, “Bg = Ag” may be deduced from this.

For t [0, T] and x [0, N], it is for sure a growing derivative/ function

Hence

This leads to the conclusion that “St f(x) = V (x, t) = u (x, t)” for every “x ∈ [0, N]” and “t ∈ [0, T]”.

IBVP exact solution

Theorem 2

. Assume “u ∈ C [0, 1]” and “v ∈ C [0, ∞)” in such a way that “u (0) = v (0)”.
Defining u0 (x) = u0 (x) − u0 (0). For “U” as a solution to (4), its also a solution to

and V is a solution to

Then, “U (x, t) = U (x, t) + V (x, t)”. Furthermore, for “T > 0”, contracting subgroups s.t., are defined

“X → X” where “X = {u ∈ C [0, 1]: u (0) = 0}” and “T(x): Y → Y”

Wherein, “Y = C [0, T]” as

Then “U (x, t) = Su0(x) + Σ x w(t) for all (x, t) ∈ [0, 1] × [0, T]”,

W: represents constraint of “v to [0, T]”.

Furthermore, if A and B represent the St and Tx generators, respectively, then

“D(A) = {g ∈ X: g ′ ∈ X}, D(B) = {g ∈ Y: g ′ ∈ Y and g ′ (0) = 0}, Ag = −ag′ for all g ∈ D(A)” and “B g = −1 a g ′ for all g ∈ D(B)”.

Convergent numerical technique for IVP and IBVP

M AND INITIAL-BOUNDARY VALUE PROBLEM

In this lesson, we’ll go through how to solve the starting value and initial boundary value issues numerically convergently. It is possible to solve the initial value issue by posing it on a smaller bounding box, and then solving it on a larger bounding box with the same answer. An improved numerical solution to the modified issue is indistinguishable from the original answer on the smaller constrained region. Decomposing the initial-boundary value issue into two problems, each of which generates a semigroup, allows for the presentation of discrete semigroup approximations.

The IVP and a Convergent Numerical Scheme

M > N and an initial value problem given on [0, M] [0, T] whose answer absolutely corresponds to the solution of (3) on [0, N] [0, T] for any subset of [0, N] [0, T] are possible for the initial value issue (3). IR+ IR+ IR+ IR+ IR+ On [0, M] [0, T], create a finite difference scheme that converges to the solution of the problem provided in (3) on [0, N] [0, T].

This finding is made easier by the following theorem.

Theorem 1.

Assume “X = C [0, M]” & “A”. Let Xn = IRn+1 whose elements are denoted as α = (α0, α1, . . . αn). The supremum standard is applied to both X and Xn spaces. Finally, defining

Pn: X → Xn as (Pnf)i = f (iM/n), i = 0, 1, . . . n.

En: Xn → X as

En(α) represents “piecewise linear function” having En(α)(iM/n) = αi. Let

Defining operations “F (τn): Xn → Xn“ as

Choosing kn = t/τ_n , it can be shown thats

Particularly, fixing “t ∈ [0, T]” & “x ∈ [0, N]”,

Where, where u (x, t) represents a solution to (3).

Proof. Since Pn is clearly linear, ||Pn || 1. The fact that (ii) of Theorem 1 is true may be deduced with norms’ definitions from the definitions of the norms & constant continuance of component X. That ||En|| 1 is simply obtained

When you consider the definitions of En and Pn as well as the uniform continuation of the element X in Theorem 1, it becomes clear that (ii) is true. Formulating differential equation leads to the definition of F(n) which is now simple. To illustrate, let’s look at the (iM/n, jn) lattice functions for “I = 0, 1, 2, …” and (“j = 0, 1, 2, … in the (x, t) plane”).

Assume V (iM/n, jτn) = ui,j . Taking into account the sup x∈ [0, M] aM(x) > 0, τn is defined clearly. Take a look at the difference equation that corresponds to the differential equation in (8) is

which can be simplified as

It is possible to calculate all ui,j by using the preceding method for ui,0. The formula for fi is f(iM/n) = fi.

Now,

Therefore, ||F(τn) || ≤ 1, holding the theorem 1’s stability condition (5).

For, f ϵ D,

Because “f ∈ D”, “Af “ is consistently periodic on “[0, M]” the R.H.S of (14) goes to 0 as n tends to infinity (n → ∞). As a result, Theorem 1 (3.6) is fulfilled.

Using Theorem 1, one must prove that for any > 0, the range “I – A” in X is dense. However, the range of I A is previously demonstrated to include all of X in Theorem 1. Theorem 1’s formulation of (I B) 1h khk may also be used.