We consider only the solution to the Cauchy problem. A system of differential equations or one equation must be converted to the form
Where 
,
–n-dimensional vectors; y– unknown vector function; x– independent argument, 
. In particular, if n= 1, then the system turns into one differential equation. The initial conditions are set as follows: 
, Where 
.
If 
in the vicinity of a point 
is continuous and has continuous partial derivatives with respect to y, then the existence and uniqueness theorem guarantees that there is only one continuous vector function 
, defined in some neighborhood of a point 
, satisfying equation (7) and the condition 
.
Let us pay attention to the fact that the neighborhood of the point 
, where the solution is determined, can be very small. When approaching the boundary of this neighborhood, the solution can go to infinity, oscillate with an infinitely increasing frequency, in general, behave so badly that it cannot be continued beyond the boundary of the neighborhood. Accordingly, such a solution cannot be tracked by numerical methods on a larger segment, if one is specified in the problem statement.
Solving the Cauchy problem on [ a; b] is a function. In numerical methods, the function is replaced by a table (Table 1).
Table 1
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Here 
,
. The distance between adjacent table nodes is usually taken to be constant: 
,
.
There are tables with variable steps. The table step is determined by the requirements of the engineering problem and not connected with the accuracy of finding a solution.
If y is a vector, then the table of solution values will take the form of a table. 2.
Table 2
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In the MATHCAD system, a matrix is used instead of a table, and it is transposed with respect to the specified table.
Solve the Cauchy problem with accuracy ε
means to get the values in the specified table 
(numbers or vectors), 
, such that 
, Where 
- exact solution. It is possible that the solution to the segment specified in the problem does not continue. Then you need to answer that the problem cannot be solved on the entire segment, and you need to get a solution on the segment where it exists, making this segment as large as possible.
It should be remembered that the exact solution 
we don’t know (otherwise why use the numerical method?). Grade 
must be justified on some other basis. As a rule, it is not possible to obtain a 100% guarantee that the assessment is being carried out. Therefore, algorithms are used to estimate the value 
, which prove effective in most engineering problems.
The general principle for solving the Cauchy problem is as follows. Line segment [ a;
b] is divided into a number of segments by integration nodes. Number of nodes k does not have to match the number of nodes m final table of decision values (Tables 1, 2). Usually, k > m. For simplicity, we will assume that the distance between nodes is constant, 
;h called the integration step. Then, according to certain algorithms, knowing the values 
at i < s, calculate the value 
. The smaller the step h, the lower the value 
will differ from the value of the exact solution 
. Step h in this partition is already determined not by the requirements of the engineering problem, but by the required accuracy of solving the Cauchy problem. In addition, it must be selected so that at one step the table. 1, 2 fit an integer number of steps h. In this case the values y, obtained as a result of calculations with steps h at points 
, are used accordingly in the table. 1 or 2.
The simplest algorithm for solving the Cauchy problem for equation (7) is the Euler method. The calculation formula is:
(8)
Let's see how the accuracy of the solution found is assessed. Let's pretend that 
is the exact solution of the Cauchy problem, and also that 
, although this is almost always not the case. Then where is the constant C depends on function 
in the vicinity of a point 
. Thus, at one step of integration (finding a solution) we get an error of the order 
. Because steps have to be taken 
, then it is natural to expect that the total error at the last point 
everything will be fine 
, i.e. order h. Therefore, Euler's method is called the first order method, i.e. the error has the order of the first power of the step h. In fact, at one step of integration the following estimate can be justified. Let 
– exact solution of the Cauchy problem with the initial condition 
. It's clear that 
does not coincide with the required exact solution 
the original Cauchy problem of equation (7). However, at small h and "good" function 
these two exact solutions will differ little. The Taylor remainder formula ensures that 
, this gives the integration step error. The final error consists not only of errors at each integration step, but also of deviations of the desired exact solution 
from exact solutions 
,
, and these deviations can become very large. However, the final estimate of the error in the Euler method for a “good” function 
still looks like 
,
.
When applying Euler's method, the calculation proceeds as follows. According to specified accuracy ε
determine the approximate step 
. Determining the number of steps 
and again approximately select the step 
. Then again we adjust it downward so that at each step the table. 1 or 2 fit an integer number of integration steps. We get a step h. According to formula (8), knowing 
And 
, we find. By found value 
And 
we find so on.
