Higher-order Differential Equations
Kanisha Dezarn
Once again, it is important to stress that Theorem 1 above is simply an extension to the Theorems on the existence and uniqueness of solutions to first order and second order linear differential equations. Furthermore, for this theorem to apply, we must have that coefficient in front of the $\fracd^ndt^n$ term is $1$.
Show that there exists a unique solution to the third order linear differential equation $t\fracdy^3dt^3+ 3 \fracd^2ydt^2 + \sin t \fracdydt + e^t y = 0$ with the initial conditions $y(1) = 1$, $y'(1) = 1$, $y''(1) = 2$.
The steps involved in converting a higher differential equation into a first order system are:
1. Rewrite the equation in its standard form, with the highest derivative on one side and all other terms on the other side.
2. Introduce new variables for each derivative in the equation, such as y' = x and y'' = z.
3. Rewrite the equation in terms of these new variables, keeping in mind that derivatives of these variables will be included in the new system.
4. Rewrite the equation as a system of first order differential equations by replacing all derivatives with their corresponding new variables.
Converting higher differential equations into first order systems has several benefits, including:
1. It simplifies the equations, making them easier to solve and analyze.
2. It allows for the use of numerical methods, which can provide more accurate solutions.
3. It can reveal hidden relationships between variables that may not be apparent in the original higher order equation.
Yes, any higher differential equation can be converted into a first order system by following the proper steps. However, the resulting system may not always be simpler or easier to solve than the original equation. In some cases, the conversion may introduce additional variables and equations, making the system more complex.
In this paper, the block pulse function method is proposed for solving high-order differential equations associated with multi-point boundary conditions. Although the orthogonal block pulse functions frequently have been applied to approximate the solution of ordinary differential equations associated with the initial conditions, the presented method provides the flexibility with respect to multi-point boundary conditions in separated or non-separated forms. This technique, which may be named the augmented block pulse function method, reduces a system of high-order boundary value problems of ordinary differential equations to a system of algebraic equations. The illustrated results confirm the computational efficiency, reliability, and simplicity of the presented method.
In this work, we proposed the augmented block pulse function method for solving a system of arbitrary-order boundary value problem associated with initial conditions or multi-point boundary conditions in separated or non-separated forms. Let us consider the following nth-order differential equations with assumption of the existence and uniqueness of the solution:
Since the differential equation may be enforced by many different conditions, we consider the general form including separated and non-separated boundary conditions but we also can consider boundary conditions or the set of conditions including some or all of these mentioned types. Let us assume
may be called separated and if the conditions are not separated will be called of non-separated type. In a more general form, the condition may also include the value of the derivatives of \(f(x)\) so
To make the article self-contained in Section 2 a short description on block pulse functions is added. In Section 3 the description of the method shows BPFs how can be applied to solve the high-order differential equations with a different kind of boundary conditions. The numerical results are illustrated in Section 4 to clarify more details of the proposed method and expectedly confirm the convergence and applicability of the method. Finally, a brief conclusion is stated in Section 5.
There are some properties for BPFs which make them popular for approximation such as orthogonality, disjointness, and completeness [22]. A function \(f(x)\) over the interval \([0,1)\), can be expanded in a BPFs series with an infinite number of terms
We consider the solution of nth-order system (1) to need at least nth-order differentiability to be able to approximate the highest-order derivative of the unknown function and prevent the discontinuity seen in (8). So we first define
The integration of BPFs has the important role to approximate differential terms and the described matrices in (13), (32), and (33) show the sparsity of the systems made by using BPFs, which affects the computational efficiency.
This assumption leads to the technique which we named the augmented block pulse function (ABPF) method. In fact, we develop the BPF method to be flexible for an approximation of the differential equations with different boundaries. We replace the expansion of \(f^(i)(x)\), \(i=0,1,\ldots,n\), into the system of (1) and (4) and then substitute the collocation points defined in (10) as follows:
It is worth noting here that we can do a few simple modifications when some of \(f^(i)(a)\), \(i=0,1,\ldots,n-1\), are given. Particularly if \(f^(i)(a)\), \(i=0,1,\ldots,n-1\), all are given, the system becomes an initial value problem and there is no need to consider any \(c_i\), \(i=m,\ldots,m+n-1\). In addition, we can keep the structure of the algorithm and input the given initial value into the described scheme. Obviously, the first state considers the value of \(f^(i)(a)\), \(i=0,1,\ldots,n-1\), precisely and the second state find them approximately such that there are good agreement between precise and approximated values. In this paper, the reported results are based on the second assumption.
In order to assess the accuracy of block pulse function method for solving higher-order differential equations with multi-point boundary conditions we will consider the following examples. The associated computations with the examples were performed using MAPLE 17 with 64 digits precision on a personal computer.
and Table 1 includes the observed absolute error by these values. The plots of the numerical solution by the proposed method with \(m=64\) versus the exact solution and the absolute error function are depicted in Figures 1 and 2, respectively, showing higher accuracy. The graph of the sixth derivative of the numerical solution by BPFs versus the exact solution for \(m=64\) is given in Figure 3. This example with other boundary conditions [7, 24] also can be reduced to a system of linear equations as described.
Plotsof the numericalsolution by the proposed method with \(m=64\) versus the exact solution and the absolute error function are depicted in Figures 4 and 5, respectively. The graph of the fourth derivative of the numerical solution by BPFs versus the exact solution for \(m=64\) is given in Figure 6. Also, Table 2 presents the observed maximum absolute error for \(m=10\) and \(m=16\), using the proposed together with the results obtained by reproducing kernel method (RKM), given in [25]. By the comparison of the results obtained using the presented method in Table 2 with the RKM, it is easily found that the present approximations are more efficient.
The plots of the numerical solution by the proposed method with \(m=32\) versus the exact solution and the absolute error function are depicted in Figures 7 and 8, respectively. The graph of second derivative of the numerical solution by BPFs versus the exact solution for \(m=32\) is given in Figure 9.
The block pulse functions provide the efficient method to solve high-order ODEs associated with the general type of multi-point boundary conditions. According to the presented method, the nth-order ODE defined in (1), which can be linear or nonlinear system with separated and non-separated boundary conditions, will be reduced to the algebraic equations by using block pulse functions and a polynomial function of degree \(n-1\). The most important privileges of the proposed method are computational efficiency due to sparse matrices, simplicity, and reliability, so one may increase the number of basis functions and consequently the accuracy will be improved.
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The ability to solve differential equations represents a key step in the modeling and understanding of complex systems. There exist several analytical and numerical methods for solving differential equations, each with their own advantages and limitations. Physics-informed neural networks (PINNs) offer an alternative perspective. Although PINNs deliver promising results, many stones remain unturned about this method. In this paper, we introduce a method that improves the efficiency of PINNs in solving differential equations. Our method is related to the formulation of the problem: Instead of training a network to solve an nth order differential equation, we propose transforming the problem into the equivalent system of n first-order equations in phase space. The target of the network is to solve all equations of the system simultaneously, effectively introducing a multitask optimization problem. We compare both approaches empirically on various problems, ranging from second-order differential equations with constant coefficients to higher-order and nonlinear problems. We also show that our approach is suited for solving partial differential equations. Our results show that the system approach performs equal or better in most experiments performed. We analyze the learning process for the few runs that did not perform well and show that the problem stems from conflicting gradients during training, effectively obstructing multitask learning. The result of this paper is a straightforward heuristic that can be incorporated into any subsequent research that builds on PINNs solving differential equations. Moreover, it also shows how to make PINNs even more efficient by implementing techniques from multitask learning literature.
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