The resulting result may not, and generally will not, have the desired accuracy. Therefore, we reduce the step by half and again apply the Euler method. We compare the results of the first application of the method and the second in identical points 
. If all discrepancies are less than the specified accuracy, then the last calculation result can be considered the answer to the problem. If not, then we reduce the step by half again and apply Euler’s method again. Now we compare the results of the last and penultimate application of the method, etc.
Euler's method is used relatively rarely due to the fact that to achieve a given accuracy ε
a large number of steps are required, in the order of 
. However, if 
has discontinuities or discontinuous derivatives, then higher order methods will produce the same error as Euler's method. That is, the same amount of calculations will be required as in the Euler method.
Of the higher order methods, the fourth order Runge–Kutta method is most often used. In it, calculations are carried out according to the formulas
This method, in the presence of continuous fourth derivatives of the function 
gives an error on one step of the order 
, i.e. in the notation introduced above, 
. In general, on the integration interval, provided that the exact solution is determined on this interval, the integration error will be of the order of 
.
The selection of the integration step occurs in the same way as described in Euler’s method, except that the initial approximate value of the step is selected from the relation 
, i.e. 
.
Most programs used to solve differential equations use automatic step selection. The gist of it is this. Let the value already be calculated 
. The value is calculated 
in steps h, chosen during calculation 
. Then two integration steps are performed with step 
, i.e. extra node is added 
in the middle between the nodes 
And 
. Two values are calculated 
And 
in nodes 
And 
. The value is calculated 
, Where p– method order. If δ
is less than the accuracy specified by the user, then it is assumed 
. If not, then choose a new step h equal and repeat the accuracy check. If during the first check δ
is much less than the specified accuracy, then an attempt is made to increase the step. For this purpose it is calculated 
at the node 
in steps h from node 
and is calculated 
in steps of 2 h from node 
. The value is calculated 
. If 
is less than the specified accuracy, then step 2 h considered acceptable. In this case, a new step is assigned 
,
,
. If 
more accuracy, then the step is left the same.
It should be taken into account that programs with automatic selection of the integration step achieve the specified accuracy only when performing one step. This occurs due to the accuracy of the approximation of the solution passing through the point 
, i.e. approximation of the solution 
. Such programs do not take into account how much the solution 
differs from the desired solution 
. Therefore, there is no guarantee that the specified accuracy will be achieved throughout the entire integration interval.
The described Euler and Runge–Kutta methods belong to the group of one-step methods. This means that to calculate 
at the point 
it is enough to know the meaning 
at the node 
. It is natural to expect that if more information about a decision is used, several previous values of the decision will be taken into account 
,
etc., then the new value 
it will be possible to find more accurately. This strategy is used in multi-step methods. To describe them, we introduce the notation 
.
Representatives of multi-step methods are the Adams–Bashforth methods:
Method k-th order gives a local order error 
or global – order 
.
These methods belong to the group of extrapolation methods, i.e. the new meaning is clearly expressed through the previous ones. Another type is interpolation methods. In them, at each step, you have to solve a nonlinear equation for a new value 
. Let's take the Adams–Moulton methods as an example:
To use these methods, you need to know several values at the beginning of the count 
(their number depends on the order of the method). These values must be obtained by other methods, for example the Runge–Kutta method with a small step (to increase accuracy). Interpolation methods in many cases turn out to be more stable and allow larger steps to be taken than extrapolation methods.
In order not to solve a nonlinear equation at each step in interpolation methods, Adams predictor-correction methods are used. The bottom line is that the extrapolation method is first applied at the step and the resulting value 
is substituted into the right side of the interpolation method. For example, in the second order method 
When solving scientific and engineering problems, it is often necessary to mathematically describe some dynamic system. This is best done in the form of differential equations ( DU) or systems of differential equations. Most often, this problem arises when solving problems related to modeling the kinetics of chemical reactions and various transfer phenomena (heat, mass, momentum) - heat transfer, mixing, drying, adsorption, when describing the movement of macro- and microparticles.
In some cases, a differential equation can be transformed into a form in which the highest derivative is expressed explicitly. This form of writing is called an equation resolved with respect to the highest derivative (in this case, the highest derivative is absent on the right side of the equation):
A solution to an ordinary differential equation is a function y(x) that, for any x, satisfies this equation in a certain finite or infinite interval. The process of solving a differential equation is called integrating a differential equation.
Historically, the first and simplest way to numerically solve the Cauchy problem for a first-order ODE is the Euler method. It is based on the approximation of the derivative by the ratio of finite increments of the dependent (y) and independent (x) variables between the nodes of a uniform grid:
where y i+1 is the desired value of the function at point x i+1.
The accuracy of Euler's method can be improved if a more accurate integration formula is used to approximate the integral - trapezoidal formula.
This formula turns out to be implicit with respect to y i+1 (this value is on both the left and right sides of the expression), that is, it is an equation with respect to y i+1, which can be solved, for example, numerically, using some iterative method (in such form, it can be considered as an iterative formula of the simple iteration method).
Composition of the course work: The course work consists of three parts. The first part contains a brief description of the methods. In the second part, the formulation and solution of the problem. In the third part - software implementation in computer language
The purpose of the course work: to study two methods for solving differential equations - the Euler-Cauchy method and the improved Euler method.
Numerical differentiation
A differential equation is an equation containing one or more derivatives. Depending on the number of independent variables, differential equations are divided into two categories.
Ordinary differential equations (ODE)
Partial differential equations.
Ordinary differential equations are those equations that contain one or more derivatives of the desired function. They can be written as
independent variable
The highest order included in equation (1) is called the order of the differential equation.
The simplest (linear) ODE is equation (1) of order resolved with respect to the derivative
A solution to the differential equation (1) is any function that, after its substitution into the equation, turns it into an identity.
The main problem associated with linear ODE is known as the Kasha problem:
Find a solution to equation (2) in the form of a function satisfying the initial condition (3)
Geometrically, this means that it is required to find the integral curve passing through the point ) when equality (2) is satisfied.
Numerical from the point of view of the Kasha problem means: it is required to construct a table of function values satisfying equation (2) and the initial condition (3) on a segment with a certain step. It is usually assumed that that is, the initial condition is specified at the left end of the segment.
The simplest numerical method for solving a differential equation is the Euler method. It is based on the idea of graphically constructing a solution to a differential equation, but this method also provides a way to find the desired function in numerical form or in a table.
Let equation (2) with the initial condition be given, that is, the Kasha problem has been posed. Let's solve the following problem first. Find in the simplest way the approximate value of the solution at a certain point where is a fairly small step. Equation (2) together with the initial condition (3) specify the direction of the tangent of the desired integral curve at the point with coordinates
The tangent equation has the form
Moving along this tangent, we obtain an approximate value of the solution at point:
Having an approximate solution at a point, you can repeat the previously described procedure: construct a straight line passing through this point with an angular coefficient, and from it find the approximate value of the solution at the point
. Note that this line is not tangent to the real integral curve, since the point is not available to us, but if it is small enough, the resulting approximate values will be close to the exact values of the solution.
Continuing this idea, let's build a system of equally spaced points
Obtaining a table of values of the required function
Euler's method consists of cyclically applying the formula
Figure 1. Graphical interpretation of Euler's method
Methods for numerical integration of differential equations, in which solutions are obtained from one node to another, are called step-by-step. Euler's method is the simplest representative of step-by-step methods. A feature of any step-by-step method is that starting from the second step, the initial value in formula (5) is itself approximate, that is, the error at each subsequent step systematically increases. The most used method for assessing the accuracy of step-by-step methods for approximate numerical solution of ODEs is the method of passing a given segment twice with a step and with a step
1.1 Improved Euler method
The main idea of this method: the next value calculated by formula (5) will be more accurate if the value of the derivative, that is, the angular coefficient of the straight line replacing the integral curve on the segment, is calculated not along the left edge (that is, at point), but at the center of the segment. But since the value of the derivative between points is not calculated, we move on to the double sections with the center, in which the point is, and the equation of the straight line takes the form:
And formula (5) takes the form
Formula (7) is applied only for , therefore, values cannot be obtained from it, therefore they are found using Euler’s method, and to obtain a more accurate result they do this: from the beginning, using formula (5) they find the value
(8)
At point and then found according to formula (7) with steps
(9)
Once found further calculations at 
produced by formula (7)
To solve differential equations, it is necessary to know the value of the dependent variable and its derivatives for certain values of the independent variable. If additional conditions are specified for one value of the unknown, i.e. independent variable., then such a problem is called the Cauchy problem. If the initial conditions are specified for two or more values of the independent variable, then the problem is called a boundary value problem. When solving differential equations of various types, the function whose values need to be determined is calculated in the form of a table.
Cauchy problem – one-step: Euler methods, Runge-Kutta methods; – multi-step: Main method, Adams method. Boundary problem – a method of reducing a boundary problem to the Cauchy problem; – finite difference method.
When solving the Cauchy problem, dif. must be specified. ur. order n or system of dif. ur. first order of n equations and n additional conditions for its solution. Additional conditions must be specified for the same value of the independent variable. When solving a boundary problem, equations must be specified. nth order or a system of n equations and n additional conditions for two or more values of the independent variable. When solving the Cauchy problem, the required function is determined discretely in the form of a table with a certain specified step . When determining each successive value, you can use information about one previous point. In this case, the methods are called one-step, or you can use information about several previous points - multi-step methods.
Given: g(x,y)y+h(x,y)=0, y=-h(x,y)/g(x,y)= f(x,y), x 0 , y( x 0)=y 0 . It is known: f(x,y), x 0 , y 0 . Determine the discrete solution: x i , y i , i=0,1,…,n. Euler's method is based on the expansion of a function into a Taylor series in the vicinity of the point x 0 . The neighborhood is described by step h. y(x 0 +h)y(x 0)+hy(x 0)+…+ (1). Euler's method takes into account only two terms of the Taylor series. Let us introduce some notation. Euler's formula will take the form: y i+1 =y i +y i, y i =hy(x i)=hf(x i,y i), y i+1 =y i +hf(x i,y i) (2), i= 0,1,2…, x i+1 =x i +h
Formula (2) is the formula of the simple Euler method.
To obtain a numerical solution, the tangent line passing through the equation is used. tangent: y=y(x 0)+y(x 0)(x-x 0), x=x 1,
y 1 =y(x 0)+f(x 0 ,y 0) (x-x 0), because
x-x 0 =h, then y 1 =y 0 +hf(x 0 ,y 0), f(x 0 ,y 0)=tg £.
Given: y=f(x,y), y(x 0)=y 0 . It is known: f(x,y), x 0 , y 0 . Determine: the dependence of y on x in the form of a tabular discrete function: x i, y i, i=0.1,…,n.
1) calculate the tangent of the angle of inclination at the starting point
tg £=y(x n ,y n)=f(x n ,y n)
2) Calculate the value y n+1 on
end of step according to Euler's formula
y n+1 =y n +f(x n ,y n) 3) Calculate the tangent of the angle of inclination
tangent at n+1 point: tg £=y(x n+1 , y n+1)=f(x n+1 , y n+1) 4) Calculate the arithmetic mean of the angles
tilt: tg £=½. 5) Using the tangent of the slope angle, we recalculate the value of the function at n+1 point: y n+1 =y n +htg £= y n +½h=y n +½h – formula of the modified Euler method. It can be shown that the resulting f-la corresponds to the expansion of the f-i in a Taylor series, including terms (up to h 2). The modified Eilnra method, unlike the simple one, is a method of second order accuracy, because the error is proportional to h 2.
Lab 1
Numerical solution methods
ordinary differential equations (4 hours)
When solving many physical and geometric problems, one has to search for an unknown function based on a given relationship between the unknown function, its derivatives and independent variables. This ratio is called differential equation , and finding a function that satisfies the differential equation is called solving a differential equation.
Ordinary differential equation called equality
, (1)in which
is an independent variable that changes in a certain segment, and
- unknown function y
(
x
)
and her first n derivatives. called order of the equation
.
The task is to find a function y that satisfies equality (1). Moreover, without stipulating this separately, we will assume that the desired solution has one or another degree of smoothness necessary for the construction and “legal” application of one or another method.
There are two types of ordinary differential equations
Equations without initial conditions
Equations with initial conditions.
Equations without initial conditions are equations of the form (1).
Equation with initial conditions is an equation of the form (1), in which it is required to find such a function
, which for some satisfies the following conditions: ,those. at the point
the function and its first derivatives take on predetermined values.Cauchy problems
When studying methods for solving differential equations using approximate methods main task counts Cauchy problem.
Let's consider the most popular method for solving the Cauchy problem - the Runge-Kutta method. This method allows you to construct formulas for calculating an approximate solution of almost any order of accuracy.
Let us derive the formulas of the Runge-Kutta method of second order accuracy. To do this, we represent the solution as a piece of a Taylor series, discarding terms with an order higher than the second. Then the approximate value of the desired function at the point x 1 can be written as:
(2)Second derivative y "( x 0 ) can be expressed through the derivative of the function f ( x , y ) , however, in the Runge-Kutta method, instead of the derivative, the difference is used
selecting parameter values accordingly
Then (2) can be rewritten as:
y 1 = y 0 + h [ β f ( x 0 , y 0 ) + α f ( x 0 + γh , y 0 + δh )], (3)
Where α , β , γ And δ – some parameters.
Considering the right-hand side of (3) as a function of the argument h , let's break it down into degrees h :
y 1 = y 0 +( α + β ) h f ( x 0 , y 0 ) + αh 2 [ γ f x ( x 0 , y 0 ) + δ f y ( x 0 , y 0 )],
and select the parameters α , β , γ And δ so that this expansion is close to (2). It follows that
α + β =1, αγ =0,5, α δ =0,5 f ( x 0 , y 0 ).
Using these equations we express β , γ And δ via parameters α , we get
y 1 = y 0 + h [(1 - α ) f ( x 0 , y 0 ) + α f ( x 0 +, y 0 + f ( x 0 , y 0 )], (4)
0 < α ≤ 1.
Now, if instead of ( x 0 , y 0 ) in (4) substitute ( x 1 , y 1 ), we get a formula for calculating y 2 – approximate value of the desired function at the point x 2 .
In the general case, the Runge-Kutta method is applied to an arbitrary partition of the segment [ x 0 , X ] on n parts, i.e. with variable pitch
x 0 , x 1 , …, x n ; h i = x i+1 – x i , x n = X. (5)
Options α are chosen equal to 1 or 0.5. Let us finally write down the calculation formulas of the second order Runge-Kutta method with variable steps for α =1:
y i+1 =y i +h i f(x i + , y i + f(x i , y i)), (6.1)
i = 0, 1,…, n -1.
And α =0,5:
y i+1 =y i + , (6.2)
i = 0, 1,…, n -1.
The most used formulas of the Runge-Kutta method are formulas of the fourth order of accuracy:
y i+1 =y i + (k 1 + 2k 2 + 2k 3 + k 4),
k 1 =f(x i , y i), k 2 = f(x i + , y i + k 1), (7)
k 3 = f(x i + , y i + k 2), k 4 = f(x i +h, y i +hk 3).
For the Runge-Kutta method, Runge's rule is applicable to estimate the error. Let y ( x ; h ) – approximate value of the solution at the point x , obtained by formulas (6.1), (6.2) or (7) with step h , A p – the order of accuracy of the corresponding formula. Then the error R ( h ) values y ( x ; h ) can be estimated using an approximate value y ( x ; 2 h ) solutions at a point x , obtained in increments 2 h :
(8)
Where p =2 for formulas (6.1) and (6.2) and p =4 for (7).
Main issues discussed in the lecture:
1. Statement of the problem
2. Euler's method
3. Runge-Kutta methods
4. Multi-step methods
5. Solution of the boundary value problem for a linear differential equation of 2nd order
6. Numerical solution of partial differential equations
1. Statement of the problem
The simplest ordinary differential equation (ODE) is a first-order equation solved with respect to the derivative: y " = f (x, y) (1). The main problem associated with this equation is known as the Cauchy problem: find a solution to equation (1) in the form of a function y (x), satisfying the initial condition: y (x0) = y0 (2).
DE of nth order y (n) = f (x, y, y",:, y(n-1)), for which the Cauchy problem is to find a solution y = y(x) that satisfies the initial conditions:
y (x0) = y0 , y" (x0) = y"0 , :, y(n-1)(x0) = y(n-1)0 , where y0 , y"0 , :, y(n- 1)0 - given numbers, can be reduced to a first order DE system.
· Euler method
The Euler method is based on the idea of graphically constructing a solution to a differential equation, but the same method also provides a numerical form of the desired function. Let equation (1) with initial condition (2) be given.
Obtaining a table of values of the desired function y (x) using the Euler method involves cyclically applying the formula: , i = 0, 1, :, n. To geometrically construct Euler’s broken line (see figure), we select the pole A(-1,0) and plot the segment PL=f(x0, y0) on the ordinate axis (point P is the origin of coordinates). Obviously, the angular coefficient of the ray AL will be equal to f(x0, y0), therefore, in order to obtain the first link of the Euler broken line, it is enough to draw straight line MM1 from point M parallel to the ray AL until it intersects with the straight line x = x1 at some point M1(x1, y1). Taking the point M1(x1, y1) as the initial one, we plot the segment PN = f (x1, y1) on the Oy axis and draw a straight line through the point M1 M1M2 | | AN until the intersection at point M2(x2, y2) with the line x = x2, etc. 
Disadvantages of the method: low accuracy, systematic accumulation of errors.
· Runge-Kutta methods
The main idea of the method: instead of using partial derivatives of the function f (x, y) in working formulas, use only this function itself, but at each step calculate its values at several points. To do this, we will look for a solution to equation (1) in the form: 
Changing α, β, r, q, we will obtain various versions of the Runge-Kutta methods.
For q=1 we obtain Euler's formula.
With q=2 and r1=r2=½ we obtain that α, β= 1 and, therefore, we have the formula: , which is called the improved Euler-Cauchy method.
For q=2 and r1=0, r2=1 we obtain that α, β = ½ and, therefore, we have the formula: - the second improved Euler-Cauchy method.
For q=3 and q=4, there are also entire families of Runge-Kutta formulas. In practice, they are used most often, because do not increase errors.
Let's consider a scheme for solving a differential equation using the Runge-Kutta method of 4th order of accuracy. Calculations when using this method are carried out according to the formulas: 
It is convenient to include them in the following table:
| x | y | y" = f (x,y) | k=h f(x,y) | Δy |
| x0 | y0 | f(x0,y0) | k1(0) | k1(0) |
| x0 + ½ h | y0 + ½ k1(0) | f(x0 + ½ h, y0 + ½ k1(0)) | k2(0) | 2k2(0) |
| x0 + ½ h | y0 + ½ k2(0) | f(x0 + ½ h, y0 + ½ k2(0)) | k3(0) | 2k3(0) |
| x0 + h | y0 + k3(0) | f(x0 + h, y0 + k3(0)) | k4(0) | k4(0) |
| Δy0 = Σ / 6 | ||||
| x1 | y1 = y0 + Δy0 | f(x1,y1) | k1(1) | k1(1) |
| x1 + ½ h | y1 + ½ k1(1) | f(x1 + ½ h, y1 + ½ k1(1)) | k2(1) | 2k2(1) |
| x1 + ½ h | y1 + ½ k2(1) | f(x1 + ½ h, y1 + ½ k2(1)) | k3(1) | 2k3(1) |
| x1 + h | y1 + k3(1) | f(x1 + h, y1 + k3(1)) | k4(1) | k4(1) |
| Δy1 = Σ / 6 | ||||
| x2 | y2 = y1 + Δy1 | etc. | until you receive all the required | y values |
· Multi-step methods
The methods discussed above are the so-called methods of step-by-step integration of a differential equation. They are characterized by the fact that the value of the solution at the next step is sought using the solution obtained only at one previous step. These are the so-called one-step methods.
The main idea of multi-step methods is to use several previous solution values when calculating the solution value at the next step. Also, these methods are called m-step methods based on the number m used to calculate previous solution values.
In the general case, to determine the approximate solution yi+1, m-step difference schemes are written as follows (m 1):
Let's consider specific formulas that implement the simplest explicit and implicit Adams methods.
Explicit 2nd order Adams method (2-step explicit Adams method)
We have a0 = 0, m = 2.
Thus, these are the calculation formulas of the explicit Adams method of the 2nd order.
For i = 1, we have an unknown y1, which we will find using the Runge-Kutta method for q = 2 or q = 4.
For i = 2, 3, : all necessary values are known.
Implicit 1st order Adams method
We have: a0 0, m = 1.
Thus, these are the calculation formulas of the implicit Adams method of the 1st order.
The main problem with implicit schemes is the following: yi+1 is included in both the right and left sides of the presented equality, so we have an equation for finding the value of yi+1. This equation is nonlinear and is written in a form suitable for an iterative solution, so we will use the simple iteration method to solve it:
If step h is chosen well, then the iterative process converges quickly.
This method is also not self-starting. So to calculate y1 you need to know y1(0). It can be found using Euler's method.
